\[
  \prod_{i\in I}\ \prod_{j\in I\setminus\{i\}}G_j \;=\; \prod_{j\in I}G_j^{\,I\setminus\{j\}}
\]

\[
  \begin{array}{cccc}
    E_i\times E_j\times E_k & x_i, & x_j, & x_k \;=\;\xi\\
    G_j\ G_k\ G_i\ G_j & g_jx_ig_k^{-1}, & g_kx_jg_i^{-1}, & g_ix_kg_j^{-1}
  \end{array}
\]

\[
  (g_jx_i,\ x_j,\ x_kg_j) \;=\; (x_ig_k^{-1},\ g_kx_jg_i^{-1},\ g_ix_k)
\]

\[
  \tau_{k,j}: E_i\to\mathrm{Isom}(G_j,G_k)
\]

\[
  \tau_{j,i}(x_k)\,\tau_{i,k}(x_j)\,\tau_{k,j}(x_i)=\mathrm{id}_{G_j}
\]

\[
  \tilde\tau_{ij}=\tilde\tau_{ij}(x_k)\quad \text{\struck{$(i,j,k\in I)$}}\quad i,j,k\in I \text{ distincts}
\]

\[
  \begin{aligned}
    g_i\gamma_ix &= (g_j,g_k)x\\
    g_i(\gamma_i,\gamma_j,\gamma_k)x &= (\gamma_j,\gamma_k)\,g_i\gamma_ix = (\gamma_jg_j,\gamma_kg_k)x
  \end{aligned}
\]

\[
  \Gamma\subset\prod_{i\in I}E_i=P .
\]

\[
  \begin{array}{lll}
    a=(a_i,a_j,a_k) & x_i=a_ig_k^{-1} & \text{détermine } g_k \text{ en fonction de } a \text{ et } x_i\\
    x=(x_i,x_j,x_k) & x_j=g_ka_jg_i^{-1} & \text{détermine } g_i \text{ en fonction de } a \text{ et } x_i,x_j\\
    \phantom{x=}\ ? & x_k=g_ia_k & \text{détermine } x_k
  \end{array}
\]

\[
  u_i(1)=1 \qquad i=1,2,3
\]

\[
  (*)\qquad xyz=1\ \Longrightarrow\ u_1(x)\,u_2(y)\,u_3(z)=1
\]

\[
  u\bigl((xy)^{-1}\bigr)^{-1}=u(x)\,u(y),
\]

\[
  \mathcal{C}'\xrightarrow{\ \varphi\ }\mathcal{C}
\]

\[
  xyz=1 .
\]

\[
  x'y'z'=1 .
\]

\[
  u_1,u_2,u_3 : G\to G'
\]

\[
  E_1\xrightarrow{\ \alpha_1\ }\mathrm{Isom}(E_2,E_3)\qquad
  E_2\xrightarrow{\ \alpha_2\ }\mathrm{Isom}(E_3,E_1)\qquad
  E_3\xrightarrow{\ \alpha_3\ }\mathrm{Isom}(E_1,E_2)
\]

\[
  \alpha_3(a_3) : a_1\mapsto a_2,\qquad \alpha_1(a_1) : a_2\mapsto a_3,\qquad \alpha_2(a_2) : a_3\mapsto a_1
\]

\[
  \begin{aligned}
    \alpha_2(a_2)\,\alpha_1(a_1) &= u_3 : E_2\to E_1\\
    \alpha_3(a_3)\,\alpha_2(a_2) &= u_1 : E_3\to E_2\\
    \alpha_1(a_1)\,\alpha_3(a_3) &= u_2 : E_1\to E_3
  \end{aligned}
\]

\[
  \boxed{u_1u_2u_3=\mathrm{id}}
  \qquad \text{NB}\quad u_3(a_2)=a_1,\quad u_1(a_3)=a_2,\quad u_2(a_1)=a_3
\]

\[
  \begin{aligned}
    &\exists!\,z\in G \text{ tel que } (x,y,z)\in\Gamma\\
    &\exists!\,z\in G \text{ tel que } (z,x,y)\in\Gamma\\
    &\exists!\,z\in G \text{ tel que } (x,z,y)\in\Gamma
  \end{aligned}
\]

\[
  \begin{aligned}
    &(e,x,\sigma_1x)\in\Gamma\\
    &(\sigma_2x,e,x)\in\Gamma\\
    &(x,\sigma_3x,e)\in\Gamma
  \end{aligned}
\]

\[
  \sigma_2\sigma_1=\mathrm{id},\quad \sigma_3\sigma_2=\mathrm{id},\quad \sigma_1\sigma_3=\mathrm{id}
\]

\[
  (x,y,z)\in\Gamma\ \Longleftrightarrow\ z=\sigma(x.y)
\]

\[
  \Gamma=\{(x,y,z)\in G^3 \mid z=\sigma(x.y)\}=\{(x,y,z)\in G^3\mid x.y=\sigma^{-1}z\}
\]

\[
  \begin{aligned}
    &\exists!\,z \text{ tel que } zx=\sigma^{-1}y\\
    &\exists!\,z \text{ tel que } xz=\sigma^{-1}y
  \end{aligned}
\]

\[
  \begin{aligned}
    &\forall x\quad \sigma^{-1}\sigma x=ex &&\text{i.e. } ex=x\\
    &\forall x\quad \sigma x.e=\sigma^{-1}x &&\text{i.e. } \forall x \text{ on a } x.e=\sigma^{-2}x\ (=x)\\
    &\forall x\quad x.\sigma x=\sigma^{-1}e\ (=e)
  \end{aligned}
\]

\[
  x.\sigma x=e\qquad (\sigma x)(\sigma\sigma x)=e
\]

\[
  (xy)z=x(yz)\ \ ??\qquad (x,y,\sigma(xy))\in\Gamma
\]

\[
  (x,y,z)\in\Gamma\ \Longrightarrow\ z=\sigma(x.y)\ \Longleftrightarrow\ (x.y).z=e\ \Longleftrightarrow\ z.(x.y)=e
\]

\[
  \begin{aligned}
    &(xy,z,\sigma((xy)z))\in\Gamma\\
    &(x,yz,\sigma(x(yz)))\in\Gamma
  \end{aligned}
  \qquad
  \begin{aligned}
    &xy=e\\
    &\Updownarrow\\
    &(xy)z=z\\
    &\Updownarrow\\
    &z(xy)=z
  \end{aligned}
\]

\[
  E_1=E_2=E_3=G\qquad \Gamma=\{(x,y,z)\mid xyz=1\}
\]

\[
  \begin{aligned}
    &f_1(x) \text{ défini par } (a,x,f_1(x))\in\Gamma\\
    &f_2(x) \text{ défini par } (f_2(x),b,x)\in\Gamma\\
    &f_3(x) \text{ défini par } (x,f_3(x),c)\in\Gamma
  \end{aligned}
\]

\[
  f_1(x)=x^{-1}a^{-1},\qquad f_2(x)=x^{-1}b^{-1},\qquad f_3(x)=x^{-1}c^{-1}
\]

\[
  x^{-1}a^{-1}=y,\qquad x=a^{-1}y^{-1}
\]

\[
  \boxed{\sigma(xy)=\sigma y\,\sigma x}
\]

\[
  \text{\struck{$z=xy$}}\qquad
  \left\{
  \begin{aligned}
    &(xy)z=1\\ &\Updownarrow\\ &(zx)y=1\\ &\Updownarrow\\ &(x^{-1}z^{-1})y^{-1}=1
  \end{aligned}
  \right.
  \qquad
  \begin{aligned}
    &z=xy \Longleftrightarrow x=zy^{-1}\\
    &y^{-1}=zx,\quad y=x^{-1}z^{-1}
  \end{aligned}
\]

\[
  \begin{aligned}
    g_1(x)&=f_3f_2(x)=(bx)c^{-1}\\
    g_2(x)&=f_1f_3(x)=(cx)a^{-1}\\
    g_3(x)&=f_2f_1(x)=(ax)b^{-1}
  \end{aligned}
\]

