Subgroup ($H$) information
| Description: | $C_7^5:(C_2^4:A_5)$ |
| Order: | \(16134720\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 7^{5} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Generators: |
$\langle(29,32,34,30,33,35,31), (1,4,6,2,5,7,3), (1,10,2,11)(3,9,5,12)(4,13,6,14) \!\cdots\! \rangle$
|
| Derived length: | $0$ |
The subgroup is the commutator subgroup (hence characteristic and normal), nonabelian, and perfect (hence nonsolvable). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_7^5.C_2^5.S_5$ |
| Order: | \(64538880\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \cdot 7^{5} \) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^5:(C_2\wr S_5\times C_3)$, of order \(193616640\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \cdot 7^{5} \) |
| $\operatorname{Aut}(H)$ | $C_7^5:(C_2\wr S_5\times C_3)$, of order \(193616640\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \cdot 7^{5} \) |
| $W$ | $C_7^5.C_2^5.S_5$, of order \(64538880\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \cdot 7^{5} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_7^5.C_2^5.S_5$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_7^5.C_2^5.S_5$ |