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G = C11⋊C20  order 220 = 22·5·11

The semidirect product of C11 and C20 acting via C20/C2=C10

metacyclic, supersoluble, monomial, Z-group

Aliases: C11⋊C20, C2.F11, C22.C10, Dic11⋊C5, C11⋊C5⋊C4, (C2×C11⋊C5).C2, SmallGroup(220,1)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C11 — C11⋊C20
Chief series
C1 — C11 — C22 — C2×C11⋊C5 — C11⋊C20
Lower central
C11 — C11⋊C20
Upper central
C1 — C2

Generators and relations for C11⋊C20
 G = < a,b | a11=b20=1, bab-1=a2 >

C1
C2
11C5
C11
11C4
11C10
C22
C11⋊C5
11C20
Dic11
C2×C11⋊C5
C11⋊C20

Character table of C11⋊C20

 class 124A4B5A5B5C5D10A10B10C10D1120A20B20C20D20E20F20G20H22
 size 111111111111111111111110111111111111111110
ρ11111111111111111111111    trivial
ρ211-1-1111111111-1-1-1-1-1-1-1-11    linear of order 2
ρ31-1-ii1111-1-1-1-11-i-ii-ii-iii-1    linear of order 4
ρ41-1i-i1111-1-1-1-11ii-ii-ii-i-i-1    linear of order 4
ρ511-1-1ζ5ζ53ζ52ζ54ζ53ζ52ζ5ζ541-ζ53-ζ52-ζ54-ζ54-ζ5-ζ5-ζ53-ζ521    linear of order 10
ρ611-1-1ζ54ζ52ζ53ζ5ζ52ζ53ζ54ζ51-ζ52-ζ53-ζ5-ζ5-ζ54-ζ54-ζ52-ζ531    linear of order 10
ρ71111ζ54ζ52ζ53ζ5ζ52ζ53ζ54ζ51ζ52ζ53ζ5ζ5ζ54ζ54ζ52ζ531    linear of order 5
ρ81111ζ53ζ54ζ5ζ52ζ54ζ5ζ53ζ521ζ54ζ5ζ52ζ52ζ53ζ53ζ54ζ51    linear of order 5
ρ911-1-1ζ52ζ5ζ54ζ53ζ5ζ54ζ52ζ531-ζ5-ζ54-ζ53-ζ53-ζ52-ζ52-ζ5-ζ541    linear of order 10
ρ1011-1-1ζ53ζ54ζ5ζ52ζ54ζ5ζ53ζ521-ζ54-ζ5-ζ52-ζ52-ζ53-ζ53-ζ54-ζ51    linear of order 10
ρ111111ζ5ζ53ζ52ζ54ζ53ζ52ζ5ζ541ζ53ζ52ζ54ζ54ζ5ζ5ζ53ζ521    linear of order 5
ρ121111ζ52ζ5ζ54ζ53ζ5ζ54ζ52ζ531ζ5ζ54ζ53ζ53ζ52ζ52ζ5ζ541    linear of order 5
ρ131-1-iiζ53ζ54ζ5ζ52-ζ54-ζ5-ζ53-ζ521ζ43ζ54ζ43ζ5ζ4ζ52ζ43ζ52ζ4ζ53ζ43ζ53ζ4ζ54ζ4ζ5-1    linear of order 20
ρ141-1i-iζ52ζ5ζ54ζ53-ζ5-ζ54-ζ52-ζ531ζ4ζ5ζ4ζ54ζ43ζ53ζ4ζ53ζ43ζ52ζ4ζ52ζ43ζ5ζ43ζ54-1    linear of order 20
ρ151-1i-iζ5ζ53ζ52ζ54-ζ53-ζ52-ζ5-ζ541ζ4ζ53ζ4ζ52ζ43ζ54ζ4ζ54ζ43ζ5ζ4ζ5ζ43ζ53ζ43ζ52-1    linear of order 20
ρ161-1i-iζ53ζ54ζ5ζ52-ζ54-ζ5-ζ53-ζ521ζ4ζ54ζ4ζ5ζ43ζ52ζ4ζ52ζ43ζ53ζ4ζ53ζ43ζ54ζ43ζ5-1    linear of order 20
ρ171-1i-iζ54ζ52ζ53ζ5-ζ52-ζ53-ζ54-ζ51ζ4ζ52ζ4ζ53ζ43ζ5ζ4ζ5ζ43ζ54ζ4ζ54ζ43ζ52ζ43ζ53-1    linear of order 20
ρ181-1-iiζ54ζ52ζ53ζ5-ζ52-ζ53-ζ54-ζ51ζ43ζ52ζ43ζ53ζ4ζ5ζ43ζ5ζ4ζ54ζ43ζ54ζ4ζ52ζ4ζ53-1    linear of order 20
ρ191-1-iiζ5ζ53ζ52ζ54-ζ53-ζ52-ζ5-ζ541ζ43ζ53ζ43ζ52ζ4ζ54ζ43ζ54ζ4ζ5ζ43ζ5ζ4ζ53ζ4ζ52-1    linear of order 20
ρ201-1-iiζ52ζ5ζ54ζ53-ζ5-ζ54-ζ52-ζ531ζ43ζ5ζ43ζ54ζ4ζ53ζ43ζ53ζ4ζ52ζ43ζ52ζ4ζ5ζ4ζ54-1    linear of order 20
ρ2110100000000000-100000000-1    orthogonal lifted from F11
ρ2210-100000000000-1000000001    symplectic faithful, Schur index 2

