Subgroup ($H$) information
| Description: | $C_{220}$ |
| Order: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Index: | \(1815\)\(\medspace = 3 \cdot 5 \cdot 11^{2} \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Generators: |
$a^{5}, b^{3}, a^{10}, d^{11}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{11}^2:C_{165}:C_{20}$ |
| Order: | \(399300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11^{3} \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^3.C_{15}.C_{10}^2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_{20}$, of order \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| $W$ | $C_5$, of order \(5\) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $33$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-11$ |
| Projective image | $C_{11}^3:(C_5\times S_3)$ |