Subgroup ($H$) information
| Description: | $\He_3$ |
| Order: | \(27\)\(\medspace = 3^{3} \) |
| Index: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(10,11,12)(13,14,15)(16,17,18)(22,23,24)(28,30,29)(34,36,35), (1,28,10) \!\cdots\! \rangle$
|
| 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: | $C_3^6:(C_3\times \SL(2,3))$ |
| Order: | \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_2.C_6^2.C_6$, of order \(314928\)\(\medspace = 2^{4} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $W$ | $\He_3$, of order \(27\)\(\medspace = 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $72$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3^6:(C_3\times \SL(2,3))$ |