Subgroup ($H$) information
| Description: | $C_{165}:C_5$ |
| Order: | \(825\)\(\medspace = 3 \cdot 5^{2} \cdot 11 \) |
| Index: | \(484\)\(\medspace = 2^{2} \cdot 11^{2} \) |
| Exponent: | \(165\)\(\medspace = 3 \cdot 5 \cdot 11 \) |
| Generators: |
$b^{22}c^{6}d^{10}, d^{22}, b^{3}, a^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 5$, and an A-group.
Ambient group ($G$) information
| Description: | $C_{11}\wr C_3:C_{10}^2$ |
| Order: | \(399300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11^{3} \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^3.C_{15}.C_{10}^2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $D_{110}:C_{20}$, of order \(4400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 11 \) |
| $W$ | $C_{11}:C_{10}$, of order \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $121$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-2$ |
| Projective image | $C_{11}^3:(S_3\times C_{10})$ |