\[
  g_1g_2g_3(x)=bcaxb^{-1}a^{-1}c^{-1}=(bca)\,x\,(cab)^{-1}=x\qquad\text{i.e. } g_1g_2g_3=\mathrm{id}
\]

\[
  g_1g_2g_3=(f_3f_2f_1)^2
\]

\[
  f_3f_2f_1(x)=(axb^{-1})^{-1}c^{-1}=bx^{-1}a^{-1}c^{-1}=bx^{-1}(ca)^{-1}=bx^{-1}b
\]

\[
  \sigma_2x=bx^{-1}b\qquad\text{de même}\qquad \sigma_3x=cx^{-1}c,\quad \sigma_1x=ax^{-1}a
\]

\[
  E_1\qquad E_2\ \underset{f_1^{-1}}{\overset{f_1}{\rightleftarrows}}\ E_3
  \qquad \text{\struck{$(x,y,z)$}}
\]

\[
  \begin{aligned}
    &(x,y,z)\\
    &(\sigma_1x,\ f_1^{-1}z,\ f_1y)=(ax^{-1}a,\ a^{-1}z^{-1},\ y^{-1}a^{-1})
  \end{aligned}
  \qquad
  \begin{aligned}
    &xyz=1\Longrightarrow\\
    &ax^{-1}aa^{-1}z^{-1}y^{-1}a^{-1}=1
  \end{aligned}
\]

\[
  \begin{aligned}
    u=(xy)z &\Longleftrightarrow (uz^{-1}=xy)\Longleftrightarrow (y^{-1}x^{-1})u=z\\
    u=x(yz) &\Longleftrightarrow (x^{-1}u=yz)\Longleftrightarrow u(\cdots
  \end{aligned}
\]

\[
  \left\{
  \begin{aligned}
    f_1(x)&=x^{-1}a^{-1}\\ f_2(x)&=x^{-1}b^{-1}\\ f_3(x)&=x^{-1}c^{-1}
  \end{aligned}
  \right.
  \qquad
  \begin{aligned}
    g_1&=f_3f_2 = x\mapsto (bx)c^{-1}\\
    g_2&=f_1f_3 = x\mapsto (cx)a^{-1}\\
    g_3&=f_2f_1 = x\mapsto (ax)b^{-1}
  \end{aligned}
\]

\[
  g_1g_2g_3=(f_3f_2f_1)^2\qquad x\mapsto \Bigl(b\Bigl(\bigl(c((ax)b^{-1})\bigr)a^{-1}\Bigr)\Bigr)c^{-1}
\]

\[
  \sigma_2x=f_3f_2f_1(x)=\bigl((ax)b^{-1}\bigr)^{-1}c^{-1}=\bigl(b(x^{-1}a^{-1})\bigr)c^{-1}
\]

\[
  \sigma_2^2x=\Bigl(b\bigl(\bigl[c((ax)b^{-1})\bigr]a^{-1}\bigr)\Bigr)c^{-1}
\]

\[
  \begin{array}{l}
    x\\ ax\\ axb^{-1}\\ caxb^{-1}\\ \text{\struck{$b$}}\,caxb^{-1}a^{-1}\\ bcaxb^{-1}\\ bcaxb^{-1}a^{-1}c^{-1}
  \end{array}
  \qquad
  \left\{
  \begin{aligned}
    &\sigma(xy)=\sigma x\,\sigma y\\
    &\tau^{d}_{x}{}^{-1}=\tau^{d}_{x^{-1}}\\
    &\tau^{g}_{x}{}^{-1}=\tau^{g}_{x^{-1}}
  \end{aligned}
  \right.
\]

\[
  u=(xy)z\qquad (bx)c^{-1}=y\qquad x=b^{-1}(yc)
\]

\[
  x=\qquad g_1g_2g_3x=x\qquad \text{\struck{$g_2g_3x=$}}
\]

\[
  \bigl(c((ax)b^{-1})\bigr)\text{\struck{$a^{-1}$}}=\bigl(b^{-1}(xc)\bigr)a
\]

\[
  \boxed{c\bigl((ax)b^{-1}\bigr)=\bigl(b^{-1}(xc)\bigr)a}
\]

\[
  (b^{-1}a^{-1})\bigl((ax)b^{-1}\bigr)=\bigl(b^{-1}(x(b^{-1}a^{-1}))\bigr)a
\]

\[
  \left\{
  \begin{aligned}
    &(ab)c=1\\
    &\text{\struck{$x\ y\ z$}}\\
    &(xy)z=1
  \end{aligned}
  \right.
  \ \Longrightarrow\
  \Bigl[\bigl((z^{-1}b^{-1})((b(y^{-1}a^{-1}))c^{-1})\bigr)\Bigr](b^{-1}x^{-1})\,\text{\struck{$b$}}=1
\]

\[
  b\mapsto z,\quad a\mapsto b,\quad x\mapsto y^{-1}a^{-1}
\]

\[
  \Bigl[\bigl(z^{-1}((y^{-1}a^{-1})(z^{-1}b^{-1}))\bigr)b\Bigr](b^{-1}x^{-1})
\]

\[
  \text{\struck{$c=b^{-1}a^{-1}$}}\quad \text{\struck{$z=ab$}}\qquad (ab)(xy)=1
\]

\[
  ab=xy\qquad \text{\struck{$(ab)(xy)=1$}}\ \Longrightarrow\
  \Bigl[(\cdots)\bigl((y^{-1}a^{-1})(\cdots\,b^{-1})\bigr)b\Bigr](b^{-1}x^{-1})
\]

\[
  (xb)(b^{-1}x^{-1})=1
\]

\[
  \begin{aligned}
    &(ab)c=1\ *\\ &(xy)z=1\ *
  \end{aligned}
  \ \overset{?}{\Longrightarrow}\ xb=(z^{-1}b^{-1})\bigl((b(y^{-1}a^{-1}))c^{-1}\bigr)
\]

\[
  \Updownarrow
\]

\[
  (bz)(xb)=\bigl(b(y^{-1}a^{-1})\bigr)c^{-1}
\]

\[
  (bz)(xb)=\bigl(b((zx)(bc))\bigr)c^{-1}
\]

\[
  \bigl((bz)(xb)\bigr)c=b(y^{-1}a^{-1})\qquad (bu)c=b\bigl((ub^{-1})(bc)\bigr)
\]

\[
  \boxed{\bigl((bz)(xb)\bigr)c=b\bigl((zx)(bc)\bigr)}
  \qquad
  \left\{
  \begin{aligned}
    &c=a^{-1}\\ &xb=y^{-1}\ *\\ &b=bz\ *
  \end{aligned}
  \right.
\]

\[
  z=1,\ c=\quad \bigl(b(xb)\bigr)c=b\bigl(x(bc)\bigr)\qquad z=1,\ y=x^{-1},\quad b=1,\ c=a^{-1}
\]

\[
  \begin{aligned}
    &\tau_xz=xz\\ &\tau_x\tau_yz=x(yz)\\ &xx'=x'x=e\\ &x(x'y)=y\\ &G\times G\to G
  \end{aligned}
\]

\[
  aa\quad bb\quad ab=ba \qquad ab\overset{?}{=}ba,\quad ab=a(aa),\ ba=(aa)a
\]

\[
  \{ae,aa,ab\}=\{e,a,b\}\qquad aa=b,\ ab=e\ \text{ou}\ aa=e,\ \text{\struck{$ab=b$}}\ \text{mais}\ ab\neq\ill{}
\]

\[
  aa=b,\ ab=e\ \Longrightarrow\ \boxed{ba=e}\qquad bb=a
\]

\[
  e,a,b,c \qquad \text{\struck{$(xy)z=$}}\ x(yz)\qquad uv=u'v'
\]

\[
  x\,y\,z\qquad \bar x\bigl((xy)z\bigr)=(yz)\qquad \bar x\bigl((xy)z\bigr)z
\]

