Subgroup ($H$) information
| Description: | $C_{11}^2:C_{30}$ |
| Order: | \(3630\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11^{2} \) |
| Index: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$d^{22}, a^{2}b^{18}c^{9}d^{16}, cd^{28}, b^{20}c^{4}d^{36}, d^{4}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{220}:F_{11}:S_3$ |
| Order: | \(145200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{2} \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^2\times C_{10}$ |
| Order: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_4\times \GL(3,2)$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_4\times \GL(3,2)$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_{30}.C_{10}.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_{11}^2.C_{60}.C_4.C_2^2$ |
| $W$ | $C_{11}^2:(S_3\times C_{10})$, of order \(7260\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11^{2} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $8$ |
| Projective image | $C_{11}^2:(C_{10}\times D_6)$ |