Subgroup ($H$) information
| Description: | $F_{61}$ |
| Order: | \(3660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 61 \) |
| Index: | \(62\)\(\medspace = 2 \cdot 31 \) |
| Exponent: | \(3660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 61 \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
33 & 48 \\
39 & 58
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
6 & 48 \\
50 & 40
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
30 & 40 \\
0 & 15
\end{array}\right) \right]$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, 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 \(226920\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 31 \cdot 61 \) |
| $\operatorname{Aut}(H)$ | $F_{61}$, of order \(3660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 61 \) |
| $W$ | $F_{61}$, of order \(3660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 61 \) |
Related subgroups
| Centralizer: | $C_1$ | |||
| Normalizer: | $F_{61}$ | |||
| Normal closure: | $\PGL(2,61)$ | |||
| Core: | $C_1$ | |||
| Minimal over-subgroups: | $\PGL(2,61)$ | |||
| Maximal under-subgroups: | $C_{61}:C_{30}$ | $C_{61}:C_{20}$ | $C_{61}:C_{12}$ | $C_{60}$ |
Other information
| Number of subgroups in this conjugacy class | $62$ |
| Möbius function | $-1$ |
| Projective image | $\PGL(2,61)$ |