Subgroup ($H$) information
| Description: | $C_{23}$ |
| Order: | \(23\) |
| Index: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Exponent: | \(23\) |
| Generators: |
$c^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $23$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
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: | $C_2^2\times C_{22}$ |
| Order: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Exponent: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Automorphism Group: | $C_{10}\times \GL(3,2)$, of order \(1680\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 7 \) |
| Outer Automorphisms: | $C_{10}\times \GL(3,2)$, of order \(1680\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 7 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
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 \(12144\)\(\medspace = 2^{4} \cdot 3 \cdot 11 \cdot 23 \) |
| $\operatorname{Aut}(H)$ | $C_{22}$, of order \(22\)\(\medspace = 2 \cdot 11 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{22}$, of order \(22\)\(\medspace = 2 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(552\)\(\medspace = 2^{3} \cdot 3 \cdot 23 \) |
| $W$ | $C_{22}$, of order \(22\)\(\medspace = 2 \cdot 11 \) |
Related subgroups
Other information
| Möbius function | $8$ |
| Projective image | $C_2^2\times F_{23}$ |