Subgroup ($H$) information
| Description: | $S_3\times D_{22}$ |
| Order: | \(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Generators: |
$ac, d^{22}, b^{2}d^{22}, b^{3}cd^{23}, d^{4}$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $D_{12}:D_{22}$ |
| Order: | \(1056\)\(\medspace = 2^{5} \cdot 3 \cdot 11 \) |
| Exponent: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_{33}\times A_4).C_5.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{11}:(C_2^2\times C_{10}\times S_3)$ |
| $\card{\operatorname{res}(S)}$ | \(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $D_6\times D_{22}$, of order \(528\)\(\medspace = 2^{4} \cdot 3 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $6$ |
| Möbius function | $2$ |
| Projective image | $D_6\times D_{22}$ |