Subgroup ($H$) information
| Description: | $C_2^6$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(1,4)(2,7)(3,5)(6,8)(11,12)(13,14), (1,5)(2,7)(3,4)(6,8)(9,10)(13,14), (2,7) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational.
Ambient group ($G$) information
| Description: | $C_2^5:S_4$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^7.(S_3\times \GL(3,2))$, of order \(129024\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $\GL(6,2)$, of order \(20158709760\)\(\medspace = 2^{15} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \) |
| $\operatorname{res}(S)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(128\)\(\medspace = 2^{7} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2^7$ | |||||||
| Normalizer: | $C_2^7$ | |||||||
| Normal closure: | $C_2^7$ | |||||||
| Core: | $C_2^2$ | |||||||
| Minimal over-subgroups: | $C_2^7$ | |||||||
| Maximal under-subgroups: | $C_2^5$ | $C_2^5$ | $C_2^5$ | $C_2^5$ | $C_2^5$ | $C_2^5$ | $C_2^5$ | $C_2^5$ |
Other information
| Number of subgroups in this autjugacy class | $42$ |
| Number of conjugacy classes in this autjugacy class | $7$ |
| Möbius function | $0$ |
| Projective image | $C_2^5:S_4$ |