Subgroup ($H$) information
| Description: | $D_4\times S_3^2$ |
| Order: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a, c^{2}d^{8}, d^{6}, c^{3}, d^{4}, d^{9}e^{3}, b^{2}e^{3}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and rational.
Ambient group ($G$) information
| Description: | $C_6^2.(D_4\times D_6)$ |
| Order: | \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_2^6.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_6^2:D_4^2$, of order \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| $\operatorname{res}(S)$ | $C_2^2\times D_6^2$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2\times D_6^2$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $12$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $D_6^2:D_6$ |