direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×Dic5, C5⋊2C12, C15⋊4C4, C10.C6, C6.2D5, C30.2C2, C2.(C3×D5), SmallGroup(60,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C3×Dic5 |
Generators and relations for C3×Dic5
G = < a,b,c | a3=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >
Character table of C3×Dic5
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 6A | 6B | 10A | 10B | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 30A | 30B | 30C | 30D | |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 2 | 2 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 6 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 6 |
| ρ7 | 1 | -1 | 1 | 1 | -i | i | 1 | 1 | -1 | -1 | -1 | -1 | -i | -i | i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | i | -i | 1 | 1 | -1 | -1 | -1 | -1 | i | i | -i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 1 | -1 | ζ3 | ζ32 | i | -i | 1 | 1 | ζ65 | ζ6 | -1 | -1 | ζ4ζ3 | ζ4ζ32 | ζ43ζ3 | ζ43ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | linear of order 12 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | -i | i | 1 | 1 | ζ6 | ζ65 | -1 | -1 | ζ43ζ32 | ζ43ζ3 | ζ4ζ32 | ζ4ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | linear of order 12 |
| ρ11 | 1 | -1 | ζ32 | ζ3 | i | -i | 1 | 1 | ζ6 | ζ65 | -1 | -1 | ζ4ζ32 | ζ4ζ3 | ζ43ζ32 | ζ43ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | linear of order 12 |
| ρ12 | 1 | -1 | ζ3 | ζ32 | -i | i | 1 | 1 | ζ65 | ζ6 | -1 | -1 | ζ43ζ3 | ζ43ζ32 | ζ4ζ3 | ζ4ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | linear of order 12 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | 2 | -1-√5/2 | -1+√5/2 | 0 | 0 | 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 | orthogonal lifted from D5 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | 2 | -1+√5/2 | -1-√5/2 | 0 | 0 | 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 | orthogonal lifted from D5 |
| ρ15 | 2 | -2 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -2 | -2 | 1-√5/2 | 1+√5/2 | 0 | 0 | 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 | symplectic lifted from Dic5, Schur index 2 |
| ρ16 | 2 | -2 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | -2 | 1+√5/2 | 1-√5/2 | 0 | 0 | 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 | symplectic lifted from Dic5, Schur index 2 |
| ρ17 | 2 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√-3 | -1-√-3 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | complex lifted from C3×D5 |
| ρ18 | 2 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√-3 | -1+√-3 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | complex lifted from C3×D5 |
| ρ19 | 2 | -2 | -1-√-3 | -1+√-3 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√-3 | 1-√-3 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ53-ζ32ζ52 | complex faithful |
| ρ20 | 2 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√-3 | -1-√-3 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | complex lifted from C3×D5 |
| ρ21 | 2 | -2 | -1-√-3 | -1+√-3 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1+√-3 | 1-√-3 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ54-ζ32ζ5 | complex faithful |
| ρ22 | 2 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√-3 | -1+√-3 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | complex lifted from C3×D5 |
| ρ23 | 2 | -2 | -1+√-3 | -1-√-3 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1-√-3 | 1+√-3 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ53-ζ3ζ52 | complex faithful |
| ρ24 | 2 | -2 | -1+√-3 | -1-√-3 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√-3 | 1+√-3 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ54-ζ3ζ5 | complex faithful |
►Regular action on 60 points
Generators in S60
(1 29 19)(2 30 20)(3 21 11)(4 22 12)(5 23 13)(6 24 14)(7 25 15)(8 26 16)(9 27 17)(10 28 18)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)
(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 36 6 31)(2 35 7 40)(3 34 8 39)(4 33 9 38)(5 32 10 37)(11 44 16 49)(12 43 17 48)(13 42 18 47)(14 41 19 46)(15 50 20 45)(21 54 26 59)(22 53 27 58)(23 52 28 57)(24 51 29 56)(25 60 30 55)
G:=sub<Sym(60)| (1,29,19)(2,30,20)(3,21,11)(4,22,12)(5,23,13)(6,24,14)(7,25,15)(8,26,16)(9,27,17)(10,28,18)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (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), (1,36,6,31)(2,35,7,40)(3,34,8,39)(4,33,9,38)(5,32,10,37)(11,44,16,49)(12,43,17,48)(13,42,18,47)(14,41,19,46)(15,50,20,45)(21,54,26,59)(22,53,27,58)(23,52,28,57)(24,51,29,56)(25,60,30,55)>;
G:=Group( (1,29,19)(2,30,20)(3,21,11)(4,22,12)(5,23,13)(6,24,14)(7,25,15)(8,26,16)(9,27,17)(10,28,18)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (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), (1,36,6,31)(2,35,7,40)(3,34,8,39)(4,33,9,38)(5,32,10,37)(11,44,16,49)(12,43,17,48)(13,42,18,47)(14,41,19,46)(15,50,20,45)(21,54,26,59)(22,53,27,58)(23,52,28,57)(24,51,29,56)(25,60,30,55) );
G=PermutationGroup([[(1,29,19),(2,30,20),(3,21,11),(4,22,12),(5,23,13),(6,24,14),(7,25,15),(8,26,16),(9,27,17),(10,28,18),(31,51,41),(32,52,42),(33,53,43),(34,54,44),(35,55,45),(36,56,46),(37,57,47),(38,58,48),(39,59,49),(40,60,50)], [(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)], [(1,36,6,31),(2,35,7,40),(3,34,8,39),(4,33,9,38),(5,32,10,37),(11,44,16,49),(12,43,17,48),(13,42,18,47),(14,41,19,46),(15,50,20,45),(21,54,26,59),(22,53,27,58),(23,52,28,57),(24,51,29,56),(25,60,30,55)]])
C3×Dic5 is a maximal subgroup of
C15⋊C8 D30.C2 C5⋊D12 C15⋊Q8 D5×C12 Dic5.A4 C35⋊3C12
C3×Dic5 is a maximal quotient of C35⋊3C12
Matrix representation of C3×Dic5 ►in GL2(𝔽19) generated by
| 7 | 0 |
| 0 | 7 |
| 1 | 17 |
| 3 | 14 |
| 10 | 10 |
| 7 | 9 |
G:=sub<GL(2,GF(19))| [7,0,0,7],[1,3,17,14],[10,7,10,9] >;
C3×Dic5 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_5 % in TeX
G:=Group("C3xDic5"); // GroupNames label
G:=SmallGroup(60,2);
// by ID
G=gap.SmallGroup(60,2);
# by ID
G:=PCGroup([4,-2,-3,-2,-5,24,771]);
// Polycyclic
G:=Group<a,b,c|a^3=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C3×Dic5 in TeX
Character table of C3×Dic5 in TeX