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G = S3×D20order 240 = 24·3·5

Direct product of S3 and D20

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

Aliases: S3×D20, C204D6, D101D6, D607C2, C121D10, C602C22, Dic33D10, D6.10D10, D302C22, C30.13C23, C51(S3×D4), C41(S3×D5), C31(C2×D20), C152(C2×D4), (C4×S3)⋊3D5, (C5×S3)⋊1D4, (S3×C20)⋊3C2, (C3×D20)⋊3C2, C3⋊D203C2, (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

C1C30 — S3×D20
C1C5C15C30C6×D5C2×S3×D5 — S3×D20
C15C30 — S3×D20
C1C2C4

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 [×6], C3, C4, C4, C22 [×9], C5, S3 [×2], S3 [×2], C6, C6 [×2], C2×C4, D4 [×4], C23 [×2], D5 [×4], C10, C10 [×2], Dic3, C12, D6, D6 [×6], C2×C6 [×2], C15, C2×D4, C20, C20, D10 [×2], D10 [×6], C2×C10, C4×S3, D12, C3⋊D4 [×2], C3×D4, C22×S3 [×2], C5×S3 [×2], C3×D5 [×2], D15 [×2], C30, D20, D20 [×3], C2×C20, C22×D5 [×2], S3×D4, C5×Dic3, C60, S3×D5 [×4], C6×D5 [×2], S3×C10, D30 [×2], C2×D20, C3⋊D20 [×2], C3×D20, S3×C20, D60, C2×S3×D5 [×2], S3×D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, D20 [×2], 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 30 54)(2 31 55)(3 32 56)(4 33 57)(5 34 58)(6 35 59)(7 36 60)(8 37 41)(9 38 42)(10 39 43)(11 40 44)(12 21 45)(13 22 46)(14 23 47)(15 24 48)(16 25 49)(17 26 50)(18 27 51)(19 28 52)(20 29 53)
(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 41)(38 42)(39 43)(40 44)
(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 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)

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

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

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

S3×D20 is a maximal subgroup of
C401D6  D40⋊S3  D60.C22  D12⋊D10  D2025D6  D2029D6  S3×D4×D5  D2014D6  D2017D6
S3×D20 is a maximal quotient of
C401D6  D40⋊S3  Dic20⋊S3  D6.1D20  D407S3  C40.2D6  D1205C2  Dic3.D20  Dic34D20  Dic3⋊D20  D102Dic6  D6.D20  D6014C4  D304Q8  D6⋊D20  C604D4  D6.9D20  C12⋊D20  D302D4  C606D4  C204Dic6  D64D20  D305D4

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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