Subgroup ($H$) information
| Description: | $S_3\times C_{11}$ |
| Order: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Index: | \(2\) |
| Exponent: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Generators: |
$a, b^{44}, b^{6}$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, a direct factor, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $S_3\times C_{22}$ |
| Order: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| Exponent: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, 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
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{10}\times D_6$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $S_3\times C_{10}$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(S)$ | $S_3\times C_{10}$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{22}$ | ||
| Normalizer: | $S_3\times C_{22}$ | ||
| Complements: | $C_2$ $C_2$ | ||
| Minimal over-subgroups: | $S_3\times C_{22}$ | ||
| Maximal under-subgroups: | $C_{33}$ | $C_{22}$ | $S_3$ |
| Autjugate subgroups: | 132.8.2.b1.b1 |
Other information
| Möbius function | $-1$ |
| Projective image | $D_6$ |