Subgroup ($H$) information
| Description: | $C_2^2\times \SL(2,8)$ |
| Order: | \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \) |
| Index: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Generators: |
$\langle(10,14)(11,13)(12,18)(15,16), (1,5,9)(2,7,4)(3,8,6)(10,11)(12,16)(13,14)(15,18), (10,11)(12,16)(13,14)(15,18), (1,3,7)(2,6,4)(5,9,8)\rangle$
|
| Derived length: | $1$ |
The subgroup is nonabelian, an A-group, and nonsolvable.
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 \(4572288\)\(\medspace = 2^{7} \cdot 3^{6} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $S_3\times {}^2G(2,3)$, of order \(9072\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7 \) |
| $W$ | $\SL(2,8)$, of order \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $126$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $\SL(2,8)^2$ |