Subgroup ($H$) information
| Description: | $C_4.D_4$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$a, c^{3}$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_4.\SOPlus(4,2)$ |
| Order: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_6^2:C_2^3$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| $\operatorname{Aut}(H)$ | $C_2\wr D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\operatorname{res}(S)$ | $D_4^2$, of order \(64\)\(\medspace = 2^{6} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_2^2:C_4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $C_4.D_4$ | ||
| Normal closure: | $C_4.\SOPlus(4,2)$ | ||
| Core: | $C_4$ | ||
| Minimal over-subgroups: | $C_4.\SOPlus(4,2)$ | ||
| Maximal under-subgroups: | $C_2\times Q_8$ | $\OD_{16}$ | $\OD_{16}$ |
Other information
| Number of subgroups in this conjugacy class | $9$ |
| Möbius function | $-1$ |
| Projective image | $S_3^2:C_4$ |