Subgroup ($H$) information
| Description: | $C_3^3:S_3^3$ |
| Order: | \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(1,5,8)(4,6,7), (1,2,8,9,5,3)(4,7)(10,12,11,14,13,15), (1,2,6)(3,4,8,9,7,5) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is normal, a semidirect factor, nonabelian, and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $C_3^3:S_3^4$ |
| Order: | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and rational.
Quotient group ($Q$) structure
| Description: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times S_4^2$, of order \(1259712\)\(\medspace = 2^{6} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_3^3.\He_3.C_6.C_2^4$ |
| $W$ | $C_3^4:D_6^2$, of order \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | not computed |
| Projective image | $C_3^3:S_3^4$ |