Subgroup ($H$) information
| Description: | $C_{11}^2:C_{15}$ |
| Order: | \(1815\)\(\medspace = 3 \cdot 5 \cdot 11^{2} \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(165\)\(\medspace = 3 \cdot 5 \cdot 11 \) |
| Generators: |
$bcd^{20}, cd^{100}, d^{10}, d^{22}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{11}^2:(D_6\times C_5^2)$ |
| Order: | \(36300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11^{2} \) |
| 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.
Quotient group ($Q$) structure
| Description: | $C_2\times C_{10}$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_4\times S_3$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Outer Automorphisms: | $C_4\times S_3$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_{15}.C_{10}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_{11}^2.C_{60}.C_4.C_2^2$ |
| $\card{W}$ | \(3630\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11^{2} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-2$ |
| Projective image | $C_3^4:(D_4\times D_6)$ |