Subgroup ($H$) information
| Description: | $Q_{16}$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\left(\begin{array}{rr}
14 & 45 \\
10 & 31
\end{array}\right), \left(\begin{array}{rr}
51 & 25 \\
70 & 51
\end{array}\right)$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $C_2\times C_{20}.S_4$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times A_4).C_2^4.C_2^4$ |
| $\operatorname{Aut}(H)$ | $D_8:C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{W}$ | \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $15$ |
| Möbius function | not computed |
| Projective image | not computed |