\[
  \begin{array}{c|ccc}
    & a & b & c\\ \hline
    a & e & c & b\\ b & & & \\ c & & &
  \end{array}
  \qquad\qquad
  \begin{array}{c|ccc}
    & a & b & c\\ \hline
    a & b & c & e\\ b & & & \\ c & & &
  \end{array}
\]

\[
  aa=e\quad ab=c\quad ac=b
\]

\[
  (xy)z=1\ \Longrightarrow\ z(xy)=1\qquad \Updownarrow\qquad y(zx)=1,\quad (zx)y=1\qquad \text{\struck{$(xy)z$}}
\]

\[
  \begin{aligned}
    &(ab)c=1\\ &(ax)y=1\\ &(yb)x=1\\ &(xy)c=1
  \end{aligned}
  \qquad
  \begin{aligned}
    &y=\sigma(ax)\\ &yb=\sigma x\\ &xy=\sigma c
  \end{aligned}
  \qquad
  \begin{aligned}
    &f_1(x)=\sigma(ax)\\ &f_2(x)=(\sigma x)b^{-1}
  \end{aligned}
\]

\[
  S=\text{\struck{$X$}}\ \{e\}\sqcup Y\sqcup\coprod_{\lambda\in\Lambda}\underbrace{Y\times_V(V_\lambda,\varphi_\lambda)}_{X_\lambda}
\]

\[
  E\times I\ \sqcup\ E\times I'
\]

\[
  E=\coprod_{\alpha\in\varepsilon}E_\alpha
\]

\[
  I\times\varepsilon\quad \text{hexagone}
\]

\[
  \coprod_{\alpha\in\varepsilon}E_\alpha\times I = E\times I \quad\text{fibré sur } I\times\varepsilon
\]

\[
  X=\prod_{\alpha\in\varepsilon}E_\alpha
\]

\[
\begin{matrix}
\xi_{14} & \xi_{15} & \xi_{16}\\
\xi_{24} & \xi_{25} & \xi_{26}\\
\xi_{34} & \xi_{35} & \xi_{36}
\end{matrix}
\]

\[
6^4 = 36 \times 36 = 2^4 3^4, \qquad 720 = 2^4 \cdot 3^2 \cdot 5, \qquad 9/5
\]

\[
\xi_{12}\ \xi_{23}\ \xi_{31} \;\Big|\; \xi_{45}\ \xi_{56}\ \xi_{64}
\]

\[
\begin{matrix}
(\alpha b, \alpha c) & (\alpha b', \alpha c')\\
(\beta c, \beta a) & (\beta c', \beta a')\\
(\gamma a, \gamma b) & (\gamma a', \gamma b')
\end{matrix}
\]

\[
\Gamma_0 = 1 \begin{pmatrix} 27 \\ 36 \\ 40 \end{pmatrix}
\]

\[
72 \cdot 36\tfrac{1}{2} = 72 \cdot \tfrac{71}{2} + 72
\]

\[
3!\, 3!\, 2! = 72
\]

\[
\{\xi_{ij}\ \xi_{jk}\ \xi_{ki}\ \hat\xi_i \ldots\}, \qquad \{\xi_{ij}\ \xi_{jk}\ \xi_{ki}\ \hat\xi_i\ \hat\xi_j \ldots\}
\]

\[
\begin{matrix}
\xi_{ip} & \xi_{jp} & \xi_{kp}\\
\xi_{iq} & \xi_{jq} & \xi_{kq}\\
\xi_{ir} & \xi_{jr} & \xi_{kr}
\end{matrix}
\]

\[
\frac{6 \cdot 5 \cdot 4}{3!\, 2}
\]

\[
\begin{matrix}
\xi_1 & \xi_2 & \xi_3 & & \xi_4' & \xi_5' & \xi_6'\\
\xi_1' & \xi_2^{*} & \xi_3' & & \xi_4^{*} & \xi_5^{*} & \xi_6^{*}\\
\xi_{23}^{*} & \xi_{31}^{*} & \xi_{12}^{*} & & \xi_{56} & \xi_{64} & \xi_{45}
\end{matrix}
\]

\[
\begin{matrix}
\xi_{14} & \xi_{15} & \xi_{16}\\
\xi_{24} & \xi_{25} & \xi_{26}\\
\xi_{34} & \xi_{35} & \xi_{36}
\end{matrix}
\]

\[
E \qquad \mathfrak{S}_3 \times \mathfrak{S}_3 \times \mathfrak{S}_3 \subset W
\]

\[
2^3 3^3, \qquad 2^4 \cdot 3 \cdot 5 = 240
\]

\[
r_6 = \eta - \xi_1 - \xi_2 - \xi_3, \qquad r_0 = 2\eta - \xi
\]

\[
E \times E' \amalg E' \times E'' \amalg E'' \times E
\]

\[
(2\eta - \xi) \cdot \xi_j = 2
\]

\[
F \to E, \qquad \coprod_{i \in E}\ \prod_{j \in E - \{i\}} F_j
\]

\[
81 \times 16 = 1296
\]

\[
9 \times 9 = 81, \qquad 3 \times 3 \times 3 = 27, \qquad 54
\]

\[
\frac{27 \cdot 16 \cdot 10}{6} = 9 \cdot 16 \cdot 5 = 720
\]

\[
r_{456} = 3(r_1 + r_2 + r_3) + 2r_4 + r_5 + r_6
\]

\[
r_{2,1} = -r_{12} = -r_1, \quad r_{2,3} = r_2, \quad r_{1,3} = r_1 + r_2, \quad r_{1,2} = r_1, \quad r_{3,2} = -r_2, \quad r_{3,1} = -r_1 - r_2 .
\]

\[
r_{5,4} = -r_{45} = -r_4, \quad r_{5,6} = r_5, \quad r_{4,6} = r_4 + r_5, \quad r_{4,5} = r_4, \quad r_{6,5} = -r_5, \quad r_{6,4} = -r_4 - r_5 .
\]

\[
\begin{matrix}
\xi_1 & \xi_2 & \xi_3 & \text{---} & \xi_4' & \xi_5' & \xi_6'\\
\xi_1' & \xi_2' & \xi_3' & \text{---} & \xi_4 & \xi_5 & \xi_6\\
\xi_{23} & \xi_{31} & \xi_{12} & \text{---} & \xi_{56} & \xi_{64} & \xi_{45}
\end{matrix}
\]

\[
J = \{\lambda_{45}, \lambda_{56}, \lambda_{61}\}, \qquad
K \cup K' = \{\underbrace{\xi_1, \xi_2, \xi_3}_{K}, \underbrace{\xi_1', \xi_2', \xi_3'}_{K'}\}
\]

\[
6! = 720 = 2^4 3^2 5, \qquad 6^4 = 2^4 3^4 = 1296
\]

\[
\begin{array}{ll}
\text{sommets} & 27\\
\text{\uncertain{bitriades}} & 36\\
\text{trihexagones} & 40\\
\text{triangles} & 45
\end{array}
\]

\[
\text{\struck{$2$}} \quad \mathfrak{S}_?\,/\,\mathfrak{S}_4 \qquad \mathbb{F}_2^{3}
\]

\[
72 \cdot 2^{4} \qquad 2^{7} \cdot 9 \qquad (\sim 16\ \text{él.})
\]

\[
\frac{8 - 2}{2} = 3 \qquad \text{\struck{$P(i)$}} \quad \text{\struck{$2$}}
\]

\[
P(i) / \text{\struck{$V$}}\,V'(i) \simeq T - \{i\}
\]

\[
0 \to k \to V'(i) \to V(i) \to 0
\]

\[
(1 + q(1 + q^{-1}))\, q^3 (q-1)^2 \qquad q = 2
\]

\[
2^3 \cdot 3 \cdot 5 = 120 \ !!! \qquad \text{\struck{$P(i)$}}
\]

\[
V \simeq k(\Delta_i)' \ \text{pour tout } i, \qquad \check{k} = k
\]

\[
\frac{72 \cdot 6!}{72 \cdot 6} = \frac{72 \cdot 720}{72 \cdot 6} = 120
\]

