Subgroup ($H$) information
| Description: | $C_5^6.(C_2\times C_{10}^2:\OD_{16})$ |
| Order: | \(50000000\)\(\medspace = 2^{7} \cdot 5^{8} \) |
| Index: | \(2\) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Generators: |
$\langle(21,24,22,25,23)(31,35,34,33,32), (1,5,4,3,2)(11,12,13,14,15)(16,20,19,18,17) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is normal, maximal, nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_5^4.D_5^4.(C_2\times Q_8)$ |
| Order: | \(100000000\)\(\medspace = 2^{8} \cdot 5^{8} \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| 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 \(12800000000\)\(\medspace = 2^{15} \cdot 5^{8} \) |
| $\operatorname{Aut}(H)$ | Group of order \(51200000000\)\(\medspace = 2^{17} \cdot 5^{8} \) |
| $\card{W}$ | 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 |