Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(56730\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 31 \cdot 61 \) |
| Exponent: | \(2\) |
| Generators: |
$\left[ \left(\begin{array}{rr}
12 & 45 \\
51 & 49
\end{array}\right) \right]$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
Ambient group ($G$) information
| Description: | $\PSL(2,61)$ |
| Order: | \(113460\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 31 \cdot 61 \) |
| Exponent: | \(56730\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 31 \cdot 61 \) |
| Derived length: | $0$ |
The ambient group is nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.
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 \(226920\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 31 \cdot 61 \) |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $D_{30}$ | ||||||||
| Normalizer: | $D_{30}$ | ||||||||
| Normal closure: | $\PSL(2,61)$ | ||||||||
| Core: | $C_1$ | ||||||||
| Minimal over-subgroups: | $D_{61}$ | $D_{31}$ | $C_{10}$ | $D_5$ | $D_5$ | $C_6$ | $S_3$ | $S_3$ | $C_2^2$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of subgroups in this conjugacy class | $1891$ |
| Möbius function | $-90$ |
| Projective image | $\PSL(2,61)$ |