Subgroup ($H$) information
| Description: | $C_3^8$ |
| Order: | \(6561\)\(\medspace = 3^{8} \) |
| Index: | \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
| Exponent: | not computed |
| Generators: |
$\langle(19,20,21)(22,23,24)(25,26,27)(28,30,29)(31,33,32)(34,36,35)(37,42,44)(38,40,45) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is characteristic (hence 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, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^{10}.S_5.\GL(2,3)$ |
| Order: | \(340122240\)\(\medspace = 2^{7} \cdot 3^{12} \cdot 5 \) |
| Exponent: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $5$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $S_5\times C_3^2:\GL(2,3)$ |
| Order: | \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Automorphism Group: | $S_5\times C_3^2:\GL(2,3)$, of order \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $5$ |
The quotient is nonabelian and nonsolvable.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(680244480\)\(\medspace = 2^{8} \cdot 3^{12} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | 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 |