Subgroup ($H$) information
| Description: | $C_3\times A_4$ |
| Order: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Index: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$a^{2}d^{2}e, b^{6}, b^{9}c^{4}, c^{3}d^{2}e$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_9:S_3\wr C_3$ |
| Order: | \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_6^2.C_6.C_2$, of order \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $S_3\times S_4$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $W$ | $S_3\times A_4$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_3$ | ||
| Normalizer: | $D_9:A_4$ | ||
| Normal closure: | $C_3\wr A_4$ | ||
| Core: | $C_3$ | ||
| Minimal over-subgroups: | $C_3\wr A_4$ | $C_9:A_4$ | $S_3\times A_4$ |
| Maximal under-subgroups: | $C_2\times C_6$ | $A_4$ | $C_3^2$ |
Other information
| Number of subgroups in this conjugacy class | $27$ |
| Möbius function | $-3$ |
| Projective image | $C_9:S_3\wr C_3$ |