Smallest permutation representation of C11⋊C20
►On 44 points
Generators in S44
(1 25 5 37 33 41 17 9 13 29 21)(2 6 34 18 14 22 26 38 42 10 30)(3 35 15 27 43 31 7 19 23 39 11)(4 16 44 8 24 12 36 28 32 20 40)

(1 2 3 4)(5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44)
 
G:=sub<Sym(44)| (1,25,5,37,33,41,17,9,13,29,21)(2,6,34,18,14,22,26,38,42,10,30)(3,35,15,27,43,31,7,19,23,39,11)(4,16,44,8,24,12,36,28,32,20,40), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44)>;
 
G:=Group( (1,25,5,37,33,41,17,9,13,29,21)(2,6,34,18,14,22,26,38,42,10,30)(3,35,15,27,43,31,7,19,23,39,11)(4,16,44,8,24,12,36,28,32,20,40), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44) );
 
G=PermutationGroup([[(1,25,5,37,33,41,17,9,13,29,21),(2,6,34,18,14,22,26,38,42,10,30),(3,35,15,27,43,31,7,19,23,39,11),(4,16,44,8,24,12,36,28,32,20,40)], [(1,2,3,4),(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44)]])
 

C11⋊C20 is a maximal subgroup of   C4.F11  C4×F11  C22⋊F11
C11⋊C20 is a maximal quotient of   C11⋊C40

Matrix representation of C11⋊C20 ►in GL10(𝔽661)

660100000000
660010000000
660001000000
660000100000
660000010000
660000001000
660000000100
660000000010
660000000001
660000000000
,
2592220259040240202590
2594022594810040240200
2590259025922240200402
0402259259259402022200
2594020025900402259222
4814022590040200259402
0048102594024024022590
2590025948140204020402
0025925900222402259402
0402025925904020481402

G:=sub<GL(10,GF(661))| [660,660,660,660,660,660,660,660,660,660,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0],[259,259,259,0,259,481,0,259,0,0,222,402,0,402,402,402,0,0,0,402,0,259,259,259,0,259,481,0,259,0,259,481,0,259,0,0,0,259,259,259,0,0,259,259,259,0,259,481,0,259,402,0,222,402,0,402,402,402,0,0,402,402,402,0,0,0,402,0,222,402,0,402,0,222,402,0,402,402,402,0,259,0,0,0,259,259,259,0,259,481,0,0,402,0,222,402,0,402,402,402] >;
 

C11⋊C20 in GAP, Magma, Sage, TeX 

C_{11}\rtimes C_{20}
 
% in TeX
 
G:=Group("C11:C20");
 
// GroupNames label
 
G:=SmallGroup(220,1);
 
// by ID
 
G=gap.SmallGroup(220,1);
 
# by ID
 
G:=PCGroup([4,-2,-5,-2,-11,40,3203,1447]);
 
// Polycyclic
 
G:=Group<a,b|a^11=b^20=1,b*a*b^-1=a^2>;
 
// generators/relations
 

Export

Subgroup lattice of C11⋊C20 in TeX
Character table of C11⋊C20 in TeX

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