Subgroup ($H$) information
| Description: | $D_4\times C_3^2$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(12,13), (10,13)(11,12), (3,7,5), (1,4,2)(6,8,9), (10,11)(12,13)\rangle$
|
| 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: | $C_3^3:S_3\times S_4$ |
| Order: | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^2.C_3^4.C_2^3$ |
| $\operatorname{Aut}(H)$ | $D_4\times \GL(2,3)$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2^4$, of order \(16\)\(\medspace = 2^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| $W$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $9$ |
| Möbius function | $0$ |
| Projective image | $C_3^3:S_3\times S_4$ |