Subgroup ($H$) information
| Description: | $C_5^4$ |
| Order: | \(625\)\(\medspace = 5^{4} \) |
| Index: | \(10000\)\(\medspace = 2^{4} \cdot 5^{4} \) |
| Exponent: | \(5\) |
| Generators: |
$\langle(1,5,4,3,2)(11,14,12,15,13)(16,20,19,18,17)(21,22,23,24,25)(26,27,28,29,30) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_5^8.\OD_{16}$ |
| Order: | \(6250000\)\(\medspace = 2^{4} \cdot 5^{8} \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial or rational has not been computed.
Quotient group ($Q$) structure
| Description: | $C_5^4.\OD_{16}$ |
| Order: | \(10000\)\(\medspace = 2^{4} \cdot 5^{4} \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Automorphism Group: | $C_5^4:(C_4^3:C_2^2)$, of order \(160000\)\(\medspace = 2^{8} \cdot 5^{4} \) |
| Outer Automorphisms: | $C_2^2\times C_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and monomial (hence solvable).
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 \(1440000000000\)\(\medspace = 2^{14} \cdot 3^{2} \cdot 5^{10} \) |
| $\operatorname{Aut}(H)$ | $\GL(4,5)$, of order \(116064000000\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 5^{6} \cdot 13 \cdot 31 \) |
| $\card{W}$ | 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 |