Subgroup ($H$) information
| Description: | $C_{12}.D_6$ |
| Order: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$ac^{3}, d^{40}, c^{2}, b, d^{15}, d^{30}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $D_{20}:S_3^2$ |
| Order: | \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_6^2.C_2^4\times F_5$ |
| $\operatorname{Aut}(H)$ | $C_4:D_6^2$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $\operatorname{res}(S)$ | $C_4:D_6^2$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $D_6^2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $10$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $1$ |
| Projective image | $D_{10}\times S_3^2$ |