\[
\begin{matrix}
\xi_1\ \xi_2\ \xi_3 & \xi_4\ \xi_5\ \xi_6 & \xi_{12}\ \xi_{23}\ \xi_{34}\\
\xi_1'\ \xi_2'\ \xi_3' & \xi_4'\ \xi_5'\ \xi_6' & \xi_{45}\ \xi_{56}\ \xi_{61}
\end{matrix}
\]

\[
\begin{matrix}
\xi_{14} & \xi_{15} & \xi_{16}\\
\xi_{24} & \xi_{25} & \xi_{26}\\
\xi_{34} & \xi_{35} & \xi_{36}
\end{matrix}
\]

\[
36 \cdot \binom{6}{3} = 720, \qquad \binom{6}{3} = \frac{6 \cdot 5 \cdot 4}{3!} = 20
\]

\[
\text{plan } A_2 : \quad \frac{720}{6} = 120
\]

\[
\mathfrak{S}_2 \cdot (\mathfrak{S}_3 \times \mathfrak{S}_3), \qquad
\mathfrak{S}_3 \cdot (\mathfrak{S}_3 \times \mathfrak{S}_3 \times \mathfrak{S}_3)
\]

\[
\begin{matrix}
\xi_{14} & \xi_{25} & \xi_{36}\\
\xi_{26} & \xi_{34} & \xi_{15}\\
\xi_{35} & \xi_{16} & \xi_{24}
\end{matrix}
\]

\[
\left\{
\begin{array}{ll}
\text{sommets} & 27\\
\text{triangles} & 45\\
\text{bibases} & 36\\
\text{trois-neufs} & 40
\end{array}
\right.
\]

\[
\begin{array}{lll}
(\xi_1\,\xi_2\,\xi_3)(\xi_4'\,\xi_5'\,\xi_6') & (\xi_4\,\xi_5\,\xi_6)(\xi_1'\,\xi_2'\,\xi_3') & (\xi_{23}\,\xi_{31}\,\xi_{12})(\xi_{56}\,\xi_{64}\,\xi_{45})\\[4pt]
(\xi_1\,\xi_1'\,\xi_{23})(\xi_{14}\,\xi_{15}\,\xi_{16}) & (\xi_{24}\,\xi_{25}\,\xi_{26})(\xi_2\,\xi_2'\,\xi_{31}) & (\xi_3\,\xi_3'\,\xi_{12})(\xi_{34}\,\xi_{35}\,\xi_{36})\\[4pt]
(\xi_4\,\xi_4'\,\xi_{56})(\xi_{14}\,\xi_{24}\,\xi_{34}) & (\xi_5\,\xi_5'\,\xi_{64})(\xi_{15}\,\xi_{25}\,\xi_{35}) & (\xi_6\,\xi_6'\,\xi_{45})(\xi_{16}\,\xi_{26}\,\xi_{36})
\end{array}
\]

\[
\Gamma = \mathrm{Ker}(V^I \to V),
\]

\[
V^\omega = V \wedge \omega
\]

\[
V^\omega \xrightarrow{\ \varphi_i\ } \Gamma, \qquad x \wedge \rho \longmapsto
\begin{cases}
\varphi_i(x \wedge \rho)_i = 0\\
\varphi_i(x \wedge \rho)_{\rho i} = x\\
\varphi_i(x \wedge \rho)_{\rho^2 i} = -x
\end{cases}
\]

\[
0 \to V^\omega \xrightarrow{\ \mathrm{diag}\ } (V^\omega)^I \to \Gamma \to 0,
\]

\[
0 \to V^\omega \to (V^\omega)^I \to V^I \to V \to 0.
\]

\[
u \psi_i = \varphi_i .
\]

\[
\varphi_{i_0}(x \wedge \rho) + \varphi_{i_1}(x \wedge \rho) + \varphi_{i_2}(x \wedge \rho) = 0
\]

\[
\psi_{i_0}(x \wedge \rho)\, \psi_{i_1}(x \wedge \rho)\, \psi_{i_2}(x \wedge \rho) = z(x, \rho, i_0) \in Z .
\]

\[
z(x, \rho, i_0) = z(x, \rho, i_1) = z(x, \rho, i_2),
\]

\[
z = z(x, \rho) = z_\rho(x).
\]

\[
\psi_{i_2}(x \wedge \rho)^{-1}\, \psi_{i_1}(x \wedge \rho)^{-1}\, \psi_{i_0}(x \wedge \rho)^{-1} = -z(x, \rho)
\]

\[
\psi_{i_2}((-x) \wedge \rho)\, \psi_{i_1}((-x) \wedge \rho)\, \psi_{i_0}((-x) \wedge \rho) = -z(x, \rho)
\]

\[
\psi_{i_2}(x \wedge \bar\rho)\, \psi_{i_1}(x \wedge \bar\rho)\, \psi_{i_0}(x \wedge \bar\rho) = -z(x, \rho)
\]

\[
z(x, \bar\rho) = -z(x, \rho) \qquad \text{i.e.} \quad z_{\bar\rho} = -z_\rho
\]

\[
z_\rho : V \longrightarrow Z, \qquad \rho \in \omega
\]

\[
\begin{cases}
\text{\struck{$z_\rho(0) = 0$ et}} \quad \text{si } j = \rho i\\
\varphi_i(x)\, \varphi_j(y)\, \varphi_i(x)^{-1}\, \varphi_j(y)^{-1} = z_\rho(x) + z_\rho(y) - z_\rho(x + y)
\end{cases}
\]

\[
(*)\quad
\begin{cases}
z_\rho(0) = 0\\
z_\rho(x) + z_\rho(y + u) + z_\rho(y + v) + z_\rho(x + u + v)\\
\qquad = z_\rho(y) + z_\rho(x + u) + z_\rho(x + v) + z_\rho(y + u + v)
\end{cases}
\]

\[
\begin{aligned}
&\lambda(x + u + v) - \lambda(x + u) - \lambda(x + v) + \lambda(x)\\
&\qquad = \lambda(y + u + v) - \lambda(y + u) - \lambda(y + v) + \lambda(y)
\end{aligned}
\]

\[
(\Delta_{u,v} \lambda)(x) = \text{c}^{\text{te}} \quad \text{(pour $u, v$ fixés)}
\]

\[
\lambda(u + v) - \lambda(u) - \lambda(v) = \mu(u, v)
\]

\[
\boxed{\lambda(x + u + v) - \lambda(x + u) - \lambda(x + v) + \lambda(x) = \mu(u, v)}
\]

\[
\begin{cases}
f(0) = 0\\
f(x + u + v) - f(x + u) - f(x + v) + f(x) = 0
\end{cases}
\]

\[
z_\rho : V \to Z
\]

\[
z_{\bar\rho}(x) = \lambda(x \wedge \bar\rho) = \lambda(-x \wedge \rho) = -\lambda(x \wedge \rho) = -z_\rho(x),
\]

\[
\lambda(x + y) - \lambda(x) - \lambda(y) = \mu(x, y) \qquad \text{($\mu$ $\mathbb{Z}$-bil.\ sym.)}
\]

\[
\lambda(2x) = 2\lambda(x) + \mu(x, x),
\]

\[
\mathfrak{Z} = V + \Gamma^2_{\mathbb{Z}}(V) \simeq (\mathbb{Z}/4\mathbb{Z})^{I} + (\mathbb{Z}/2\mathbb{Z})^{P_2(I)}
\]

\[
q(x) =
\begin{cases}
0 & \text{si } x = 0\\
1 & \text{si } x \neq 0
\end{cases}
\]

\[
\psi_{i_0}(x)\, \psi_{i_1}(x)\, \psi_{i_2}(x) = z \qquad \text{(él.\ non nul de $Z = \mathbb{Z}/2\mathbb{Z}$)}
\]

\[
S = \tau \times \varepsilon, \qquad
A = \tau \times (\varepsilon \setminus \omega(\tau)), \qquad
\widetilde{S} = \tau \times \varepsilon \times \omega(\tau)
\]

