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G = S3×C20order 120 = 23·3·5

Direct product of C20 and S3

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

Aliases: S3×C20, C606C2, D6.C10, C122C10, Dic3C20, C10.14D6, Dic32C10, C30.19C22, C31(C2×C20), C159(C2×C4), C4(C5×Dic3), C2.1(S3×C10), C6.2(C2×C10), C20(C5×Dic3), (S3×C10).2C2, (C5×Dic3)⋊5C2, SmallGroup(120,22)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C20
C1C3C6C30S3×C10 — S3×C20
C3 — S3×C20
C1C20

Generators and relations for S3×C20
 G = < a,b,c | a20=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C4
3C22
3C10
3C10
3C2×C4
3C20
3C2×C10
3C2×C20

Smallest permutation representation of S3×C20
On 60 points
Generators in S60
(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 21 48)(2 22 49)(3 23 50)(4 24 51)(5 25 52)(6 26 53)(7 27 54)(8 28 55)(9 29 56)(10 30 57)(11 31 58)(12 32 59)(13 33 60)(14 34 41)(15 35 42)(16 36 43)(17 37 44)(18 38 45)(19 39 46)(20 40 47)
(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)

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

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

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

S3×C20 is a maximal subgroup of   D6.Dic5  D205S3  D60⋊C2  D6.D10

60 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D5A5B5C5D 6 10A10B10C10D10E···10L12A12B15A15B15C15D20A···20H20I···20P30A30B30C30D60A···60H
order122234444555561010101010···1012121515151520···2020···203030303060···60
size1133211331111211113···32222221···13···322222···2

60 irreducible representations

dim1111111111222222
type++++++
imageC1C2C2C2C4C5C10C10C10C20S3D6C4×S3C5×S3S3×C10S3×C20
kernelS3×C20C5×Dic3C60S3×C10C5×S3C4×S3Dic3C12D6S3C20C10C5C4C2C1
# reps11114444416112448

Matrix representation of S3×C20 in GL2(𝔽41) generated by

210
021
,
02
2040
,
12
040
G:=sub<GL(2,GF(41))| [21,0,0,21],[0,20,2,40],[1,0,2,40] >;

S3×C20 in GAP, Magma, Sage, TeX

S_3\times C_{20}
% in TeX

G:=Group("S3xC20");
// GroupNames label

G:=SmallGroup(120,22);
// by ID

G=gap.SmallGroup(120,22);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-3,106,2004]);
// Polycyclic

G:=Group<a,b,c|a^20=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of S3×C20 in TeX

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