Subgroup ($H$) information
| Description: | $C_{11}^2\times C_{110}$ |
| Order: | \(13310\)\(\medspace = 2 \cdot 5 \cdot 11^{3} \) |
| Index: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Generators: |
$a^{10}, b^{3}, c, d^{11}, d^{5}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the socle, abelian (hence metabelian and an A-group), and elementary for $p = 11$ (hence hyperelementary).
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.
Quotient group ($Q$) structure
| Description: | $C_5\times S_3$ |
| Order: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Automorphism Group: | $C_4\times S_3$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Outer Automorphisms: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^3.C_{15}.C_{10}^2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_4\times C_{10}.\PSL(3,11)$ |
| $W$ | $C_5\times S_3$, of order \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-3$ |
| Projective image | $C_{11}^3:(C_5\times S_3)$ |