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: |
$\langle(9,15)(10,16)(11,14)(12,13), (1,12,3,9)(2,11,4,10)(5,16,7,14)(6,15,8,13) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $C_2^6.(C_4\times S_4)$ |
| Order: | \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
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 \(36864\)\(\medspace = 2^{12} \cdot 3^{2} \) |
| $W$ | $C_2^6:(C_4\times S_3)$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_2^5.(C_4\times S_4)$ |