Subgroup ($H$) information
| Description: | $C_4:Q_8$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(34\)\(\medspace = 2 \cdot 17 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a, b^{6}c^{51}, c^{51}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.
Ambient group ($G$) information
| Description: | $C_{68}.D_8$ |
| Order: | \(1088\)\(\medspace = 2^{6} \cdot 17 \) |
| Exponent: | \(136\)\(\medspace = 2^{3} \cdot 17 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{17}:((C_2^4\times C_8).C_2^5)$ |
| $\operatorname{Aut}(H)$ | $D_4^2:D_4$, of order \(512\)\(\medspace = 2^{9} \) |
| $\operatorname{res}(S)$ | $D_4^2$, of order \(64\)\(\medspace = 2^{6} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(64\)\(\medspace = 2^{6} \) |
| $W$ | $C_2^2:C_4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $17$ |
| Möbius function | $1$ |
| Projective image | $D_{34}:C_4$ |