Subgroup ($H$) information
| Description: | $(C_3\times \He_3):C_4$ |
| Order: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(11,16,12)(14,17,18), (10,13,15)(11,12,16)(14,17,18), (10,17,16)(11,13,18) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $(C_3^3\times \He_3):C_4$ |
| Order: | \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_3:(C_3:S_3).C_6^2.C_{12}.C_2^4$, of order \(3359232\)\(\medspace = 2^{9} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_3^3:C_3^2.Q_8.D_6$, of order \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \) |
| $\card{\operatorname{res}(S)}$ | \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_3^2:S_3$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $9$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_3^5:C_4$ |