\[
\tau \times (\varepsilon \setminus \omega(\tau)) \setminus \omega(\tau') \qquad
\tau\times\varepsilon \cdot \tau\times\varepsilon' \qquad
\tau \times \varepsilon \times \varepsilon'
\]

\[
15 \cdot 6 \cdot 4 \cdot 2 \cdot 1 = 720 = 6!
\]

\[
\{1\,3\}\,\{2\,4\}\,\{3\,5\}\,\{4\,1\}\,\{5\,2\} \qquad
\{1\,5\}\,\{3\,2\}\,\{5\,4\}\,\{2\,1\}\,\{4\,3\}
\]

\[
10\,P = 720, \qquad P = 72, \qquad P_0 = 12 = \text{\uncertain{nb des struct.\ \ill{}}}, \qquad 4!
\]

\[
\begin{aligned}
&\text{l'une est} && \overline{\tau'} = L(\tau'') \setminus \tau\\
&\text{l'autre} && \overline{\tau''} = L(\tau') \setminus \tau
\end{aligned}
\]

\[
X \longrightarrow I \times J
\]

\[
X_\alpha \xrightarrow{\sim} \prod_{\beta \in A\setminus\{\alpha\}} I_\beta
\]

\[
S \simeq \coprod_{\alpha} \prod_{\beta\in A\setminus\{\alpha\}} I_\beta \quad \text{\struck{$I_\alpha$}}
\]

\[
S \simeq \underbrace{I_2 \times I_3}_{X_1} \amalg \underbrace{I_3 \times I_1}_{X_2} \amalg \underbrace{I_1 \times I_2}_{X_3}
\]

\[
X_\alpha = \text{ens.\ des } A \text{ tels que } p(A) = \Lambda \setminus \{\alpha\}
\]

\[
A, B \in S \text{ liés ssi } A \neq B \text{ et}
\]

\[
\begin{aligned}
&\text{ou bien } p(A) = p(B), && A \cap B = \emptyset\\
&\text{ou bien } p(A) \neq p(B), && A \cap B \neq \emptyset
\end{aligned}
\]

\[
\simeq \mathfrak{S}_3 \cdot (\mathfrak{S}_3 \times \mathfrak{S}_3 \times \mathfrak{S}_3)
\]

\[
H \simeq I_\gamma \times \Lambda_\gamma,
\]

\[
\overline{T_{\alpha\beta\gamma}} = T_{\beta,\alpha,\gamma}
\]

\[
\begin{cases}
\text{pour sommets } = \mathfrak{P}_2(B)\\
\text{\struck{$A$, $B$}} \ u, v \in \mathfrak{P}_2(B) \text{ liés ssi } u \cap v = \emptyset
\end{cases}
\]

\[
  \mathrm{bit}(\tau) = \mathrm{bit}(\bar\tau) = \mathrm{bit}(\tau') = \mathrm{bit}(\tau'')
  = \mathrm{bit}(\bar\tau') = \mathrm{bit}(\bar\tau'')
\]

\[
  A_4 \Leftrightarrow A_5 \Leftrightarrow P_6 \Leftrightarrow \text{bitriangle épinglé}
\]

\[
  \varphi_X : x \mapsto \mathrm{Om}_X(x) : S \setminus X \to \mathrm{Tria}(X)
\]

\[
  X_1 \xrightarrow{\ \sim\ } I_1 \times I_0, \qquad X_2 = I_2 \times I_0
\]

\[
  S \setminus X = X_1 \amalg X_2 \xrightarrow{\ \pi\ } I_0
\]

\[
  \mathrm{Tria}' \to \mathrm{Hex} \to \mathrm{triHex}
\]

\[
  \mathrm{Tria}' \xrightarrow{\ 3\ } \mathrm{Hex} \xrightarrow{\ 3\ } \mathrm{triHex}
\]

\[
  t'' = P - h = \{a'', b'', c''\}
\]

\[
  I = \coprod_{\alpha \in \Lambda} I_\alpha \xrightarrow{\ p\ } \Lambda .
\]

\[
  h \leftrightarrow (I_\gamma,\ I_\alpha \setminus \{i\})
\]

\[
  H \leftrightarrow (I_\alpha, \{0,1\})
\]

\[
  \omega(H) \simeq \omega(I_\alpha) \simeq I_\alpha \setminus \{i\}, \qquad
  \omega(h) \simeq \omega(I_\gamma)
\]

\[
  \text{Sommets} = \tau \times \varepsilon, \qquad \text{côtés} = \tau \times \varepsilon'
\]

\[
  (\tau, \varepsilon', \varepsilon, -\varphi \circ \sigma)
\]

\[
  \mathrm{Aut}(h) \subset \mathrm{Aut}(\tau) \times (\pm 1)^2
\]

\[
  \alpha\beta = \mathrm{sg}(u) \quad\text{ou}\quad \alpha\beta\,\mathrm{sg}(u) = 1
\]

\[
  (\pm 1)^{3\prime} = \{ (\alpha,\beta,\gamma) \in (\pm 1)^3 \mid \alpha\beta\gamma = 1 \}.
\]

\[
  \mathrm{Cube}_d \simeq (\mathrm{Ens})_{\underbrace{2,2,\dots,2}_{d}} \Big)
\]

\[
  P_6 \simeq (\mathrm{Ens})_3 \times (\mathrm{Ens})_2
\]

\[
  \boxed{H_d \simeq H_{d^*}} \quad (= t)
\]

\[
  D_6 \simeq \mathfrak{S}_2 \cdot \mathbb{Z}/6\mathbb{Z} \simeq \mathfrak{S}_2 \times \mathfrak{S}_3
\]

\[
  \mathrm{Car}_{d}(X)\simeq \mathrm{tria\,pt}(X),\qquad
  \mathrm{Car}_{s}(X)\simeq \mathrm{tria\,ép}(X)
\]

\begin{align*}
  \mathrm{Car}_{d*}(X) &\simeq \mathrm{tria\,pt}(X),\\
  \mathrm{Car}_{s*}(X) &\simeq \mathrm{arc}(X)\simeq \mathrm{tria\,ép}(X)
\end{align*}

\[
  \mathrm{Car}_{\cdot}(X)\simeq \text{ens des couples d'une triade et deux triangles sécants dans } X.
\]

\[
  \mathrm{Pav}(C)^{\circ} \;\overset{\mathrm{canon}}{\simeq}\; \mathrm{Pav}(C^{*})
\]

\[
  \mathrm{Pl}(\mathrm{Pav}(C),\mathcal{X}) \xrightarrow{\ \sim\ } \mathrm{Pl}(C,\mathcal{X})
  \qquad\text{(plongements)}
\]

\[
  \mathrm{Iso}(\mathrm{Pav}(C),\mathcal{X}) \simeq \mathrm{Pl}(C,\mathcal{X}).
\]

\[
  \Gamma^{\circ}=\prod\Gamma_i^{\circ},\quad\text{i.e.}\quad \Gamma=\Bigl(\prod\Gamma_i^{\circ}\Bigr)^{\circ}.
\]

\[
  (ba')\text{---}(bb'),\qquad (aa')\text{---}(ab'),
\]

\[
  \varphi_{\beta}\varphi_{\alpha}=-\mathrm{id}_{C_{\alpha}},\qquad \varphi_{\alpha}\varphi_{\beta}=-\mathrm{id}_{C_{\beta}}
\]

\[
  \mathrm{or}(C)\simeq\Bigl(\underbrace{\bigwedge_{\alpha\in D}C_{\alpha}}_{\substack{D^{*}=\text{ens des diagonales}\\ \text{du carré dual}}}\Bigr)\times D \;\simeq\; D^{*}\times D .
\]

\[
  S\setminus t_0=\coprod_{\alpha\in t_0}\widetilde{E}_{\alpha}=\widetilde{E}_a\amalg\widetilde{E}_b\amalg\widetilde{E}_c ,
\]

\[
  (1)\qquad S\simeq t_0\amalg\coprod_{\alpha\in t_0}\widetilde{E}_{\alpha}.
\]

