Subgroup ($H$) information
| Description: | $D_{11}\times D_{12}$ |
| Order: | \(528\)\(\medspace = 2^{4} \cdot 3 \cdot 11 \) |
| Index: | \(3\) |
| Exponent: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| Generators: |
$a, c^{12}, c^{33}, c^{66}, b^{3}, c^{88}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_{132}:D_6$ |
| Order: | \(1584\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PSU(3,2).C_{66}.C_5.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{66}.C_{10}.C_2^4$ |
| $\card{\operatorname{res}(S)}$ | \(10560\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $S_3\times D_{22}$, of order \(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $12$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $-1$ |
| Projective image | $C_{66}:D_6$ |