Subgroup ($H$) information
| Description: | $D_5$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Index: | \(92\)\(\medspace = 2^{2} \cdot 23 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$ab^{2}, c^{46}$
|
| Derived length: | $2$ |
The subgroup is normal, a direct factor, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{10}.D_{46}$ |
| Order: | \(920\)\(\medspace = 2^{3} \cdot 5 \cdot 23 \) |
| Exponent: | \(460\)\(\medspace = 2^{2} \cdot 5 \cdot 23 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{23}:C_4$ |
| Order: | \(92\)\(\medspace = 2^{2} \cdot 23 \) |
| Exponent: | \(92\)\(\medspace = 2^{2} \cdot 23 \) |
| Automorphism Group: | $C_2\times F_{23}$, of order \(1012\)\(\medspace = 2^{2} \cdot 11 \cdot 23 \) |
| Outer Automorphisms: | $C_{22}$, of order \(22\)\(\medspace = 2 \cdot 11 \) |
| Derived length: | $2$ |
The quotient 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{115}.C_{44}.C_2^3$, of order \(40480\)\(\medspace = 2^{5} \cdot 5 \cdot 11 \cdot 23 \) |
| $\operatorname{Aut}(H)$ | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\operatorname{res}(S)$ | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1012\)\(\medspace = 2^{2} \cdot 11 \cdot 23 \) |
| $W$ | $D_5$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_{10}.D_{46}$ |