\[
  \sigma|t_0=\mathrm{id}_{t_0},\qquad \sigma|\widetilde{E}_{\alpha}=\sigma_{\alpha}\quad\text{pour }\alpha\in t_0 .
\]

\[
  \widetilde{T}\xrightarrow{\ \widetilde{\varphi}_{\alpha}\ }\widetilde{E}_{\alpha}\qquad(\alpha\in t)
\]

\[
  \widetilde{T}\xrightarrow{\ \widetilde{\varphi}\ }\prod_{\alpha\in t_0}\widetilde{E}_{\alpha}
  \quad\bigl(=\widetilde{E}_{a}\times\widetilde{E}_{b}\times\widetilde{E}_{c}\bigr).
\]

\[
  \widetilde{T}\xrightarrow{\ \varphi_u\ }\prod_{i\in u}\widetilde{E}_i\quad\bigl(\simeq\widetilde{E}_{\alpha}\times\widetilde{E}_{\beta}\bigr)
\]

\[
  T\simeq\underbrace{\{t_0\}}_{\substack{\text{triangle}\\ \text{de base}}}\amalg\underbrace{\coprod_{\alpha\in t_0}E_{\alpha}}_{\substack{\text{triangles}\\ \text{verticaux}}}\amalg\underbrace{\widetilde{T}}_{\substack{\text{triangles (horiz.)}\\ \text{supérieurs}}}
\]

\[
  E_{\alpha}=\widetilde{E}_{\alpha}/\sigma_{\alpha}
\]

\[
  u_s^{\gamma,\beta}:\widetilde{E}_{\beta}(s)\xrightarrow{\ \sim\ }\widetilde{E}_{\gamma}(s).
\]

\[
  S=\{\alpha\}\amalg\widetilde{S}'\qquad\text{où }\widetilde{S}'=S-\{\alpha\}
\]

\[
  A\simeq\widetilde{S}'\amalg\underbrace{\widetilde{S}'/\sigma_{S'}}_{S'} .
\]

\[
  \widetilde{\Gamma}\xrightarrow{\ \widetilde{\varphi}_{\alpha\beta}\ }
  \widetilde{E}_\alpha\times\widetilde{E}_\beta
\]

\[
  \Gamma\xrightarrow{\ \varphi_\alpha\ }E_\alpha ,\qquad
  \Gamma\xrightarrow{\ \varphi\ }\prod_{\alpha\in t_0}E_\alpha
\]

\[
  \varphi_\alpha:\Gamma\to E_\alpha ,\qquad
  \varphi:\Gamma\to\prod_{\alpha\in t_0}E_\alpha ,
\]

\[
  \widetilde{E}_\alpha\xrightarrow{\ p_\alpha\ }E_\alpha\qquad(\alpha\in t_0)
\]

\[
  \Gamma\xrightarrow{\ \varphi_{\alpha\beta}\ }E_\alpha\times E_\beta
\]

\[
  \bigl(\Gamma\xrightarrow{\ \varphi_\alpha\ }\Gamma_\alpha\bigr)_{\alpha\in t_0}
\]

\[
  \bigl(\Gamma\xrightarrow{\ \varphi_\alpha\ }\Gamma_\alpha\bigr)_{\alpha\in t_0}
\]

\[
  \sigma_{\underline{t}}=\sigma_{\{a,b,c\}}:S\to S
\]

\[
  \sigma_{\underline{t}}\,|\,\underline{t}=\mathrm{id}_{\underline{t}},\qquad
  \sigma_{\underline{t}}\,|\,E_a^{\underline{t}}=\sigma_a^{\underline{t}}
\]

\[
  \mathrm{int}(g).\sigma_{\underline{t}}=g\,\sigma_{\underline{t}}\,g^{-1}
  =\sigma_{g(\underline{t})}
\]

\[
  u\,a=a,\quad u\,b=b,\quad u\,c=c
\]

\[
  u\in\widetilde{E}_b,\ v\in\widetilde{E}_c \qquad
  \{s,u,v\}\in\widetilde{T}\subset T
\]

\[
  u'\in\widetilde{E}_b,\ v'\in\widetilde{E}_c \qquad
  \{s',u',v'\}\in\widetilde{T}\subset T
\]

\[
  [\,(b,u,u')\in T,\quad(c,v,v')\in T\,]
\]

\[
  (t,u,v_1)\in\widetilde{\Gamma}
\]

\[
  (t',v,u_1)\in\widetilde{\Gamma}\subset T\quad(\widetilde{E}_b)
\]

\[
  \sigma_{\underline{\sigma}}\,s=u \quad\text{car } \{s,u,v\}\in T,\ v\in\underline{\sigma}
\]

\[
  \sigma_{\underline{\tau}}\,u=t \quad\text{car } \{t,u,w\}\in T,\ w\in\underline{\tau}
\]

\[
  W=\sigma_{\underline{\tau}}\,\sigma_{\underline{\sigma}}
\]

\[
  V\,U\,\underline{s}=\underline{t}.
\]

\[
  \widetilde{E}_b=\underset{B'}{\widetilde{E}_b(s)}\amalg
  \underset{B''}{\widetilde{E}_b(t)}
\]

\[
  \widetilde{E}_b(\bar s_1)\cap\widetilde{E}_b(s)\neq\emptyset,\qquad
  E_b(\bar s_1)\cap\widetilde{E}_b(t)\neq\emptyset .
\]

\[
  \forall\,r\in A=\widetilde{E}_a,\ \text{\uncertain{ou}}\quad
  \widetilde{E}_b(r)=B'\ \text{ou}\ \widetilde{E}_b(r)=B''
\]

\[
  A'=\{r\in A\mid\widetilde{E}_b(r)=B'\},\qquad
  A''=\{r\in A\mid\widetilde{E}_b(r)=B''\}=\complement_A A'
\]

\[
  \widetilde{E}_c(r_1)=\widetilde{E}_c(r'_2)
\]

\[
  \widetilde{E}_c(r_1)=\widetilde{E}_c(r_2)
\]

\[
  \mathrm{trep}=6(1+2c)(1+c)=6(1+3c+2c^2)
\]

\[
  \mathrm{bitrep}=2c^2\,\mathrm{trep}=12c^2(1+2c)(1+c)
\]

\[
  =12c^2(1+3c+2c^2)
\]

\[
  \mathcal{G}=\mathcal{G}^{\circ}\quad\text{d'ordre}\quad 12c^2(1+2c)(1+c)
\]

\[
  =12.4.5.3=2^4.3^2.5=720
\]

\[
  \mathcal{G}\simeq\mathfrak{S}_2.(\mathfrak{S}_3\times\mathfrak{S}_3),
\]

\[
  12c^2(1+2c)(1+c)=12.1.3.2=72
\]

\[
  =\frac16\,c^2(1+2c)(1+c)
\]

\[
  xyz=1
\]

\[
  \varphi_a:(x,y,z)\longmapsto(x,\,a^{-1}z^{-1},\,a^{-1}y^{-1})
\]

\[
  xyz=1\Longrightarrow x\,a^{-1}z^{-1}a^{-1}y^{-1}=1
\]

\[
  \varphi_c\quad(x,y,z)\longmapsto(x,\,z+c,\,y+c)
\]

\[
  \varphi_a\varphi_b:\ (x,y,z)\longmapsto(x,\,y+c,\,z+c)\qquad c=a+b
\]

\[
  [\,\mathcal{G}_a\to\mathrm{Aut}_{E_a}(\widetilde{E}_a)\ \text{est injectif}\,]
\]

\[
  \alpha,\beta,\gamma\in[-1,2]\qquad\alpha\leqslant\beta\leqslant\gamma
\]

\[
  (1)\quad \boxed{c'_{\beta\gamma}=N_\beta\,c_{\beta\gamma}}\qquad\text{i.e.}\qquad
  \boxed{c_{\beta\gamma}=\frac{c'_{\beta\gamma}}{c'_{\beta\beta}}}
\]

