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: | \(2\) |
| Exponent: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(22,23,24)(27,28,30)(32,35,33)(37,39,38), (1,21,12,35,3,23,11,33,2,24,13,34,5,25,14,32) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is normal, maximal, nonabelian, and nonsolvable. Whether it is a direct factor, a semidirect factor, or rational has not been computed.
Ambient group ($G$) information
| Description: | $A_5^8.C_2\wr C_4.C_2^3$ |
| Order: | \(85996339200000000\)\(\medspace = 2^{25} \cdot 3^{8} \cdot 5^{8} \) |
| 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: | $C_2$ |
| 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
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(687970713600000000\)\(\medspace = 2^{28} \cdot 3^{8} \cdot 5^{8} \) |
| $\operatorname{Aut}(H)$ | 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 |