Subgroup ($H$) information
| Description: | $C_{597}:C_{99}$ |
| Order: | \(59103\)\(\medspace = 3^{3} \cdot 11 \cdot 199 \) |
| Index: | \(2\) |
| Exponent: | \(19701\)\(\medspace = 3^{2} \cdot 11 \cdot 199 \) |
| Generators: |
$a^{132}, b^{3}, a^{18}, b^{199}, a^{110}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, a Hall subgroup, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $C_{1194}:C_{99}$ |
| 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: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{199}:(C_{11}:(C_{18}\times S_3))$ |
| $\operatorname{Aut}(H)$ | $C_{199}:(C_{11}:(C_{18}\times S_3))$ |
| $W$ | $C_{199}:C_{99}$, of order \(19701\)\(\medspace = 3^{2} \cdot 11 \cdot 199 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_{199}:C_{198}$ |