Subgroup ($H$) information
| Description: | $A_5^8.C_2\wr C_4.C_2^2$ |
| Order: | \(42998169600000000\)\(\medspace = 2^{24} \cdot 3^{8} \cdot 5^{8} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(27,30,28), (28,30,29), (32,34,35), (6,8,7), (6,10,7), (16,17,19), (26,29,28) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), nonabelian, and nonsolvable. Whether it is a direct factor, a semidirect factor, or rational has not been computed.
Ambient group ($G$) information
| Description: | $A_5^8.C_4^2.D_4^2$ |
| Order: | \(171992678400000000\)\(\medspace = 2^{26} \cdot 3^{8} \cdot 5^{8} \) |
| Exponent: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and nonsolvable. Whether it is rational has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(343985356800000000\)\(\medspace = 2^{27} \cdot 3^{8} \cdot 5^{8} \) |
| $\operatorname{Aut}(H)$ | Group of order \(343985356800000000\)\(\medspace = 2^{27} \cdot 3^{8} \cdot 5^{8} \) |
| $\card{W}$ | 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 |