Subgroup ($H$) information
| Description: | $C_3^7.D_6$ |
| Order: | \(26244\)\(\medspace = 2^{2} \cdot 3^{8} \) |
| Index: | \(3\) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(19,20,22)(21,25,23)(24,26,27), (1,2,5)(4,9,7,13,11,17)(8,14,16,18,12,15) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $C_3^8.D_6$ |
| Order: | \(78732\)\(\medspace = 2^{2} \cdot 3^{9} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5\times C_{11}^2:C_{40}$, of order \(8503056\)\(\medspace = 2^{4} \cdot 3^{12} \) |
| $\operatorname{Aut}(H)$ | $\SOPlus(4,4)$, of order \(472392\)\(\medspace = 2^{3} \cdot 3^{10} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^7.D_6$ |
| Normal closure: | $C_3^8.D_6$ |
| Core: | $C_3^7.C_6$ |
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | not computed |