Subgroup ($H$) information
| Description: | $C_5\times Q_8$ |
| Order: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Index: | \(44730\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 71 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rr}
25 & 0 \\
0 & 25
\end{array}\right), \left(\begin{array}{rr}
70 & 0 \\
0 & 70
\end{array}\right), \left(\begin{array}{rr}
24 & 50 \\
4 & 47
\end{array}\right), \left(\begin{array}{rr}
1 & 46 \\
55 & 70
\end{array}\right)$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $C_5\times \SL(2,71)$ |
| Order: | \(1789200\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 71 \) |
| Exponent: | \(178920\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 71 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times \PSL(2,71).C_2$ |
| $\operatorname{Aut}(H)$ | $C_4\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $7455$ |
| Möbius function | $0$ |
| Projective image | $\PSL(2,71)$ |