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G = S3×Dic5order 120 = 23·3·5

Direct product of S3 and Dic5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

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

C1C15 — S3×Dic5
C1C5C15C30C3×Dic5 — S3×Dic5
C15 — S3×Dic5
C1C2

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 >

3C2
3C2
3C22
5C4
15C4
3C10
3C10
15C2×C4
5C12
5Dic3
3Dic5
3C2×C10
5C4×S3
3C2×Dic5

Character table of S3×Dic5

 class 12A2B2C34A4B4C4D5A5B610A10B10C10D10E10F12A12B15A15B30A30B
 size 1133255151522222666610104444
ρ1111111111111111111111111    trivial
ρ211-1-1111-1-111111-1-1-1-1111111    linear of order 2
ρ311-1-11-1-11111111-1-1-1-1-1-11111    linear of order 2
ρ411111-1-1-1-1111111111-1-11111    linear of order 2
ρ51-11-11i-ii-i11-1-1-11-1-11i-i11-1-1    linear of order 4
ρ61-1-111i-i-ii11-1-1-1-111-1i-i11-1-1    linear of order 4
ρ71-1-111-iii-i11-1-1-1-111-1-ii11-1-1    linear of order 4
ρ81-11-11-ii-ii11-1-1-11-1-11-ii11-1-1    linear of order 4
ρ92200-1-2-20022-122000011-1-1-1-1    orthogonal lifted from D6
ρ102200-1220022-1220000-1-1-1-1-1-1    orthogonal lifted from S3
ρ11222220000-1+5/2-1-5/22-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/200-1-5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ1222-2-220000-1+5/2-1-5/22-1+5/2-1-5/21-5/21+5/21-5/21+5/200-1-5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D10
ρ1322-2-220000-1-5/2-1+5/22-1-5/2-1+5/21+5/21-5/21+5/21-5/200-1+5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D10
ρ14222220000-1-5/2-1+5/22-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/200-1+5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ152-22-220000-1-5/2-1+5/2-21+5/21-5/2-1-5/21-5/21+5/2-1+5/200-1+5/2-1-5/21+5/21-5/2    symplectic lifted from Dic5, Schur index 2
ρ162-2-2220000-1+5/2-1-5/2-21-5/21+5/21-5/2-1-5/2-1+5/21+5/200-1-5/2-1+5/21-5/21+5/2    symplectic lifted from Dic5, Schur index 2
ρ172-2-2220000-1-5/2-1+5/2-21+5/21-5/21+5/2-1+5/2-1-5/21-5/200-1+5/2-1-5/21+5/21-5/2    symplectic lifted from Dic5, Schur index 2
ρ182-22-220000-1+5/2-1-5/2-21-5/21+5/2-1+5/21+5/21-5/2-1-5/200-1-5/2-1+5/21-5/21+5/2    symplectic lifted from Dic5, Schur index 2
ρ192-200-1-2i2i00221-2-20000i-i-1-111    complex lifted from C4×S3
ρ202-200-12i-2i00221-2-20000-ii-1-111    complex lifted from C4×S3
ρ214400-20000-1+5-1-5-2-1+5-1-50000001+5/21-5/21-5/21+5/2    orthogonal lifted from S3×D5
ρ224400-20000-1-5-1+5-2-1-5-1+50000001-5/21+5/21+5/21-5/2    orthogonal lifted from S3×D5
ρ234-400-20000-1+5-1-521-51+50000001+5/21-5/2-1+5/2-1-5/2    symplectic faithful, Schur index 2
ρ244-400-20000-1-5-1+521+51-50000001-5/21+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

1000
0100
00060
00160
,
1000
0100
00060
00600
,
0100
601700
00600
00060
,
253000
283600
00110
00011
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

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