Subgroup ($H$) information
| Description: | $C_6^3.(S_3\times D_4)$ |
| Order: | \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$ab^{3}c^{4}d^{10}g^{4}, c^{2}g^{4}, d^{3}e^{2}g^{2}, d^{6}e^{6}, e^{4}g^{4}, b^{3}c^{3}, e^{3}f^{2}g, c^{4}d^{4}fg^{2}, d^{6}, g^{2}, c^{3}g^{3}$
|
| Derived length: | $3$ |
The subgroup is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^4.C_4^2:D_6^2$ |
| Order: | \(186624\)\(\medspace = 2^{8} \cdot 3^{6} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian, solvable, and rational. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_{12}^2.C_6.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_3^4.C_2^6.C_2^6$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $9$ |
| Möbius function | not computed |
| Projective image | not computed |