Subgroup ($H$) information
| Description: | $\He_3:A_4$ |
| Order: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$a^{2}c^{2}, d^{3}, c^{3}, bc^{4}, d^{4}e, d^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $(C_3\times \He_3):S_4$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_3:A_4.C_6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_3\times C_6^2:S_3^2$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| $\operatorname{res}(S)$ | $C_3^3:A_4$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3\) |
| $W$ | $C_3^3:A_4$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $(C_3\times \He_3):S_4$ |