Subgroup ($H$) information
| Description: | $D_{35}$ |
| Order: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Index: | \(2556\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 71 \) |
| Exponent: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
14 & 62 \\
14 & 57
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
37 & 63 \\
30 & 30
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
51 & 65 \\
58 & 28
\end{array}\right) \right]$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $\PSL(2,71)$ |
| Order: | \(178920\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 71 \) |
| Exponent: | \(89460\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 71 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PGL(2,71)$, of order \(357840\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 71 \) |
| $\operatorname{Aut}(H)$ | $F_5\times F_7$, of order \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| $W$ | $D_{35}$, of order \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_1$ | ||
| Normalizer: | $D_{35}$ | ||
| Normal closure: | $\PSL(2,71)$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $\PSL(2,71)$ | ||
| Maximal under-subgroups: | $C_{35}$ | $D_7$ | $D_5$ |
Other information
| Number of subgroups in this conjugacy class | $2556$ |
| Möbius function | $-1$ |
| Projective image | $\PSL(2,71)$ |