Subgroup ($H$) information
| Description: | $C_3^3:S_3$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$b^{3}c^{3}, e^{6}, c^{2}d^{2}e^{6}, e^{14}, d^{2}$
|
| Derived length: | $3$ |
The subgroup is normal, a direct factor, nonabelian, and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $C_6^3:S_3^2$ |
| Order: | \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2\times S_4$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $3$ |
The quotient is nonabelian, monomial (hence solvable), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^2.C_3^4.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_3^3.S_3^2$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| $W$ | $C_3^3:S_3$, of order \(162\)\(\medspace = 2 \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $24$ |
| Projective image | $C_6^3:S_3^2$ |