Subgroup ($H$) information
| Description: | $C_3^2:S_3^2$ |
| Order: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(10,12,15)(11,16,14)(13,17,18), (3,6)(4,5), (2,5,4), (10,15,12)(13,17,18), (3,6)(7,9)(10,17,15,18,12,13)(14,16), (7,9,8)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3^4:(C_3\times D_{12})$ |
| Order: | \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5.C_6.C_2^5$, of order \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_3^2:\GL(2,3)\times D_6$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| $W$ | $C_3:S_3^2$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $18$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3^4:(C_3\times D_{12})$ |