Subgroup ($H$) information
| Description: | $C_{20}$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Index: | \(11346\)\(\medspace = 2 \cdot 3 \cdot 31 \cdot 61 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
4 & 21 \\
0 & 19
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
22 & 25 \\
0 & 5
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
20 & 36 \\
0 & 37
\end{array}\right) \right]$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $\PGL(2,61)$ |
| Order: | \(226920\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 31 \cdot 61 \) |
| Exponent: | \(113460\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 31 \cdot 61 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, almost simple, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PGL(2,61)$, of order \(226920\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 31 \cdot 61 \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{60}$ | ||
| Normalizer: | $D_{60}$ | ||
| Normal closure: | $\PGL(2,61)$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_{61}:C_{20}$ | $C_{60}$ | $D_{20}$ |
| Maximal under-subgroups: | $C_{10}$ | $C_4$ |
Other information
| Number of subgroups in this conjugacy class | $1891$ |
| Möbius function | $0$ |
| Projective image | $\PGL(2,61)$ |