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## G = S3×Dic5order 120 = 23·3·5

### Direct product of S3 and Dic5

Aliases: S3×Dic5, D6.D5, C6.2D10, C10.2D6, Dic153C2, C30.2C22, C54(C4×S3), C155(C2×C4), (C5×S3)⋊2C4, (S3×C10).C2, C2.2(S3×D5), C31(C2×Dic5), (C3×Dic5)⋊1C2, SmallGroup(120,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — S3×Dic5
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — S3×Dic5
 Lower central C15 — S3×Dic5
 Upper central C1 — C2

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

Smallest permutation representation of S3×Dic5
On 60 points
Generators in S60
(1 27 40)(2 28 31)(3 29 32)(4 30 33)(5 21 34)(6 22 35)(7 23 36)(8 24 37)(9 25 38)(10 26 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 39)(22 40)(23 31)(24 32)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(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 48 26 43)(22 47 27 42)(23 46 28 41)(24 45 29 50)(25 44 30 49)

G:=sub<Sym(60)| (1,27,40)(2,28,31)(3,29,32)(4,30,33)(5,21,34)(6,22,35)(7,23,36)(8,24,37)(9,25,38)(10,26,39)(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)(21,39)(22,40)(23,31)(24,32)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(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)(21,48,26,43)(22,47,27,42)(23,46,28,41)(24,45,29,50)(25,44,30,49)>;

G:=Group( (1,27,40)(2,28,31)(3,29,32)(4,30,33)(5,21,34)(6,22,35)(7,23,36)(8,24,37)(9,25,38)(10,26,39)(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)(21,39)(22,40)(23,31)(24,32)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(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)(21,48,26,43)(22,47,27,42)(23,46,28,41)(24,45,29,50)(25,44,30,49) );

G=PermutationGroup([(1,27,40),(2,28,31),(3,29,32),(4,30,33),(5,21,34),(6,22,35),(7,23,36),(8,24,37),(9,25,38),(10,26,39),(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),(21,39),(22,40),(23,31),(24,32),(25,33),(26,34),(27,35),(28,36),(29,37),(30,38),(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),(21,48,26,43),(22,47,27,42),(23,46,28,41),(24,45,29,50),(25,44,30,49)])

S3×Dic5 is a maximal subgroup of
D6.F5  D12⋊D5  D125D5  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

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