Subgroup ($H$) information
| Description: | $D_4\times C_3^2$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(8,9), (7,10)(8,9), (7,8)(9,10), (2,5,6), (1,3,4)(2,6,5)\rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, nilpotent (hence solvable, supersolvable, and monomial), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $S_4\times S_3^2$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_6^2:D_6$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $D_4\times \GL(2,3)$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| $W$ | $C_2^4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $-2$ |
| Projective image | $S_4\times S_3^2$ |