metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊2D4, C3⋊2D20, Dic3⋊D5, D10⋊2S3, D30⋊3C2, C6.5D10, C10.5D6, C30.5C22, (C6×D5)⋊2C2, C5⋊1(C3⋊D4), C2.5(S3×D5), (C5×Dic3)⋊3C2, SmallGroup(120,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊D20
G = < a,b,c | a3=b20=c2=1, bab-1=cac=a-1, cbc=b-1 >
Character table of C3⋊D20
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | |
| size | 1 | 1 | 10 | 30 | 2 | 6 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 2 | -2 | 0 | 0 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | 2 | 0 | -1 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | -2 | 0 | -1 | 0 | 2 | 2 | -1 | 1 | 1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 0 | 0 | 2 | 2 | -1-√5/2 | -1+√5/2 | 2 | 0 | 0 | -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 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | 2 | -1+√5/2 | -1-√5/2 | 2 | 0 | 0 | -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 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | -2 | -1+√5/2 | -1-√5/2 | 2 | 0 | 0 | -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 D10 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | -2 | -1-√5/2 | -1+√5/2 | 2 | 0 | 0 | -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 D10 |
| ρ12 | 2 | -2 | 0 | 0 | 2 | 0 | -1-√5/2 | -1+√5/2 | -2 | 0 | 0 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | -ζ4ζ53+ζ4ζ52 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | ζ4ζ53-ζ4ζ52 | 1+√5/2 | 1-√5/2 | orthogonal lifted from D20 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | 0 | -1+√5/2 | -1-√5/2 | -2 | 0 | 0 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | -ζ43ζ54+ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | ζ43ζ54-ζ43ζ5 | 1-√5/2 | 1+√5/2 | orthogonal lifted from D20 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | 0 | -1-√5/2 | -1+√5/2 | -2 | 0 | 0 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | ζ4ζ53-ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | 1+√5/2 | 1-√5/2 | orthogonal lifted from D20 |
| ρ15 | 2 | -2 | 0 | 0 | 2 | 0 | -1+√5/2 | -1-√5/2 | -2 | 0 | 0 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | ζ43ζ54-ζ43ζ5 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | 1-√5/2 | 1+√5/2 | orthogonal lifted from D20 |
| ρ16 | 2 | -2 | 0 | 0 | -1 | 0 | 2 | 2 | 1 | -√-3 | √-3 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ17 | 2 | -2 | 0 | 0 | -1 | 0 | 2 | 2 | 1 | √-3 | -√-3 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ18 | 4 | -4 | 0 | 0 | -2 | 0 | -1-√5 | -1+√5 | 2 | 0 | 0 | 1-√5 | 1+√5 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | orthogonal faithful, Schur index 2 |
| ρ19 | 4 | 4 | 0 | 0 | -2 | 0 | -1-√5 | -1+√5 | -2 | 0 | 0 | -1+√5 | -1-√5 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | orthogonal lifted from S3×D5 |
| ρ20 | 4 | 4 | 0 | 0 | -2 | 0 | -1+√5 | -1-√5 | -2 | 0 | 0 | -1-√5 | -1+√5 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | orthogonal lifted from S3×D5 |
| ρ21 | 4 | -4 | 0 | 0 | -2 | 0 | -1+√5 | -1-√5 | 2 | 0 | 0 | 1+√5 | 1-√5 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | orthogonal faithful, Schur index 2 |
►On 60 points
Generators in S60
(1 24 57)(2 58 25)(3 26 59)(4 60 27)(5 28 41)(6 42 29)(7 30 43)(8 44 31)(9 32 45)(10 46 33)(11 34 47)(12 48 35)(13 36 49)(14 50 37)(15 38 51)(16 52 39)(17 40 53)(18 54 21)(19 22 55)(20 56 23)
(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 54)(22 53)(23 52)(24 51)(25 50)(26 49)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)
G:=sub<Sym(60)| (1,24,57)(2,58,25)(3,26,59)(4,60,27)(5,28,41)(6,42,29)(7,30,43)(8,44,31)(9,32,45)(10,46,33)(11,34,47)(12,48,35)(13,36,49)(14,50,37)(15,38,51)(16,52,39)(17,40,53)(18,54,21)(19,22,55)(20,56,23), (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)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)>;
G:=Group( (1,24,57)(2,58,25)(3,26,59)(4,60,27)(5,28,41)(6,42,29)(7,30,43)(8,44,31)(9,32,45)(10,46,33)(11,34,47)(12,48,35)(13,36,49)(14,50,37)(15,38,51)(16,52,39)(17,40,53)(18,54,21)(19,22,55)(20,56,23), (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)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55) );
G=PermutationGroup([[(1,24,57),(2,58,25),(3,26,59),(4,60,27),(5,28,41),(6,42,29),(7,30,43),(8,44,31),(9,32,45),(10,46,33),(11,34,47),(12,48,35),(13,36,49),(14,50,37),(15,38,51),(16,52,39),(17,40,53),(18,54,21),(19,22,55),(20,56,23)], [(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),(21,54),(22,53),(23,52),(24,51),(25,50),(26,49),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43),(33,42),(34,41),(35,60),(36,59),(37,58),(38,57),(39,56),(40,55)]])
C3⋊D20 is a maximal subgroup of
D20⋊S3 D6.D10 C12.28D10 S3×D20 Dic5.D6 D5×C3⋊D4 D10⋊D6 C9⋊D20 C32⋊7D20 C3⋊D60 C32⋊3D20 D10.2S4 A4⋊D20
C3⋊D20 is a maximal quotient of
C3⋊D40 C6.D20 C15⋊SD16 C3⋊Dic20 D10⋊Dic3 D30⋊4C4 C6.Dic10 C9⋊D20 C32⋊7D20 C3⋊D60 C32⋊3D20 A4⋊D20
Matrix representation of C3⋊D20 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 46 |
| 0 | 0 | 49 | 59 |
| 7 | 32 | 0 | 0 |
| 29 | 2 | 0 | 0 |
| 0 | 0 | 27 | 35 |
| 0 | 0 | 14 | 34 |
| 0 | 60 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 12 | 1 |
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