Subgroup ($H$) information
| Description: | not computed |
| Order: | \(390625\)\(\medspace = 5^{8} \) |
| Index: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | not computed |
| Generators: |
$\langle(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,38,40,37,39), (26,27,28,29,30) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $5$-Sylow subgroup (hence a Hall subgroup), and a $p$-group (hence elementary and hyperelementary). Whether it is a direct factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_5^8.D_4^2.C_2^2$ |
| Order: | \(100000000\)\(\medspace = 2^{8} \cdot 5^{8} \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial 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: | $(C_2^2\times D_4^2).D_4$, of order \(2048\)\(\medspace = 2^{11} \) |
| Outer Automorphisms: | $C_2^4$, of order \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $4$ |
| 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 \(1600000000\)\(\medspace = 2^{12} \cdot 5^{8} \) |
| $\operatorname{Aut}(H)$ | 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 |