Subgroup ($H$) information
| Description: | $C_3^3:S_3^2$ |
| Order: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,7,4)(2,8,5)(3,9,6)(10,11,18)(12,13,14)(15,16,17)(19,23,24)(20,21,25) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.
Ambient group ($G$) information
| Description: | $C_3^6.(S_3\times Q_8)$ |
| Order: | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Quotient set structure
Since this subgroup has trivial core, the ambient group $G$ acts faithfully and transitively on the set of cosets of $H$. The resulting permutation representation is isomorphic to 36T13949.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.Q_8.C_3^3.C_2^2$, of order \(629856\)\(\medspace = 2^{5} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_3^3:C_2.\SL(3,3)\times \AGL(2,3)$, of order \(131010048\)\(\medspace = 2^{9} \cdot 3^{9} \cdot 13 \) |
| $W$ | $C_3^3:S_3^2$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $108$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $C_3^6.(S_3\times Q_8)$ |