Subgroup ($H$) information
| Description: | $C_7^5:S_6$ |
| Order: | \(12101040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{5} \) |
| Index: | \(3\) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Generators: |
$\langle(8,11,9,13,12,14,10)(22,26,27,24,28,25,23)(36,37,40,38,39,41,42), (1,6,3,2,7,4,5) \!\cdots\! \rangle$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, and nonsolvable. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_7^5:S_6.C_3$ |
| Order: | \(36303120\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \cdot 7^{5} \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| 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, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^5:(C_6\times S_6)$, of order \(72606240\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \cdot 7^{5} \) |
| $\operatorname{Aut}(H)$ | $C_7^5:(C_6\times S_6)$, of order \(72606240\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \cdot 7^{5} \) |
| $W$ | $C_7^5:S_6.C_3$, of order \(36303120\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \cdot 7^{5} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_7^5:S_6.C_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_7^5:S_6.C_3$ |