Subgroup ($H$) information
| Description: | $C_{11}^3$ |
| Order: | \(1331\)\(\medspace = 11^{3} \) |
| Index: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Exponent: | \(11\) |
| Generators: |
$b, d, cd^{6}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_{11}^3:C_{110}$ |
| Order: | \(146410\)\(\medspace = 2 \cdot 5 \cdot 11^{4} \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_{110}$ |
| Order: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Automorphism Group: | $C_2\times C_{20}$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Outer Automorphisms: | $C_2\times C_{20}$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5,11$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^3.C_{11}^2.C_{10}^2$ |
| $\operatorname{Aut}(H)$ | $\GL(3,11)$, of order \(2124276000\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{3} \cdot 7 \cdot 11^{3} \cdot 19 \) |
| $W$ | $C_{110}$, of order \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
Related subgroups
| Centralizer: | $C_{11}^3$ | ||
| Normalizer: | $C_{11}^3:C_{110}$ | ||
| Complements: | $C_{110}$ | ||
| Minimal over-subgroups: | $C_{11}^3:C_{11}$ | $C_{11}^3:C_5$ | $C_{11}^3:C_2$ |
| Maximal under-subgroups: | $C_{11}^2$ | $C_{11}^2$ | $C_{11}^2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_{11}^3:C_{110}$ |