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G = C3⋊D20order 120 = 23·3·5

The semidirect product of C3 and D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C152D4, C32D20, Dic3⋊D5, D102S3, D303C2, C6.5D10, C10.5D6, C30.5C22, (C6×D5)⋊2C2, C51(C3⋊D4), C2.5(S3×D5), (C5×Dic3)⋊3C2, SmallGroup(120,12)

Series: Derived Chief Lower central Upper central

C1C30 — C3⋊D20
C1C5C15C30C6×D5 — C3⋊D20
C15C30 — C3⋊D20
C1C2

Generators and relations for C3⋊D20
 G = < a,b,c | a3=b20=c2=1, bab-1=cac=a-1, cbc=b-1 >

10C2
30C2
3C4
5C22
15C22
10C6
10S3
2D5
6D5
15D4
5D6
5C2×C6
3C20
3D10
2C3×D5
2D15
5C3⋊D4
3D20

Character table of C3⋊D20

 class 12A2B2C345A5B6A6B6C10A10B15A15B20A20B20C20D30A30B
 size 1110302622210102244666644
ρ1111111111111111111111    trivial
ρ2111-11-1111111111-1-1-1-111    linear of order 2
ρ311-111-1111-1-11111-1-1-1-111    linear of order 2
ρ411-1-111111-1-11111111111    linear of order 2
ρ52-2002022-200-2-2220000-2-2    orthogonal lifted from D4
ρ62220-1022-1-1-122-1-10000-1-1    orthogonal lifted from S3
ρ722-20-1022-11122-1-10000-1-1    orthogonal lifted from D6
ρ8220022-1-5/2-1+5/2200-1+5/2-1-5/2-1-5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ9220022-1+5/2-1-5/2200-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ1022002-2-1+5/2-1-5/2200-1-5/2-1+5/2-1+5/2-1-5/21-5/21+5/21+5/21-5/2-1+5/2-1-5/2    orthogonal lifted from D10
ρ1122002-2-1-5/2-1+5/2200-1+5/2-1-5/2-1-5/2-1+5/21+5/21-5/21-5/21+5/2-1-5/2-1+5/2    orthogonal lifted from D10
ρ122-20020-1-5/2-1+5/2-2001-5/21+5/2-1-5/2-1+5/24ζ534ζ52ζ43ζ5443ζ543ζ5443ζ5ζ4ζ534ζ521+5/21-5/2    orthogonal lifted from D20
ρ132-20020-1+5/2-1-5/2-2001+5/21-5/2-1+5/2-1-5/243ζ5443ζ54ζ534ζ52ζ4ζ534ζ52ζ43ζ5443ζ51-5/21+5/2    orthogonal lifted from D20
ρ142-20020-1-5/2-1+5/2-2001-5/21+5/2-1-5/2-1+5/2ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ54ζ534ζ521+5/21-5/2    orthogonal lifted from D20
ρ152-20020-1+5/2-1-5/2-2001+5/21-5/2-1+5/2-1-5/2ζ43ζ5443ζ5ζ4ζ534ζ524ζ534ζ5243ζ5443ζ51-5/21+5/2    orthogonal lifted from D20
ρ162-200-10221--3-3-2-2-1-1000011    complex lifted from C3⋊D4
ρ172-200-10221-3--3-2-2-1-1000011    complex lifted from C3⋊D4
ρ184-400-20-1-5-1+52001-51+51+5/21-5/20000-1-5/2-1+5/2    orthogonal faithful, Schur index 2
ρ194400-20-1-5-1+5-200-1+5-1-51+5/21-5/200001+5/21-5/2    orthogonal lifted from S3×D5
ρ204400-20-1+5-1-5-200-1-5-1+51-5/21+5/200001-5/21+5/2    orthogonal lifted from S3×D5
ρ214-400-20-1+5-1-52001+51-51-5/21+5/20000-1+5/2-1-5/2    orthogonal faithful, Schur index 2

Smallest permutation representation of C3⋊D20
On 60 points
Generators in S60
(1 42 28)(2 29 43)(3 44 30)(4 31 45)(5 46 32)(6 33 47)(7 48 34)(8 35 49)(9 50 36)(10 37 51)(11 52 38)(12 39 53)(13 54 40)(14 21 55)(15 56 22)(16 23 57)(17 58 24)(18 25 59)(19 60 26)(20 27 41)
(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 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 43)(22 42)(23 41)(24 60)(25 59)(26 58)(27 57)(28 56)(29 55)(30 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)

G:=sub<Sym(60)| (1,42,28)(2,29,43)(3,44,30)(4,31,45)(5,46,32)(6,33,47)(7,48,34)(8,35,49)(9,50,36)(10,37,51)(11,52,38)(12,39,53)(13,54,40)(14,21,55)(15,56,22)(16,23,57)(17,58,24)(18,25,59)(19,60,26)(20,27,41), (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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,43)(22,42)(23,41)(24,60)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)>;

G:=Group( (1,42,28)(2,29,43)(3,44,30)(4,31,45)(5,46,32)(6,33,47)(7,48,34)(8,35,49)(9,50,36)(10,37,51)(11,52,38)(12,39,53)(13,54,40)(14,21,55)(15,56,22)(16,23,57)(17,58,24)(18,25,59)(19,60,26)(20,27,41), (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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,43)(22,42)(23,41)(24,60)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44) );

G=PermutationGroup([(1,42,28),(2,29,43),(3,44,30),(4,31,45),(5,46,32),(6,33,47),(7,48,34),(8,35,49),(9,50,36),(10,37,51),(11,52,38),(12,39,53),(13,54,40),(14,21,55),(15,56,22),(16,23,57),(17,58,24),(18,25,59),(19,60,26),(20,27,41)], [(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),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,43),(22,42),(23,41),(24,60),(25,59),(26,58),(27,57),(28,56),(29,55),(30,54),(31,53),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,45),(40,44)])

C3⋊D20 is a maximal subgroup of
D20⋊S3  D6.D10  C12.28D10  S3×D20  Dic5.D6  D5×C3⋊D4  D10⋊D6  C9⋊D20  C327D20  C3⋊D60  C323D20  D10.2S4  A4⋊D20
C3⋊D20 is a maximal quotient of
C3⋊D40  C6.D20  C15⋊SD16  C3⋊Dic20  D10⋊Dic3  D304C4  C6.Dic10  C9⋊D20  C327D20  C3⋊D60  C323D20  A4⋊D20

Matrix representation of C3⋊D20 in GL4(𝔽61) generated by

1000
0100
00146
004959
,
73200
29200
002735
001434
,
06000
60000
00600
00121
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,1,49,0,0,46,59],[7,29,0,0,32,2,0,0,0,0,27,14,0,0,35,34],[0,60,0,0,60,0,0,0,0,0,60,12,0,0,0,1] >;

C3⋊D20 in GAP, Magma, Sage, TeX

C_3\rtimes D_{20}
% in TeX

G:=Group("C3:D20");
// GroupNames label

G:=SmallGroup(120,12);
// by ID

G=gap.SmallGroup(120,12);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-5,61,26,168,2404]);
// Polycyclic

G:=Group<a,b,c|a^3=b^20=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3⋊D20 in TeX
Character table of C3⋊D20 in TeX

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