Subgroup ($H$) information
| Description: | $\He_3$ |
| Order: | \(27\)\(\medspace = 3^{3} \) |
| Index: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Exponent: | \(3\) |
| Generators: |
$a^{2}b^{2}d^{2}e^{2}g^{3}, c^{2}de^{2}fg^{8}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^6:(C_6\times S_3)$ |
| Order: | \(26244\)\(\medspace = 2^{2} \cdot 3^{8} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.S_3^2$, of order \(472392\)\(\medspace = 2^{3} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $W$ | $C_3^2$, of order \(9\)\(\medspace = 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $108$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_3^6:(C_6\times S_3)$ |