Subgroup ($H$) information
| Description: | $D_{16}:C_2^2$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$ab^{7}c^{2}d^{3}, c^{5}d^{6}, b^{6}c^{2}d^{6}, c^{8}d^{4}$
|
| Nilpotency class: | $4$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_3\times Q_{16}^2:C_2^2$ |
| Order: | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $7$ |
| Derived length: | $3$ |
The ambient group is nonabelian and elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_4^2.C_2^4.C_2^4$ |
| $\operatorname{Aut}(H)$ | $D_8:C_4\times S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_4^3.C_2^3$, of order \(512\)\(\medspace = 2^{9} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $D_4\times D_8$, of order \(128\)\(\medspace = 2^{7} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $C_3 \times ((C_4:D_8) . C_2^3)$ |