Subgroup ($H$) information
| Description: | $D_{231}$ |
| Order: | \(462\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 11 \) |
| Index: | \(6841300\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 37 \cdot 43^{2} \) |
| Exponent: | \(462\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 11 \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
555 & 1313 \\
1768 & 1124
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
1372 & 846 \\
1301 & 1828
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
1747 & 1773 \\
1044 & 823
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
1747 & 1777 \\
384 & 549
\end{array}\right) \right]$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $\PSL(2,1849)$ |
| Order: | \(3160680600\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 37 \cdot 43^{2} \) |
| Exponent: | \(36752100\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 37 \cdot 43 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(12642722400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 37 \cdot 43^{2} \) |
| $\operatorname{Aut}(H)$ | $C_{231}.C_{30}.C_2^2$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $3420650$ |
| Möbius function | not computed |
| Projective image | not computed |