Subgroup ($H$) information
| Description: | $C_3^4$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(7558272\)\(\medspace = 2^{7} \cdot 3^{10} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(1,3,2)(22,24,23)(25,27,26)(34,36,35), (16,18,17)(19,21,20)(28,29,30)(31,32,33) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
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 or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^8.C_2^5:(\He_3^2:C_4)$ |
| Order: | \(612220032\)\(\medspace = 2^{7} \cdot 3^{14} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_3^8.(C_2\times A_4^2:C_4)$ |
| Order: | \(7558272\)\(\medspace = 2^{7} \cdot 3^{10} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Automorphism Group: | Group of order \(2176782336\)\(\medspace = 2^{12} \cdot 3^{12} \) |
| Outer Automorphisms: | $F_9:C_2^2$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $4$ |
The quotient is nonabelian and solvable. Whether it is monomial has not been computed.
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 \(132239526912\)\(\medspace = 2^{10} \cdot 3^{17} \) |
| $\operatorname{Aut}(H)$ | $C_2.\PSL(4,3).C_2$ |
| $\card{W}$ | 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 |