Subgroup ($H$) information
| Description: | $(C_2^2\times C_4^2).S_4$ |
| Order: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Index: | \(2\) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(1,6,2,5)(3,8,4,7)(9,14)(10,13)(11,16)(12,15)(17,23,18,24)(19,21,20,22) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is normal, maximal, a semidirect factor, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $(C_2\times C_4^2).\GL(2,\mathbb{Z}/4)$ |
| Order: | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4^2:A_4.C_2^5.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_4^2:A_4.D_4.C_2^3$ |
| $\card{\operatorname{res}(S)}$ | \(12288\)\(\medspace = 2^{12} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $(C_4^2:A_4) \rtimes (C_2\times C_4)$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $-1$ |
| Projective image | $(C_4^2:A_4) \rtimes (C_2\times C_4)$ |