Subgroup ($H$) information
| Description: | $D_{14}$ |
| Order: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Index: | \(351\)\(\medspace = 3^{3} \cdot 13 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
13 & 9 \\
17 & 0
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
23 & 23 \\
21 & 10
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
14 & 19 \\
1 & 17
\end{array}\right) \right]$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $\PSL(2,27)$ |
| Order: | \(9828\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7 \cdot 13 \) |
| Exponent: | \(546\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PSL(2,27).C_6$, of order \(58968\)\(\medspace = 2^{3} \cdot 3^{4} \cdot 7 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_2\times F_7$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| $W$ | $D_7$, of order \(14\)\(\medspace = 2 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_2$ | |||
| Normalizer: | $D_{14}$ | |||
| Normal closure: | $\PSL(2,27)$ | |||
| Core: | $C_1$ | |||
| Minimal over-subgroups: | $\PSL(2,27)$ | |||
| Maximal under-subgroups: | $C_{14}$ | $D_7$ | $D_7$ | $C_2^2$ |
Other information
| Number of subgroups in this conjugacy class | $351$ |
| Möbius function | $-1$ |
| Projective image | $\PSL(2,27)$ |