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G = S3×D20  order 240 = 24·3·5

Direct product of S3 and D20

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

Aliases: S3×D20, C20⋊4D6, D10⋊1D6, D60⋊7C2, C12⋊1D10, C60⋊2C22, Dic3⋊3D10, D6.10D10, D30⋊2C22, C30.13C23, C5⋊1(S3×D4), C4⋊1(S3×D5), C3⋊1(C2×D20), C15⋊2(C2×D4), (C4×S3)⋊3D5, (C5×S3)⋊1D4, (S3×C20)⋊3C2, (C3×D20)⋊3C2, C3⋊D20⋊3C2, (C6×D5)⋊1C22, C6.13(C22×D5), C10.13(C22×S3), (C5×Dic3)⋊4C22, (S3×C10).10C22, (C2×S3×D5)⋊2C2, C2.16(C2×S3×D5), SmallGroup(240,137)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — S3×D20
Chief series
C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — S3×D20
Lower central
C15 — C30 — S3×D20
Upper central
C1 — C2 — C4

Generators and relations for S3×D20
 G = < a,b,c,d | a3=b2=c20=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 616 in 108 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, S3, C6, C6, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×D4, C20, C20, D10, D10, C2×C10, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C5×S3, C3×D5, D15, C30, D20, D20, C2×C20, C22×D5, S3×D4, C5×Dic3, C60, S3×D5, C6×D5, S3×C10, D30, C2×D20, C3⋊D20, C3×D20, S3×C20, D60, C2×S3×D5, S3×D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, D20, C22×D5, S3×D4, S3×D5, C2×D20, C2×S3×D5, S3×D20

Smallest permutation representation of S3×D20
►On 60 points
Generators in S60
(1 25 47)(2 26 48)(3 27 49)(4 28 50)(5 29 51)(6 30 52)(7 31 53)(8 32 54)(9 33 55)(10 34 56)(11 35 57)(12 36 58)(13 37 59)(14 38 60)(15 39 41)(16 40 42)(17 21 43)(18 22 44)(19 23 45)(20 24 46)

(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 41)(40 42)

(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 23)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)
 
G:=sub<Sym(60)| (1,25,47)(2,26,48)(3,27,49)(4,28,50)(5,29,51)(6,30,52)(7,31,53)(8,32,54)(9,33,55)(10,34,56)(11,35,57)(12,36,58)(13,37,59)(14,38,60)(15,39,41)(16,40,42)(17,21,43)(18,22,44)(19,23,45)(20,24,46), (21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,41)(40,42), (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)(21,23)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)>;
 
G:=Group( (1,25,47)(2,26,48)(3,27,49)(4,28,50)(5,29,51)(6,30,52)(7,31,53)(8,32,54)(9,33,55)(10,34,56)(11,35,57)(12,36,58)(13,37,59)(14,38,60)(15,39,41)(16,40,42)(17,21,43)(18,22,44)(19,23,45)(20,24,46), (21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,41)(40,42), (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)(21,23)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55) );
 
G=PermutationGroup([[(1,25,47),(2,26,48),(3,27,49),(4,28,50),(5,29,51),(6,30,52),(7,31,53),(8,32,54),(9,33,55),(10,34,56),(11,35,57),(12,36,58),(13,37,59),(14,38,60),(15,39,41),(16,40,42),(17,21,43),(18,22,44),(19,23,45),(20,24,46)], [(21,43),(22,44),(23,45),(24,46),(25,47),(26,48),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,55),(34,56),(35,57),(36,58),(37,59),(38,60),(39,41),(40,42)], [(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),(21,23),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33),(41,47),(42,46),(43,45),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55)]])
 

S3×D20 is a maximal subgroup of
 C40⋊1D6  D40⋊S3  D60.C22  D12⋊D10  D20⋊25D6  D20⋊29D6  S3×D4×D5  D20⋊14D6  D20⋊17D6
S3×D20 is a maximal quotient of
 C40⋊1D6  D40⋊S3  Dic20⋊S3  D6.1D20  D40⋊7S3  C40.2D6  D120⋊5C2  Dic3.D20  Dic3⋊4D20  Dic3⋊D20  D10⋊2Dic6  D6.D20  D60⋊14C4  D30⋊4Q8  D6⋊D20  C60⋊4D4  D6.9D20  C12⋊D20  D30⋊2D4  C60⋊6D4  C20⋊4Dic6  D6⋊4D20  D30⋊5D4

39 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B6A6B6C10A10B10C10D10E10F 12 15A15B20A20B20C20D20E20F20G20H30A30B60A60B60C60D
order12222222344556661010101010101215152020202020202020303060606060
size113310103030226222202022666644422226666444444

39 irreducible representations

dim1111112222222224444
type+++++++++++++++++++
imageC1C2C2C2C2C2S3D4D5D6D6D10D10D10D20S3×D4S3×D5C2×S3×D5S3×D20
kernelS3×D20C3⋊D20C3×D20S3×C20D60C2×S3×D5D20C5×S3C4×S3C20D10Dic3C12D6S3C5C4C2C1
# reps1211121221222281224

Matrix representation of S3×D20 ►in GL4(𝔽61) generated by

1000
0100
006060
0010
,
1000
0100
0010
006060
,
363200
23400
0010
0001
,
1000
196000
0010
0001
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,60,1,0,0,60,0],[1,0,0,0,0,1,0,0,0,0,1,60,0,0,0,60],[36,2,0,0,32,34,0,0,0,0,1,0,0,0,0,1],[1,19,0,0,0,60,0,0,0,0,1,0,0,0,0,1] >;
 

S3×D20 in GAP, Magma, Sage, TeX 

S_3\times D_{20}
 
% in TeX
 
G:=Group("S3xD20");
 
// GroupNames label
 
G:=SmallGroup(240,137);
 
// by ID
 
G=gap.SmallGroup(240,137);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,218,50,490,6917]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^2=c^20=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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