Subgroup ($H$) information
| Description: | $\SL(2,8)$ |
| Order: | \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| Index: | \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Generators: |
$\langle(1,5,9)(2,7,4)(3,8,6), (1,3,7)(2,6,4)(5,9,8)\rangle$
|
| Derived length: | $0$ |
The subgroup is normal, a direct factor, nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), 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).
Quotient group ($Q$) structure
| Description: | $\SL(2,8)$ |
| Order: | \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Automorphism Group: | ${}^2G(2,3)$, of order \(1512\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 7 \) |
| Outer Automorphisms: | $C_3$, of order \(3\) |
| Derived length: | $0$ |
The quotient is nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.
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 \(4572288\)\(\medspace = 2^{7} \cdot 3^{6} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | ${}^2G(2,3)$, of order \(1512\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 7 \) |
| $W$ | $\SL(2,8)$, of order \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
Related subgroups
| Centralizer: | $\SL(2,8)$ | ||
| Normalizer: | $\SL(2,8)^2$ | ||
| Complements: | $\SL(2,8)$ $\SL(2,8)$ | ||
| Minimal over-subgroups: | $\GL(2,8)$ | $C_3\times \SL(2,8)$ | $C_2\times \SL(2,8)$ |
| Maximal under-subgroups: | $F_8$ | $D_9$ | $D_7$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $-504$ |
| Projective image | $\SL(2,8)^2$ |