Subgroup ($H$) information
| Description: | $C_{10}\times D_{22}$ |
| Order: | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Index: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Generators: |
$a^{5}, c^{33}, a^{2}, b^{5}, c^{6}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{66}:C_{10}^2$ |
| Order: | \(6600\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2.C_{165}.C_{60}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $(C_{11}\times A_4).C_{20}.C_2^2$ |
| $W$ | $F_{11}$, of order \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_{33}:C_{10}$ |