\[
  (2)\quad \boxed{c'_{\beta\gamma}=c'_{\alpha,\gamma}\,\nu_{\alpha,\beta,\gamma}}
\]

\[
  c'_{\beta\gamma}=N_\beta\,c_{\beta\gamma}
\]

\[
  c_{-1,\gamma}=\text{nb de triangles \add{de $\tau_\gamma$}}
  =(1+c_\gamma)(1+2c_\gamma)
\]

\[
  c'_{-1,\gamma}=\text{nb de triangles épinglés de }\tau_\gamma
  =6(1+c_\gamma)(1+2c_\gamma)
\]

\[
  N_\gamma=N_\beta\,N_{\beta\gamma}
\]

\[
  \nu_{\beta-1,\beta,\gamma}\quad\text{et}\quad
  N_{\gamma-1,\gamma}\ \text{pour}\ \gamma\geqslant0
\]

\[
  \nu_{-1,0,\gamma}=2c_\gamma^2\quad
  \begin{cases}\gamma=0 & 2\\ \gamma=1 & 8=2^3\\ \gamma=2 & 32=2^5\end{cases}
\]

\[
  \nu_{0,1,\gamma}=c_\gamma-1\quad
  \begin{cases}\gamma=1 & 1\\ \gamma=2 & 3\end{cases}
\]

\[
  \nu_{1,2,\gamma}=c_\gamma-2=2\ \text{si}\ \gamma=2
\]

\[
  c'_{-1,\gamma}=6(1+c_\gamma)(1+2c_\gamma)\qquad
  6\cdot\left\{\begin{array}{l}1.1\\2.3\\3.5\\5.9\end{array}\right.
\]

\[
  \begin{array}{c|c|c|c|c}
  \beta\backslash\gamma & -1 & 0 & 1 & 2\\\hline
  -1 & 6=2.3 & 36=2^2.3^2 & 90=2.3^2.5 & 270=2.3^3.5\\\hline
  0 & 0 & 2^3.3^2 & 2^4.3^2.5 & 2^6.3^3.5\\\hline
  1 & 0 & 0 & 2^4.3^2.5 & 2^6.3^4.5\\
    &   &   & =6!=720 & \\\hline
  2 & 0 & 0 & 0 & 2^7.3^4.5
  \end{array}
\]

\[
  c'_{-1,\gamma}=6(1+c_\gamma)(1+2c_\gamma)\qquad\gamma\geqslant-1
\]

\[
  c'_{0,\gamma}=c'_{-1,\gamma}\,\nu_{-1,0,\gamma}
  =12c_\gamma^2(1+c_\gamma)(1+2c_\gamma)\qquad\gamma\geqslant0
\]

\[
  c'_{1,\gamma}=c'_{0,\gamma}\,\nu_{0,1,\gamma}
  =12c_\gamma^2(c_\gamma+1)(c_\gamma-1)(2c_\gamma+1)\qquad\gamma\geqslant1
\]

\[
  c'_{2,\gamma}=c'_{1\gamma}\,\nu_{1,2,\gamma}
  =12c_\gamma^2(c_\gamma+1)(c_\gamma-1)(c_\gamma-2)(2c_\gamma+1)
\]

\[
  \gamma\geqslant2\ \text{i.e.}\ \gamma=2
\]

\[
  c_{0,\gamma}=\frac1{72}\,c_{0,\gamma}
  =\frac16\,c_\gamma^2(c_\gamma+1)(c_\gamma-1)(2c_\gamma+1)
\]

\[
  c_{1,\gamma}=\frac1{720}\,c'_{1,\gamma}
  =\frac1{60}\,c_\gamma^2(c_\gamma-1)(c_\gamma+1)(2c_\gamma+1)
\]

\[
  c_{2,\gamma}=1\ \text{—}
\]

\[
  \begin{array}{c|c|c|c|c}
  \beta\backslash\gamma & -1 & 0 & 1 & 2\\\hline
  -1 & 1 & 6=2.3 & 15=3.5 & 45=3^2.5\\\hline
  0 & 0 & 1 & 10=2.5 & 120=2^3.3.5\\\hline
  1 & 0 & 0 & 1 & 36=2^2.3^2\\\hline
  2 & 0 & 0 & 0 & 1
  \end{array}
\]

\[
  \begin{array}{l|c|c|c|c}
   & c=0 & 1 & 2 & 4\\\hline
  \text{Nb de triangles incidents à un sommet : } c+1=e & 1 & 2 & 3 & 5\\
  \text{Nb de sommets —— : } 2(c+1) & 2 & 4 & 6 & 10\\
  \text{Nb de sommets : } 3(1+2c)=3+6c & 3 & 9 & 15 & 27
  \end{array}
\]

\[
  \begin{array}{l|c|c|c|c}
  \text{Nb des arêtes : } 3(c+1)(1+2c) & 3 & 18 & 45 & 135\\
  \quad =3(1+3c+2c^2) & & & & \\\hline
  \text{Nb de triangles } (c+1)(1+2c)=1+3c+2c^2 & 1 & 6 & 15 & 45\\
  \text{Nb de triangles verticaux } =3c & 0 & 3 & 6 & 12\\
  \text{\phantom{Nb de triangles} transversaux } 2c^2 & 0 & 2 & 8 & 32
  \end{array}
\]

\[
  2c^2=2\,2^{2\nu}=2^{2\nu+1}\qquad(\nu=-\infty,0,1,2)
\]

\[
  \begin{array}{l|c|c|c|c}
  \text{NB des sommets antiliés}
   & 0 & 4=2^2 & 8=2^3 & 16=2^4\\
  \text{à un sommet } 4c=2^{\nu+2} & & =2^e & =2^e & =2^{e-1}\\\hline
  \text{Ordre de } \mathrm{Aut}(S,A) : N_\omega & 6=2.3 & 72=2^3.3^2 & \cdot & 51840\\
   & & & & =2^7.3^4.5
  \end{array}
\]

\[
  \begin{array}{l|c|c|c|c}
  \text{Nb de } \tau_{-1}\ \text{(triangles) plongés}
   & 1 & 6=2.3 & 15=3.5 & 45=3^2 5\\
  \quad (c+1)(2c+1)=1+3c+2c^2 & & & & \\
  \text{Nb de } \tau_0\ \text{(bitriangles plongés)}
   & 0 & 1 & 10 & 120\\
  \quad \tfrac16\,c(c+1)(2c+1) & & & & \\
  \text{Nb de } \tau_1\ \text{(tritriangles) plongés}
   & 0 & 0 & 1 & 36=2^2 3^2\\
  \quad \tfrac1{30}\,c(c+1)(2c+1)(c-1) & & & &
  \end{array}
\]

\[
  \begin{array}{c|c|c|c}
  6 & 36=2^2 3^2 & 90=2.3^2.5 & 270=2.3^3.5\\\hline
  0 & 72=2^3.3^2 & 360=2^3.3^2.5 & 2160=2^4.3^3.5\\\hline
  0 & 0 & 720=2^4.3^2 & 2^6.3^4.5\\\hline
  0 & 0 & - & 2^7.3^4.5
  \end{array}
\]

\[
\begin{array}{lll}
x & y & z \\
x & y+u+w, & z+u+w \\
x_0 & y_0 + w & z_0 + w
\end{array}
\]

\[
0 \to \widetilde{\Gamma} \to \widetilde{E}_1 \times \widetilde{E}_2 \to \qquad \widetilde{\Gamma} - \widetilde{E}_2
\]

\[
(x,y) \longmapsto \text{\struck{$x * y$}} \ (= -x\cdot y)
\]

\[
q_{12}(a\cdot\tilde{y}) = q_1\,\varepsilon(a)\,q_2\,\ldots \qquad 2^4/h = 2^2
\]

\[
\tau_2\tau_1 = \mathrm{id} \qquad \tau_3\tau_2 = \mathrm{id} \qquad \tau_1\tau_3 = \mathrm{id} \qquad \tau^2 = \mathrm{id}
\]

