Subgroup ($H$) information
| Description: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Index: | \(2480\)\(\medspace = 2^{4} \cdot 5 \cdot 31 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
12 & 24 \\
3 & 19
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
2 & 20 \\
26 & 28
\end{array}\right) \right]$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, and rational.
Ambient group ($G$) information
| Description: | $\PSL(2,31)$ |
| Order: | \(14880\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 31 \) |
| Exponent: | \(7440\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 31 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PGL(2,31)$, of order \(29760\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 31 \) |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_1$ | ||||
| Normalizer: | $S_3$ | ||||
| Normal closure: | $\PSL(2,31)$ | ||||
| Core: | $C_1$ | ||||
| Minimal over-subgroups: | $A_5$ | $A_5$ | $D_{15}$ | $S_4$ | $S_4$ |
| Maximal under-subgroups: | $C_3$ | $C_2$ |
Other information
| Number of subgroups in this conjugacy class | $2480$ |
| Möbius function | $4$ |
| Projective image | $\PSL(2,31)$ |