Subgroup ($H$) information
| Description: | $C_3^4$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(7,8,9)(10,12,11)(19,20,21)(22,23,24)(31,33,32)(34,35,36), (7,9,8)(10,12,11) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Frattini subgroup (hence characteristic and normal), the socle, 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: | $C_3^4.S_3^4:S_3$ |
| Order: | \(629856\)\(\medspace = 2^{5} \cdot 3^{9} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $S_3^4:S_3$ |
| Order: | \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $S_3^5.D_4$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.(S_3^5.D_4)$, of order \(45349632\)\(\medspace = 2^{8} \cdot 3^{11} \) |
| $\operatorname{Aut}(H)$ | $C_2.\PSL(4,3).C_2$, of order \(24261120\)\(\medspace = 2^{9} \cdot 3^{6} \cdot 5 \cdot 13 \) |
| $W$ | $C_2^2\wr C_2$, of order \(32\)\(\medspace = 2^{5} \) |
Related subgroups
| Centralizer: | $C_3^8.C_3$ |
| Normalizer: | $C_3^4.S_3^4:S_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4.S_3^4:S_3$ |