Subgroup ($H$) information
| Description: | $C_3^5:D_{12}$ |
| Order: | \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,2,3,5,9,14)(4,7,12,16,17,18)(6,11,15,13,8,10)(19,20,21,23)(22,24), (20,23,22) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $\He_3^2.\SOPlus(4,2)$ |
| Order: | \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5.C_3^4.C_2^5.C_2$, of order \(1259712\)\(\medspace = 2^{6} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_3^5.C_6.C_2^5$, of order \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \) |
| $W$ | $C_3^5:D_{12}$, of order \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^5:D_{12}$ |
| Normal closure: | $\He_3^2.\SOPlus(4,2)$ |
| Core: | $C_3^5:C_6$ |
Other information
| Number of subgroups in this autjugacy class | $9$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $\He_3^2.\SOPlus(4,2)$ |