Subgroup ($H$) information
| Description: | $C_5:Q_{32}$ |
| Order: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Index: | \(6162\)\(\medspace = 2 \cdot 3 \cdot 13 \cdot 79 \) |
| Exponent: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rr}
29 & 28 \\
32 & 50
\end{array}\right), \left(\begin{array}{rr}
11 & 50 \\
15 & 18
\end{array}\right), \left(\begin{array}{rr}
77 & 6 \\
65 & 2
\end{array}\right), \left(\begin{array}{rr}
39 & 39 \\
67 & 65
\end{array}\right), \left(\begin{array}{rr}
53 & 52 \\
63 & 35
\end{array}\right), \left(\begin{array}{rr}
78 & 0 \\
0 & 78
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $\SL(2,79):C_2$ |
| Order: | \(985920\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \) |
| Exponent: | \(492960\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 13 \cdot 79 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times \PSL(2,79).C_2$ |
| $\operatorname{Aut}(H)$ | $D_{80}:C_4^2$, of order \(2560\)\(\medspace = 2^{9} \cdot 5 \) |
| $W$ | $D_{80}$, of order \(160\)\(\medspace = 2^{5} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $C_{160}:C_2$ | ||
| Normal closure: | $\SL(2,79)$ | ||
| Core: | $C_2$ | ||
| Minimal over-subgroups: | $\SL(2,79)$ | $C_{160}:C_2$ | |
| Maximal under-subgroups: | $C_{80}$ | $C_5:Q_{16}$ | $Q_{32}$ |
Other information
| Number of subgroups in this conjugacy class | $3081$ |
| Möbius function | $1$ |
| Projective image | $\PGL(2,79)$ |