Subgroup ($H$) information
| Description: | $C_{199}:C_{99}$ |
| Order: | \(19701\)\(\medspace = 3^{2} \cdot 11 \cdot 199 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(19701\)\(\medspace = 3^{2} \cdot 11 \cdot 199 \) |
| Generators: |
$a^{110}, a^{18}, b^{2}, a^{132}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a Hall subgroup, and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).
Ambient group ($G$) information
| Description: | $C_{398}:C_{198}$ |
| Order: | \(78804\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \cdot 199 \) |
| Exponent: | \(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6\times F_{199}$, of order \(236412\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 11 \cdot 199 \) |
| $\operatorname{Aut}(H)$ | $C_3\times F_{199}$, of order \(118206\)\(\medspace = 2 \cdot 3^{3} \cdot 11 \cdot 199 \) |
| $W$ | $C_{199}:C_{66}$, of order \(13134\)\(\medspace = 2 \cdot 3 \cdot 11 \cdot 199 \) |
Related subgroups
Other information
| Möbius function | $2$ |
| Projective image | $C_{398}:C_{66}$ |