Subgroup ($H$) information
| Description: | not computed |
| Order: | \(110592\)\(\medspace = 2^{12} \cdot 3^{3} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(16,20)(19,23)(21,24), (16,20)(18,22), (4,12)(9,10), (3,8,6)(4,10,12)(5,7,11) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $A_4^3.C_2^4:\GL(2,\mathbb{Z}/4)$ |
| Order: | \(2654208\)\(\medspace = 2^{15} \cdot 3^{4} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $1$ |
| Derived length: | $3$ |
The quotient is nonabelian, monomial (hence solvable), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.C_3^3.C_2^6.C_6.C_2^5$, of order \(21233664\)\(\medspace = 2^{18} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $A_4:S_4^2.S_4$, of order \(165888\)\(\medspace = 2^{11} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $A_4^3.C_2^4:\GL(2,\mathbb{Z}/4)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $A_4^3.C_2^5:S_4$ |