Subgroup ($H$) information
| Description: | $C_{1194}:C_{22}$ |
| Order: | \(26268\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \cdot 199 \) |
| Index: | \(3\) |
| Exponent: | \(13134\)\(\medspace = 2 \cdot 3 \cdot 11 \cdot 199 \) |
| Generators: |
$a^{18}, b^{2}, a^{132}, b^{199}, a^{99}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and 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: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6\times F_{199}$, of order \(236412\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 11 \cdot 199 \) |
| $\operatorname{Aut}(H)$ | $C_{199}:(C_2^2\times C_{198})$ |
| $W$ | $C_{199}:C_{66}$, of order \(13134\)\(\medspace = 2 \cdot 3 \cdot 11 \cdot 199 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_{199}:C_{66}$ |