Subgroup ($H$) information
| Description: | not computed |
| Order: | \(16796160000\)\(\medspace = 2^{12} \cdot 3^{8} \cdot 5^{4} \) |
| Index: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | not computed |
| Generators: |
$\langle(22,28,26,24)(23,25,27,30)(32,35,34)(33,37,39)(36,38,40), (1,3,4,8,7)(2,9,6,10,5) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, and perfect (hence nonsolvable). 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: | $A_6^4.D_4.C_2^3.D_4$ |
| Order: | \(8599633920000\)\(\medspace = 2^{21} \cdot 3^{8} \cdot 5^{4} \) |
| Exponent: | \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2^6:D_4$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^9.D_4^2$, of order \(32768\)\(\medspace = 2^{15} \) |
| Outer Automorphisms: | $C_2^5:D_4$, of order \(256\)\(\medspace = 2^{8} \) |
| Derived length: | $3$ |
The quotient is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(17199267840000\)\(\medspace = 2^{22} \cdot 3^{8} \cdot 5^{4} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |