Subgroup ($H$) information
| Description: | $C_3^4:Q_8$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(2,9,5)(4,8,10)(7,12,11)(13,15,17)(14,18,19)(16,21,20), (13,16,19)(14,15,21) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $C_3^6.\SL(2,3)$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_6.C_6^2.D_6$, of order \(1889568\)\(\medspace = 2^{5} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $C_3^4:\GL(2,3):S_4$, of order \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
| $W$ | $C_3^4:\SL(2,3)$, of order \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
Related subgroups
| Centralizer: | $C_1$ | ||
| Normalizer: | $C_3^4:\SL(2,3)$ | ||
| Normal closure: | $C_3^6.Q_8$ | ||
| Core: | $C_3^4$ | ||
| Minimal over-subgroups: | $C_3^6.Q_8$ | $C_3^4:\SL(2,3)$ | |
| Maximal under-subgroups: | $C_3^4:C_4$ | $\PSU(3,2)$ | $\PSU(3,2)$ |
Other information
| Number of subgroups in this autjugacy class | $9$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_3^6.\SL(2,3)$ |