Subgroup ($H$) information
| Description: | $C_{199}:C_3$ |
| Order: | \(597\)\(\medspace = 3 \cdot 199 \) |
| Index: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(597\)\(\medspace = 3 \cdot 199 \) |
| Generators: |
$a^{132}, b^{3}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 3$.
Ambient group ($G$) information
| Description: | $C_3\times F_{199}$ |
| Order: | \(118206\)\(\medspace = 2 \cdot 3^{3} \cdot 11 \cdot 199 \) |
| 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: | $C_3\times C_{66}$ |
| Order: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Automorphism Group: | $C_{10}\times \GL(2,3)$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Outer Automorphisms: | $C_{10}\times \GL(2,3)$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{199}:(C_{11}:(C_{18}\times S_3))$ |
| $\operatorname{Aut}(H)$ | $F_{199}$, of order \(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \) |
| $W$ | $F_{199}$, of order \(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \) |
Related subgroups
| Centralizer: | $C_3$ | |||||
| Normalizer: | $C_3\times F_{199}$ | |||||
| Minimal over-subgroups: | $C_{199}:C_{33}$ | $C_{597}:C_3$ | $C_{199}:C_9$ | $C_{199}:C_9$ | $C_{199}:C_9$ | $C_{199}:C_6$ |
| Maximal under-subgroups: | $C_{199}$ | $C_3$ |
Other information
| Möbius function | $3$ |
| Projective image | $C_3\times F_{199}$ |