Subgroup ($H$) information
| Description: | $(C_3\times C_{18}).C_6^2$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$ad^{3}, f^{3}, g^{7}, e^{3}f^{3}, d^{2}, g^{3}, e^{2}g^{6}, cf^{4}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $S_4\times C_3^4.S_3^2$ |
| Order: | \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4\times C_3^4.S_3^2$, of order \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $C_3^4.D_6^2$, of order \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| $W$ | $C_{1011}:C_{56}$, of order \(56616\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 337 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $S_4\times C_3^4.S_3^2$ |