Subgroup ($H$) information
| Description: | $C_{467}$ |
| Order: | \(467\) |
| Index: | \(233\) |
| Exponent: | \(467\) |
| Generators: |
$b$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), the Fitting subgroup (hence nilpotent, solvable, supersolvable, and monomial), the socle, maximal, a semidirect factor, cyclic (hence abelian, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $467$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{467}:C_{233}$ |
| Order: | \(108811\)\(\medspace = 233 \cdot 467 \) |
| Exponent: | \(108811\)\(\medspace = 233 \cdot 467 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 233$.
Quotient group ($Q$) structure
| Description: | $C_{233}$ |
| Order: | \(233\) |
| Exponent: | \(233\) |
| Automorphism Group: | $C_{232}$, of order \(232\)\(\medspace = 2^{3} \cdot 29 \) |
| Outer Automorphisms: | $C_{232}$, of order \(232\)\(\medspace = 2^{3} \cdot 29 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{467}$, of order \(217622\)\(\medspace = 2 \cdot 233 \cdot 467 \) |
| $\operatorname{Aut}(H)$ | $C_{466}$, of order \(466\)\(\medspace = 2 \cdot 233 \) |
| $W$ | $C_{233}$, of order \(233\) |
Related subgroups
| Centralizer: | $C_{467}$ |
| Normalizer: | $C_{467}:C_{233}$ |
| Complements: | $C_{233}$ |
| Minimal over-subgroups: | $C_{467}:C_{233}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_{467}:C_{233}$ |