Subgroup ($H$) information
| Description: | not computed |
| Order: | \(59049\)\(\medspace = 3^{10} \) |
| Index: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Exponent: | not computed |
| Generators: |
$\langle(4,6,5)(10,12,11)(13,15,14)(19,21,20)(25,26,27)(28,29,30)(31,32,33)(34,35,36) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is normal, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian. 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: | $C_3^8.C_3^4:\SL(2,3)$ |
| Order: | \(12754584\)\(\medspace = 2^{3} \cdot 3^{13} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $\PU(3,2)$ |
| Order: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $4$ |
The quotient is nonabelian and solvable.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(459165024\)\(\medspace = 2^{5} \cdot 3^{15} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_3^6:\PU(3,2)$, of order \(157464\)\(\medspace = 2^{3} \cdot 3^{9} \) |
Related subgroups
| Centralizer: | $C_3^4$ |
| Normalizer: | $C_3^8.C_3^4:\SL(2,3)$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_3^8.C_3^4:\SL(2,3)$ |