Subgroup ($H$) information
| Description: | $C_3^4$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(104976\)\(\medspace = 2^{4} \cdot 3^{8} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(7,9,8)(10,11,12)(19,21,20)(22,23,24)(31,32,33)(34,35,36), (1,3,2)(4,5,6) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the socle (hence characteristic and 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: | $C_3^8.C_3^3:\GL(2,3)$ |
| Order: | \(8503056\)\(\medspace = 2^{4} \cdot 3^{12} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $6$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_3^6:(C_3\times \GL(2,3))$ |
| Order: | \(104976\)\(\medspace = 2^{4} \cdot 3^{8} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Automorphism Group: | $C_3^6.(S_3\times \GL(2,3))$, of order \(209952\)\(\medspace = 2^{5} \cdot 3^{8} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $5$ |
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3\times C_3^5.C_3^4.Q_8.S_3^2$, of order \(17006112\)\(\medspace = 2^{5} \cdot 3^{12} \) |
| $\operatorname{Aut}(H)$ | $C_2.\PSL(4,3).C_2$, of order \(24261120\)\(\medspace = 2^{9} \cdot 3^{6} \cdot 5 \cdot 13 \) |
| $\card{W}$ | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_3^8.C_3^3.C_2$ |
| Normalizer: | $C_3^8.C_3^3:\GL(2,3)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |