Subgroup ($H$) information
| Description: | $C_2^6$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(3969\)\(\medspace = 3^{4} \cdot 7^{2} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(2,5)(3,8)(4,7)(6,9)(10,17)(11,15)(12,13)(16,18), (2,3)(4,6)(5,8)(7,9), (2,8) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $2$-Sylow subgroup (hence a Hall subgroup), a $p$-group (hence elementary and hyperelementary), and rational.
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 \(4572288\)\(\medspace = 2^{7} \cdot 3^{6} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $\GL(6,2)$, of order \(20158709760\)\(\medspace = 2^{15} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \) |
| $W$ | $C_7^2$, of order \(49\)\(\medspace = 7^{2} \) |
Related subgroups
| Centralizer: | $C_2^6$ | ||
| Normalizer: | $C_2^6.C_7^2$ | ||
| Normal closure: | $\SL(2,8)^2$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_2^3\times F_8$ | $C_2^3:F_8$ | $C_2^3:F_8$ |
| Maximal under-subgroups: | $C_2^5$ | $C_2^5$ |
Other information
| Number of subgroups in this autjugacy class | $81$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $\SL(2,8)^2$ |