Subgroup ($H$) information
| Description: | $D_5$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Index: | \(161\)\(\medspace = 7 \cdot 23 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$a, b^{161}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Hall subgroup, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{23}\times D_{35}$ |
| Order: | \(1610\)\(\medspace = 2 \cdot 5 \cdot 7 \cdot 23 \) |
| Exponent: | \(1610\)\(\medspace = 2 \cdot 5 \cdot 7 \cdot 23 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{22}\times F_5\times F_7$ |
| $\operatorname{Aut}(H)$ | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\operatorname{res}(S)$ | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| $W$ | $D_5$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
Related subgroups
| Centralizer: | $C_{23}$ | |
| Normalizer: | $D_5\times C_{23}$ | |
| Normal closure: | $D_{35}$ | |
| Core: | $C_5$ | |
| Minimal over-subgroups: | $D_5\times C_{23}$ | $D_{35}$ |
| Maximal under-subgroups: | $C_5$ | $C_2$ |
Other information
| Number of subgroups in this conjugacy class | $7$ |
| Möbius function | $1$ |
| Projective image | $C_{23}\times D_{35}$ |