Subgroup ($H$) information
| Description: | $C_3^4:\SL(2,3)$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Index: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(1,20)(2,21,3,19)(4,24,5,23)(6,22)(7,27,8,25)(9,26)(10,28)(11,30,12,29) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and solvable.
Ambient group ($G$) information
| Description: | $C_3^6.(S_3\times \GL(2,3))$ |
| Order: | \(209952\)\(\medspace = 2^{5} \cdot 3^{8} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $5$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.(S_3\times \GL(2,3))$, of order \(209952\)\(\medspace = 2^{5} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $C_3^4:C_3:\GL(2,3)$, of order \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| $W$ | $C_3^4:(Q_8^2.C_2^3)$, of order \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | not computed | ||
| Normalizer: | $C_3^5:\GL(2,3)$ | ||
| Normal closure: | $C_3^6.Q_8.C_3^2$ | ||
| Core: | $C_3^4$ | ||
| Minimal over-subgroups: | $C_3^6.\SL(2,3)$ | $C_3^4.Q_8.C_3^2$ | $C_3^4.Q_8.S_3$ |
Other information
| Number of subgroups in this autjugacy class | $18$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^6.(S_3\times \GL(2,3))$ |