Subgroup ($H$) information
| Description: | $C_{4632}$ |
| Order: | \(4632\)\(\medspace = 2^{3} \cdot 3 \cdot 193 \) |
| Index: | \(3\) |
| Exponent: | \(4632\)\(\medspace = 2^{3} \cdot 3 \cdot 193 \) |
| Generators: |
$a^{6}, a^{12}, b^{3}, b^{193}, a^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), maximal, a semidirect factor, and cyclic (hence abelian, elementary ($p = 2,3,193$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{4632}:C_3$ |
| Order: | \(13896\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 193 \) |
| Exponent: | \(4632\)\(\medspace = 2^{3} \cdot 3 \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 3$, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(A_4\times C_2^4):D_6$, of order \(889344\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 193 \) |
| $\operatorname{Aut}(H)$ | $C_2^3\times C_{192}$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $W$ | $C_3$, of order \(3\) |
Related subgroups
| Centralizer: | $C_{4632}$ | ||
| Normalizer: | $C_{4632}:C_3$ | ||
| Complements: | $C_3$ $C_3$ $C_3$ | ||
| Minimal over-subgroups: | $C_{4632}:C_3$ | ||
| Maximal under-subgroups: | $C_{2316}$ | $C_{1544}$ | $C_{24}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_{193}:C_3$ |