Subgroup ($H$) information
| Description: | $S_3\times C_{99}$ |
| Order: | \(594\)\(\medspace = 2 \cdot 3^{3} \cdot 11 \) |
| Index: | \(2\) |
| Exponent: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Generators: |
$a, c^{9}, c^{66}, c^{88}, b^{2}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $S_3\times D_{99}$ |
| Order: | \(1188\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 11 \) |
| Exponent: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3\times C_{99}).C_{30}.C_2^2$ |
| $\operatorname{Aut}(H)$ | $D_6\times C_{30}$, of order \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_6\times C_{30}$, of order \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(99\)\(\medspace = 3^{2} \cdot 11 \) |
| $W$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $S_3\times D_{99}$ |