Subgroup ($H$) information
| Description: | $Q_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\left(\begin{array}{rrr}
2 & 0 & 0 \\
0 & 4 & 0 \\
3 & 4 & 3
\end{array}\right), \left(\begin{array}{rrr}
4 & 0 & 0 \\
0 & 1 & 0 \\
0 & 3 & 4
\end{array}\right), \left(\begin{array}{rrr}
1 & 2 & 2 \\
0 & 4 & 0 \\
4 & 0 & 4
\end{array}\right)$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), and rational.
Ambient group ($G$) information
| Description: | $C_2\times C_5^2:Q_8$ |
| Order: | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{10}^2:\Unitary(2,3)$, of order \(9600\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\operatorname{res}(S)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $25$ |
| Möbius function | $1$ |
| Projective image | $C_2\times C_5^2:Q_8$ |