Subgroup ($H$) information
| Description: | not computed |
| Order: | \(167961600000000\)\(\medspace = 2^{16} \cdot 3^{8} \cdot 5^{8} \) |
| Index: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | not computed |
| Generators: |
$\langle(18,20,19)(22,25,24)(26,30)(28,29)(31,33)(34,35)(36,39,40,37,38), (23,24,25) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, an A-group, 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: | $A_5^8.C_2\wr C_4.C_2^2$ |
| Order: | \(42998169600000000\)\(\medspace = 2^{24} \cdot 3^{8} \cdot 5^{8} \) |
| 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: | $D_4^2:C_2^2$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_4.D_4^2:C_2^4$, of order \(4096\)\(\medspace = 2^{12} \) |
| Outer Automorphisms: | $C_2^5$, of order \(32\)\(\medspace = 2^{5} \) |
| Derived length: | $3$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), 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 \(343985356800000000\)\(\medspace = 2^{27} \cdot 3^{8} \cdot 5^{8} \) |
| $\operatorname{Aut}(H)$ | 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 |