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

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

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

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

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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