Subgroup ($H$) information
| Description: | $C_{23}:C_{11}$ |
| Order: | \(253\)\(\medspace = 11 \cdot 23 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(253\)\(\medspace = 11 \cdot 23 \) |
| Generators: |
$a^{2}, c^{2}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a Hall subgroup, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 11$.
Ambient group ($G$) information
| Description: | $C_2^2\times F_{23}$ |
| Order: | \(2024\)\(\medspace = 2^{3} \cdot 11 \cdot 23 \) |
| Exponent: | \(506\)\(\medspace = 2 \cdot 11 \cdot 23 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4\times F_{23}$, of order \(12144\)\(\medspace = 2^{4} \cdot 3 \cdot 11 \cdot 23 \) |
| $\operatorname{Aut}(H)$ | $F_{23}$, of order \(506\)\(\medspace = 2 \cdot 11 \cdot 23 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $F_{23}$, of order \(506\)\(\medspace = 2 \cdot 11 \cdot 23 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $F_{23}$, of order \(506\)\(\medspace = 2 \cdot 11 \cdot 23 \) |
Related subgroups
Other information
| Möbius function | $-8$ |
| Projective image | $C_2^2\times F_{23}$ |