Subgroup ($H$) information
| Description: | $C_3^4:C_3$ |
| Order: | \(243\)\(\medspace = 3^{5} \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(3\) |
| Generators: |
$cd^{4}fgh, d^{2}h, ef^{2}g^{2}$
|
| 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: | $\He_3^2.(C_2\times D_4)$ |
| Order: | \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^2.C_3^4.D_4.C_2^3$, of order \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_3^5.(S_3\times C_3^2:\GL(2,3))$, of order \(629856\)\(\medspace = 2^{5} \cdot 3^{9} \) |
| $W$ | $C_3:S_3^3$, of order \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $8$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $\He_3^2.(C_2\times D_4)$ |