Subgroup ($H$) information
| Description: | $C_2^5:A_4$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,12,23)(2,11,24)(3,10,22)(4,9,21)(5,15,17)(6,16,18)(7,13,19)(8,14,20) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_4^3:C_2^2:S_4$ |
| 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: | $C_2\times D_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2\wr C_2^2$, of order \(64\)\(\medspace = 2^{6} \) |
| Outer Automorphisms: | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4^2:A_4.C_2^4.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2^6.\POPlus(4,3)$, of order \(36864\)\(\medspace = 2^{12} \cdot 3^{2} \) |
| $\card{W}$ | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |