Subgroup ($H$) information
| Description: | $D_8$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(930\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 31 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
27 & 7 \\
2 & 4
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
28 & 22 \\
8 & 3
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
11 & 28 \\
8 & 12
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
21 & 18 \\
14 & 15
\end{array}\right) \right]$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence metabelian).
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 \(29760\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 31 \) |
| $\operatorname{Aut}(H)$ | $D_8:C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| $W$ | $D_8$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
| Centralizer: | $C_2$ | |
| Normalizer: | $D_{16}$ | |
| Normal closure: | $\PSL(2,31)$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $D_{16}$ | |
| Maximal under-subgroups: | $C_8$ | $D_4$ |
| Autjugate subgroups: | 14880.a.930.b1.a2 |
Other information
| Number of subgroups in this conjugacy class | $465$ |
| Möbius function | $0$ |
| Projective image | $\PSL(2,31)$ |