\[
\sigma_x\sigma_y\sigma_x^{-1}\sigma_y^{-1} = \sigma_{\sigma_x y}\,\sigma_y^{-1}
\]

\[
\underbrace{\sigma_x\sigma_{x'}}_{\pi}\ \underbrace{\sigma_y\sigma_{y'}}_{\sigma_{\pi y}\sigma_{\pi y'}}\ \sigma_x\sigma_{x'}\ \sigma_y\sigma_{y'}
\]

\[
\pi_{x,x'}\,\pi_{y,y'} = \pi_{x,x'}\,\pi_{\pi_{x,x'}y,\ \ldots}
\]

\[
12 \cdot 4^2 \cdot \text{\struck{$6$}}\,8 \cdot 9 \cdot 5 = 2^6 \cdot 3^3 \cdot 5
\]

\[
2^7 \cdot 3^4 \cdot 5 \qquad 128 \times 405 = 51840 \qquad /\,2\cdot 3
\]

\[
\begin{array}{c|c|c|c|c}
\beta \backslash \alpha & -1 & 0 & 1 & 2 \\ \hline
-\infty & (\mathrm{Ens})_3 & (\mathrm{Ens})_{3,3} & (\mathrm{Ens})_6 & * \\
-1 & (\mathrm{Ens})_3 & (\mathrm{Ens})_3 \times (\mathrm{Ens})_2 & (\mathrm{Ens})_{2,2,2} & \text{Trialité} \\
0 & \emptyset & (\mathrm{Ens})_{3,3} & (\mathrm{Ens})_{3,3} & (\mathrm{Ens})_{3,3} \times (\mathrm{Ens})_{3,3} \\
1 & \emptyset & \emptyset & (\mathrm{Ens})_6 & (\mathrm{Ens})_6 \times (\mathrm{Ens})_2
\end{array}
\]

\[
\begin{array}{c|c}
-1,\,0,\,1 & (\mathrm{Ens})_3 \\
-1,\,0,\,2 & (\mathrm{Ens})_3 \times (\mathrm{Ens})_3 \\
-1,\,1,\,2 & (\mathrm{Ens})_{2,2,2} \times (\mathrm{Ens})_2 \\
0,\,1,\,2 & (\mathrm{Ens})_{3,3} \times (\mathrm{Ens})_2 \\
-1,\,0,\,1,\,2 & (\mathrm{Ens})_3 \times (\mathrm{Ens})_2
\end{array}
\]

\[
\begin{array}{lll}
\sigma_{t_0} : & x \leftrightarrow x', \quad y \leftrightarrow y', \quad z \leftrightarrow z' \\
\sigma_t : & x \leftrightarrow y', \quad y \leftrightarrow z', \quad z \leftrightarrow x' \\
\sigma_{t'} : & x \leftrightarrow z', \quad y \leftrightarrow x', \quad z \leftrightarrow y'
\end{array}
\]

\[
\sigma_{t_0}\sigma_t = \sigma_t\sigma_{t'} = \sigma_{t'}\sigma_{t_0} = \pi_a, \qquad \pi^3 = \mathrm{id}
\]

\[
[\sigma_{t_0},\sigma_t] = [\sigma_t,\sigma_{t'}] = [\sigma_{t'},\sigma_{t_0}] = \pi^2 = \pi^{-1}
\]

\[
\sigma_t\sigma_{t_0} = \sigma_{t_0}\sigma_{t'} = \sigma_{t'}\sigma_t = \pi^{-1}
\]

\[
\begin{array}{ll}
(s,t)\ (s',t') & (x,u)\ (x',u') \\
(t,u)\ (t',u') & (x,t)\ (x',t) \\
(u,s)\ (u',s') &
\end{array}
\]

\[
\begin{array}{ll}
(y',u)\ (y,u') & (z',t')\ (z,t) \\
(y',s')\ (y,s) & (z',s)\ (z,s') \\
(y',x)\ (y,x') & (z',x)\ (z,x') \\
 & (z',y)\ (z,y')
\end{array}
\]

\[
\text{\struck{$3(1+2c)$}} \qquad \tfrac12(3+6c)(2+2c) = 3(1+2c)(1+c) = 3(1+3c+2c^2)
\]

\[
\boxed{\sigma_t\sigma_s\sigma_u = \sigma_{t_0} \text{ sur } \widetilde{E}_a \smallsetminus \{s,s'\}}
\]

\[
\sigma_t\sigma_s = \sigma_{t_0}\sigma_u = \sigma_u\sigma_{t_0}
\]

\[
\sigma_t\sigma_s\sigma_t^{-1} = \sigma_{\sigma_t s} = \sigma_u
\]

\[
\sigma_t\sigma_s = \text{\struck{\ldots}}\ \sigma_u\sigma_t \quad \text{i.e.} \quad \underbrace{\sigma_u\sigma_t\sigma_s}_{=\ \sigma_{t_0} \text{ sur } S\smallsetminus\{t_0\}} = \sigma_t
\]

\[
\sigma_t\sigma_y \neq \sigma_y\sigma_t : \ s \to u \to x \qquad\qquad \sigma_u\sigma_z = \sigma_z\sigma_u : \ s \to t \to x'
\]

\[
\varphi(x) = (\varphi_i(x))_{i\in I} \in E = \prod X_i ,
\]

\[
\rho(\xi_i(z_0))\,(x_j)_j = (y_j)_j , \quad y_j = x_j \text{ si } j \neq i, \quad y_i = z_0 x_i \text{ sinon}.
\]

\[
(\sigma_{i_0})_E = \prod_{i \in I \smallsetminus i_0} \rho(\xi_i(z_0)) = \rho\Bigl(\sum_{i\in I\smallsetminus i_0} \xi_i(z_0)\Bigr) = \rho(z)\,\rho(\xi_{i_0}(z_0))
\]

\[
(\sigma_g)_E = \rho\bigl(g + \delta\eta(g)\bigr)
\]

\[
V \xrightarrow{\ u\ } V, \qquad u(x) = x + \delta\eta(x)
\]

\[
x = \delta\eta x
\]

\[
\text{\struck{$z$}} = \delta\eta z \quad \text{i.e.} \quad \text{\struck{$z_0 = \eta\delta z_0$}} \quad (\text{comme } z = \delta z_0)
\]

\[
z_0 = \eta z \quad \text{i.e.} \quad \eta z \neq 0
\]

\[
\text{\struck{\ill{}}} = 2^{c_\nu + 1} = 2^2 = 4,
\]

\[
\text{\struck{Im}}\ \delta(\mathbb{F}_2) = \{0, z\}, \quad \text{comme} \ \ldots
\]

\[
(\mathrm{Ens})_{2,2,2} \qquad (\mathrm{Ens})_2 \times (\mathrm{Ens})_4
\]

\[
\mathfrak{S}_3 \cdot (\pm 1)^3 \simeq \mathfrak{S}_2 \times \mathfrak{S}_4
\]

\[
\mathfrak{S}_4 \cdot \text{\struck{$\mathbb{Z}$}}(\pm 1)^4 \simeq W_{B_4} \quad (\text{groupe de Weyl de } SO(9))
\]

\[
\zeta_1 \overset{\lambda_{12}}{\text{------}} \zeta_1^*
\]

\[
\cong (\mathrm{Ens})_2 \times (\mathrm{Ens})_3 \times (\mathrm{Ens})_3
\]

\[
\tfrac12(720 \times 6 \times 3) = 2^4 \cdot 3^4 \cdot 5 = N : 2^3 \ \text{---}
\]

\[
\simeq \underbrace{(\mathrm{Ens})_2 \times (\mathrm{Ens})_2}_{\text{bicarrés}} \times \underbrace{(\mathrm{Ens})_2}_{\text{\uncertain{complétant l'enveloppe quasi-trialitaire}}} \simeq \mathrm{Tors}\bigl((\pm1)^3\bigr)
\]

\[
\text{\struck{$(\mathrm{Ens})$}}_{2,2} \times \mathrm{Ens}_3 \simeq \text{Carré} \times (\mathrm{Ens})_3
\]