direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×Dic5, D6.D5, C6.2D10, C10.2D6, Dic15⋊3C2, C30.2C22, C5⋊4(C4×S3), C15⋊5(C2×C4), (C5×S3)⋊2C4, (S3×C10).C2, C2.2(S3×D5), C3⋊1(C2×Dic5), (C3×Dic5)⋊1C2, SmallGroup(120,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — S3×Dic5 |
Generators and relations for S3×Dic5
G = < a,b,c,d | a3=b2=c10=1, d2=c5, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Character table of S3×Dic5
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6 | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 30A | 30B | |
| size | 1 | 1 | 3 | 3 | 2 | 5 | 5 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 4 | 4 | 4 | 4 | |
| 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 | -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 |
| ρ4 | 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 |
| ρ5 | 1 | -1 | 1 | -1 | 1 | i | -i | i | -i | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | i | -i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | 1 | i | -i | -i | i | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | i | -i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | -1 | -1 | 1 | 1 | -i | i | i | -i | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -i | i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | -i | i | -i | i | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -i | i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ12 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D10 |
| ρ13 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D10 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ15 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -2 | 1+√5/2 | 1-√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | -1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ16 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ17 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | 1-√5/2 | 1+√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | -1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ19 | 2 | -2 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 2 | 2 | 1 | -2 | -2 | 0 | 0 | 0 | 0 | i | -i | -1 | -1 | 1 | 1 | complex lifted from C4×S3 |
| ρ20 | 2 | -2 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 2 | 2 | 1 | -2 | -2 | 0 | 0 | 0 | 0 | -i | i | -1 | -1 | 1 | 1 | complex lifted from C4×S3 |
| ρ21 | 4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | -2 | -1+√5 | -1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from S3×D5 |
| ρ22 | 4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | -2 | -1-√5 | -1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from S3×D5 |
| ρ23 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | symplectic faithful, Schur index 2 |
| ρ24 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | symplectic faithful, Schur index 2 |
►On 60 points
Generators in S60
(1 21 40)(2 22 31)(3 23 32)(4 24 33)(5 25 34)(6 26 35)(7 27 36)(8 28 37)(9 29 38)(10 30 39)(11 51 47)(12 52 48)(13 53 49)(14 54 50)(15 55 41)(16 56 42)(17 57 43)(18 58 44)(19 59 45)(20 60 46)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 41)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 31)(28 32)(29 33)(30 34)(51 56)(52 57)(53 58)(54 59)(55 60)
(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 56 6 51)(2 55 7 60)(3 54 8 59)(4 53 9 58)(5 52 10 57)(11 40 16 35)(12 39 17 34)(13 38 18 33)(14 37 19 32)(15 36 20 31)(21 42 26 47)(22 41 27 46)(23 50 28 45)(24 49 29 44)(25 48 30 43)
G:=sub<Sym(60)| (1,21,40)(2,22,31)(3,23,32)(4,24,33)(5,25,34)(6,26,35)(7,27,36)(8,28,37)(9,29,38)(10,30,39)(11,51,47)(12,52,48)(13,53,49)(14,54,50)(15,55,41)(16,56,42)(17,57,43)(18,58,44)(19,59,45)(20,60,46), (1,6)(2,7)(3,8)(4,9)(5,10)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,41)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,31)(28,32)(29,33)(30,34)(51,56)(52,57)(53,58)(54,59)(55,60), (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), (1,56,6,51)(2,55,7,60)(3,54,8,59)(4,53,9,58)(5,52,10,57)(11,40,16,35)(12,39,17,34)(13,38,18,33)(14,37,19,32)(15,36,20,31)(21,42,26,47)(22,41,27,46)(23,50,28,45)(24,49,29,44)(25,48,30,43)>;
G:=Group( (1,21,40)(2,22,31)(3,23,32)(4,24,33)(5,25,34)(6,26,35)(7,27,36)(8,28,37)(9,29,38)(10,30,39)(11,51,47)(12,52,48)(13,53,49)(14,54,50)(15,55,41)(16,56,42)(17,57,43)(18,58,44)(19,59,45)(20,60,46), (1,6)(2,7)(3,8)(4,9)(5,10)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,41)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,31)(28,32)(29,33)(30,34)(51,56)(52,57)(53,58)(54,59)(55,60), (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), (1,56,6,51)(2,55,7,60)(3,54,8,59)(4,53,9,58)(5,52,10,57)(11,40,16,35)(12,39,17,34)(13,38,18,33)(14,37,19,32)(15,36,20,31)(21,42,26,47)(22,41,27,46)(23,50,28,45)(24,49,29,44)(25,48,30,43) );
G=PermutationGroup([[(1,21,40),(2,22,31),(3,23,32),(4,24,33),(5,25,34),(6,26,35),(7,27,36),(8,28,37),(9,29,38),(10,30,39),(11,51,47),(12,52,48),(13,53,49),(14,54,50),(15,55,41),(16,56,42),(17,57,43),(18,58,44),(19,59,45),(20,60,46)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,48),(18,49),(19,50),(20,41),(21,35),(22,36),(23,37),(24,38),(25,39),(26,40),(27,31),(28,32),(29,33),(30,34),(51,56),(52,57),(53,58),(54,59),(55,60)], [(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)], [(1,56,6,51),(2,55,7,60),(3,54,8,59),(4,53,9,58),(5,52,10,57),(11,40,16,35),(12,39,17,34),(13,38,18,33),(14,37,19,32),(15,36,20,31),(21,42,26,47),(22,41,27,46),(23,50,28,45),(24,49,29,44),(25,48,30,43)]])
S3×Dic5 is a maximal subgroup of
D6.F5 D12⋊D5 D12⋊5D5 C4×S3×D5 C30.C23 Dic3.D10 Dic15⋊S3 Dic5.6S4
S3×Dic5 is a maximal quotient of
D6.Dic5 D6⋊Dic5 C6.Dic10 Dic15⋊S3
Matrix representation of S3×Dic5 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 60 | 0 |
| 0 | 1 | 0 | 0 |
| 60 | 17 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 25 | 30 | 0 | 0 |
| 28 | 36 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,60],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,60,0],[0,60,0,0,1,17,0,0,0,0,60,0,0,0,0,60],[25,28,0,0,30,36,0,0,0,0,11,0,0,0,0,11] >;
S3×Dic5 in GAP, Magma, Sage, TeX
S_3\times {\rm Dic}_5 % in TeX
G:=Group("S3xDic5"); // GroupNames label
G:=SmallGroup(120,9);
// by ID
G=gap.SmallGroup(120,9);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-5,20,168,2404]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^10=1,d^2=c^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of S3×Dic5 in TeX
Character table of S3×Dic5 in TeX