Copied to
clipboard

G = C11⋊C20order 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

C1C11 — C11⋊C20
C1C11C22C2×C11⋊C5 — C11⋊C20
C11 — C11⋊C20
C1C2

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

11C5
11C4
11C10
11C20

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
ρ511-1-1ζ5ζ53ζ52ζ54ζ53ζ52ζ5ζ541535254545553521    linear of order 10
ρ611-1-1ζ54ζ52ζ53ζ5ζ52ζ53ζ54ζ51525355545452531    linear of order 10
ρ71111ζ54ζ52ζ53ζ5ζ52ζ53ζ54ζ51ζ52ζ53ζ5ζ5ζ54ζ54ζ52ζ531    linear of order 5
ρ81111ζ53ζ54ζ5ζ52ζ54ζ5ζ53ζ521ζ54ζ5ζ52ζ52ζ53ζ53ζ54ζ51    linear of order 5
ρ911-1-1ζ52ζ5ζ54ζ53ζ5ζ54ζ52ζ531554535352525541    linear of order 10
ρ1011-1-1ζ53ζ54ζ5ζ52ζ54ζ5ζ53ζ521545525253535451    linear of order 10
ρ111111ζ5ζ53ζ52ζ54ζ53ζ52ζ5ζ541ζ53ζ52ζ54ζ54ζ5ζ5ζ53ζ521    linear of order 5
ρ121111ζ52ζ5ζ54ζ53ζ5ζ54ζ52ζ531ζ5ζ54ζ53ζ53ζ52ζ52ζ5ζ541    linear of order 5
ρ131-1-iiζ53ζ54ζ5ζ5254553521ζ43ζ54ζ43ζ5ζ4ζ52ζ43ζ52ζ4ζ53ζ43ζ53ζ4ζ54ζ4ζ5-1    linear of order 20
ρ141-1i-iζ52ζ5ζ54ζ5355452531ζ4ζ5ζ4ζ54ζ43ζ53ζ4ζ53ζ43ζ52ζ4ζ52ζ43ζ5ζ43ζ54-1    linear of order 20
ρ151-1i-iζ5ζ53ζ52ζ5453525541ζ4ζ53ζ4ζ52ζ43ζ54ζ4ζ54ζ43ζ5ζ4ζ5ζ43ζ53ζ43ζ52-1    linear of order 20
ρ161-1i-iζ53ζ54ζ5ζ5254553521ζ4ζ54ζ4ζ5ζ43ζ52ζ4ζ52ζ43ζ53ζ4ζ53ζ43ζ54ζ43ζ5-1    linear of order 20
ρ171-1i-iζ54ζ52ζ53ζ552535451ζ4ζ52ζ4ζ53ζ43ζ5ζ4ζ5ζ43ζ54ζ4ζ54ζ43ζ52ζ43ζ53-1    linear of order 20
ρ181-1-iiζ54ζ52ζ53ζ552535451ζ43ζ52ζ43ζ53ζ4ζ5ζ43ζ5ζ4ζ54ζ43ζ54ζ4ζ52ζ4ζ53-1    linear of order 20
ρ191-1-iiζ5ζ53ζ52ζ5453525541ζ43ζ53ζ43ζ52ζ4ζ54ζ43ζ54ζ4ζ5ζ43ζ5ζ4ζ53ζ4ζ52-1    linear of order 20
ρ201-1-iiζ52ζ5ζ54ζ5355452531ζ43ζ5ζ43ζ54ζ4ζ53ζ43ζ53ζ4ζ52ζ43ζ52ζ4ζ5ζ4ζ54-1    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 29 19 41 37 25 11 23 7 33 15)(2 20 38 12 8 16 30 42 26 24 34)(3 39 9 31 27 35 21 13 17 43 5)(4 10 28 22 18 6 40 32 36 14 44)
(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)| (1,29,19,41,37,25,11,23,7,33,15)(2,20,38,12,8,16,30,42,26,24,34)(3,39,9,31,27,35,21,13,17,43,5)(4,10,28,22,18,6,40,32,36,14,44), (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( (1,29,19,41,37,25,11,23,7,33,15)(2,20,38,12,8,16,30,42,26,24,34)(3,39,9,31,27,35,21,13,17,43,5)(4,10,28,22,18,6,40,32,36,14,44), (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([(1,29,19,41,37,25,11,23,7,33,15),(2,20,38,12,8,16,30,42,26,24,34),(3,39,9,31,27,35,21,13,17,43,5),(4,10,28,22,18,6,40,32,36,14,44)], [(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

׿
×
𝔽