Subgroup ($H$) information
| Description: | $C_2^5:D_4$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Index: | \(3\) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(1,4)(2,7)(3,5)(6,8)(11,12)(13,14), (2,7)(6,8), (1,2)(3,8)(4,7)(5,6)(11,13) \!\cdots\! \rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), metabelian, 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 \(129024\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2^{15}.C_2^4.\PSL(2,7)$ |
| $\operatorname{res}(S)$ | $C_2^5:\GL(3,2)$, of order \(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_2^4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
| Centralizer: | $C_2^4$ | ||||
| Normalizer: | $C_2^5:D_4$ | ||||
| Normal closure: | $C_2^5:S_4$ | ||||
| Core: | $C_2^7$ | ||||
| Minimal over-subgroups: | $C_2^5:S_4$ | ||||
| Maximal under-subgroups: | $C_2^7$ | $C_2^4:D_4$ | $C_2^5:C_4$ | $C_2^3\wr C_2$ | $C_2^3\wr C_2$ |
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_2^4:S_4$ |