Subgroup ($H$) information
| Description: | $C_5^4.D_5^4.C_2^3:C_8$ |
| Order: | \(400000000\)\(\medspace = 2^{10} \cdot 5^{8} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Generators: |
$\langle(11,14,13,15)(22,24,25,23)(27,28,30,29)(32,35)(33,34)(37,39,40,38), (1,40,12,7,5,36,13,10,2,39,11,9,3,38,15,6) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is normal, nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_5^4.D_5^4.(C_2^5.D_4)$ |
| Order: | \(1600000000\)\(\medspace = 2^{12} \cdot 5^{8} \) |
| Exponent: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial 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
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 \(25600000000\)\(\medspace = 2^{16} \cdot 5^{8} \) |
| $\operatorname{Aut}(H)$ | Group of order \(12800000000\)\(\medspace = 2^{15} \cdot 5^{8} \) |
| $\card{W}$ | 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 |