Subgroup ($H$) information
| Description: | $C_2^3\times F_8$ |
| Order: | \(448\)\(\medspace = 2^{6} \cdot 7 \) |
| Index: | \(567\)\(\medspace = 3^{4} \cdot 7 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$\langle(2,9,5,7,4,6,8)(10,16)(11,13)(12,15)(17,18), (2,4,6,7,8,3,9), (2,6)(3,4) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $\SL(2,8)^2$ |
| Order: | \(254016\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 7^{2} \) |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Derived length: | $0$ |
The ambient group is nonabelian, an A-group, and perfect (hence nonsolvable).
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | ${}^2G(2,3)\wr C_2$, of order \(4572288\)\(\medspace = 2^{7} \cdot 3^{6} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $F_8:C_3\times \GL(3,2)$, of order \(28224\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
| $W$ | $C_7\times F_8$, of order \(392\)\(\medspace = 2^{3} \cdot 7^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $162$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $\SL(2,8)^2$ |