Subgroup ($H$) information
| Description: | $D_5$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Index: | \(482664\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13^{2} \cdot 17 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
144 & 33 \\
60 & 60
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
166 & 126 \\
127 & 122
\end{array}\right) \right]$
|
| 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: | $\PGL(2,169)$ |
| Order: | \(4826640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17 \) |
| Exponent: | \(185640\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, almost simple, and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PSL(2,169).C_2^2$, of order \(9653280\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17 \) |
| $\operatorname{Aut}(H)$ | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $W$ | $D_5$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $D_{10}$ | ||
| Normal closure: | $\PSL(2,169)$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $D_{85}$ | $A_5$ | $D_{10}$ |
| Maximal under-subgroups: | $C_5$ | $C_2$ |
Other information
| Number of subgroups in this conjugacy class | $241332$ |
| Möbius function | $0$ |
| Projective image | $\PGL(2,169)$ |