Subgroup ($H$) information
| Description: | $C_3^3:C_6$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Index: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^3:S_3^2.S_4$ |
| Order: | \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $6$ |
The ambient group 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:\SOPlus(4,2).S_4$, of order \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_3^3.S_3^2$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| $W$ | $C_3^3:D_6$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_1$ | ||
| Normalizer: | $C_3^3:D_6$ | ||
| Normal closure: | $C_3^3:C_3^2:\GL(2,3)$ | ||
| Core: | $C_3$ | ||
| Minimal over-subgroups: | $C_3^4:C_6$ | $C_3^3:D_6$ | |
| Maximal under-subgroups: | $C_3\wr C_3$ | $C_3^2:C_6$ | $C_3^2:S_3$ |
Other information
| Number of subgroups in this conjugacy class | $72$ |
| Möbius function | $0$ |
| Projective image | $C_3^3:S_3^2.S_4$ |