Subgroup ($H$) information
| Description: | $C_3^2:C_{16}$ |
| Order: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Generators: |
$a, c^{4}, b^{2}c^{8}, a^{2}, a^{4}, c^{6}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $D_4.F_9$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^2\times F_9).C_2^3$ |
| $\operatorname{Aut}(H)$ | $F_9:C_4$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| $\operatorname{res}(S)$ | $F_9:C_4$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2\times F_9$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_2$ | |
| Normalizer: | $C_3^2:\OD_{32}$ | |
| Normal closure: | $C_3^2:\OD_{32}$ | |
| Core: | $C_3^2:C_8$ | |
| Minimal over-subgroups: | $C_3^2:\OD_{32}$ | |
| Maximal under-subgroups: | $C_3^2:C_8$ | $C_{16}$ |
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_6^2:C_8$ |