Subgroup ($H$) information
| Description: | not computed |
| Order: | \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | not computed |
| Generators: |
$\langle(3,15)(5,10), (5,10,15)(8,11,12), (1,2)(4,16)(6,14)(7,11)(8,12)(9,13), (2,9,13) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is the commutator subgroup (hence characteristic and 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: | $A_4^2.S_4\wr C_2.C_4$ |
| Order: | \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| 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: | $C_2^2\times C_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2^3:S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Outer Automorphisms: | $C_2^3:S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.C_3^4.C_2^5.C_2^3$ |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $A_4^2\wr C_2.D_4$, of order \(331776\)\(\medspace = 2^{12} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $A_4^2.S_4\wr C_2.C_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $A_4^2.S_4\wr C_2.C_4$ |