Subgroup ($H$) information
| Description: | $C_3^4$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(4,5,6)(10,12,11)(16,17,18)(22,24,23)(28,29,30)(34,36,35), (10,11,12)(13,14,15) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_3^5:\PSU(3,2)$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_3^4.(Q_8.A_4).D_6$, of order \(2519424\)\(\medspace = 2^{7} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_2.\PSL(4,3).C_2$, of order \(24261120\)\(\medspace = 2^{9} \cdot 3^{6} \cdot 5 \cdot 13 \) |
| $W$ | $C_3$, of order \(3\) |
Related subgroups
| Centralizer: | $C_3^5$ | |||||
| Normalizer: | $C_3^5:C_3$ | |||||
| Normal closure: | $C_3^4.C_3^3$ | |||||
| Core: | $C_1$ | |||||
| Minimal over-subgroups: | $C_3^5$ | $C_3^4:C_3$ | ||||
| Maximal under-subgroups: | $C_3^3$ | $C_3^3$ | $C_3^3$ | $C_3^3$ | $C_3^3$ | $C_3^3$ |
Other information
| Number of subgroups in this autjugacy class | $96$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $C_3^5:\PSU(3,2)$ |