Subgroup ($H$) information
| Description: | $C_{11}^2:C_6$ |
| Order: | \(726\)\(\medspace = 2 \cdot 3 \cdot 11^{2} \) |
| Index: | \(550\)\(\medspace = 2 \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Generators: |
$d^{55}, c, d^{10}, b^{22}$
|
| 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: | $C_{11}\wr C_3:C_{10}^2$ |
| 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.
Quotient group ($Q$) structure
| Description: | $C_{110}:C_5$ |
| Order: | \(550\)\(\medspace = 2 \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Automorphism Group: | $F_5\times F_{11}$, of order \(2200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11 \) |
| Outer Automorphisms: | $C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 5$, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^3.C_{15}.C_{10}^2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $F_{121}:C_2$, of order \(29040\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \) |
| $W$ | $C_{11}^2:(C_5\times S_3)$, of order \(3630\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11^{2} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $55$ |
| Projective image | $C_{11}^3:(S_3\times C_5^2)$ |