Subgroup ($H$) information
| Description: | $C_2^4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(2,8)(6,7)(9,10)(11,12), (1,5)(2,7)(3,4)(6,8), (1,4)(2,6)(3,5)(7,8), (1,5)(2,8)(3,4)(6,7)(9,11)(10,12)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a semidirect factor, 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: | $C_2^4:S_4$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and rational.
Quotient group ($Q$) structure
| Description: | $S_4$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6:(S_3\times \GL(3,2))$, of order \(64512\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $A_8$, of order \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $\operatorname{res}(S)$ | $S_3^2$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(256\)\(\medspace = 2^{8} \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^6$ | ||
| Normalizer: | $C_2^4:S_4$ | ||
| Complements: | $S_4$ | ||
| Minimal over-subgroups: | $C_2^2:A_4$ | $C_2^2\wr C_2$ | $C_2^5$ |
| Maximal under-subgroups: | $C_2^3$ | $C_2^3$ |
Other information
| Number of subgroups in this autjugacy class | $7$ |
| Number of conjugacy classes in this autjugacy class | $7$ |
| Möbius function | $-12$ |
| Projective image | $C_2^4:S_4$ |