Subgroup ($H$) information
| Description: | $C_3^2$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Index: | \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(1,6,34)(2,5,35)(3,4,36)(7,20,31)(8,21,32)(9,19,33)(10,14,17)(11,15,16) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_3^6:(C_3\times \SL(2,3))$ |
| Order: | \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_2.C_6^2.C_6$, of order \(314928\)\(\medspace = 2^{4} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $S_3\times C_3^2$ | ||
| Normalizer: | $S_3\times C_3^2$ | ||
| Normal closure: | $C_3^6:(C_3\times \SL(2,3))$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_3^3$ | $C_3\times C_6$ | |
| Maximal under-subgroups: | $C_3$ | $C_3$ | $C_3$ |
Other information
| Number of subgroups in this autjugacy class | $972$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3^6:(C_3\times \SL(2,3))$ |