Subgroup ($H$) information
| Description: | $A_4^2.S_4\wr C_2.C_4$ |
| Order: | \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| Index: | \(2\) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(5,6)(9,10), (15,23,20)(16,24,19), (15,16)(23,24), (1,6,9)(2,5,10)(15,23,20) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{10}.C_3^4:(C_2\times D_4)$ |
| Order: | \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \) |
| 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: | $C_2$ |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_2^8.C_3^4.C_2^5.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2^8.C_3^4.C_2^5.C_2^3$ |
| $W$ | $C_2\times A_4^2\wr C_2.D_4$, of order \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^{10}.C_3^4:(C_2\times D_4)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2\times A_4^2\wr C_2.D_4$ |