Subgroup ($H$) information
| Description: | $C_2^6:S_4$ |
| Order: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,14,4,16)(2,13,3,15) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is characteristic (hence normal), nonabelian, monomial (hence solvable), and rational. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_2^7.(C_2\times S_4)$ |
| Order: | \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^7.(D_4\times S_4)$, of order \(24576\)\(\medspace = 2^{13} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_2^6.S_4^2$, of order \(36864\)\(\medspace = 2^{12} \cdot 3^{2} \) |
| $W$ | $C_2^5:(C_2\times S_4)$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^2$ | ||||
| Normalizer: | $C_2^7.(C_2\times S_4)$ | ||||
| Minimal over-subgroups: | $C_2^7:S_4$ | $C_2^6.A_4.C_2^2$ | |||
| Maximal under-subgroups: | $C_2^6:A_4$ | $C_2^6:D_4$ | $C_2^2\wr S_3$ | $C_2^4:S_4$ | $C_2^4:S_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $D_5^3.C_2^2$ |