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G = S3xC2xC10order 120 = 23·3·5

Direct product of C2xC10 and S3

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

Aliases: S3xC2xC10, C15:4C23, C30:4C22, C6:(C2xC10), C3:(C22xC10), (C2xC30):7C2, (C2xC6):3C10, SmallGroup(120,45)

Series: Derived Chief Lower central Upper central

C1C3 — S3xC2xC10
C1C3C15C5xS3S3xC10 — S3xC2xC10
C3 — S3xC2xC10
C1C2xC10

Generators and relations for S3xC2xC10
 G = < a,b,c,d | a2=b10=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 108 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C5, S3, C6, C23, C10, C10, D6, C2xC6, C15, C2xC10, C2xC10, C22xS3, C5xS3, C30, C22xC10, S3xC10, C2xC30, S3xC2xC10
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C2xC10, C22xS3, C5xS3, C22xC10, S3xC10, S3xC2xC10

Smallest permutation representation of S3xC2xC10
On 60 points
Generators in S60
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 49)(12 50)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 40)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)
(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 49 39)(2 50 40)(3 41 31)(4 42 32)(5 43 33)(6 44 34)(7 45 35)(8 46 36)(9 47 37)(10 48 38)(11 30 51)(12 21 52)(13 22 53)(14 23 54)(15 24 55)(16 25 56)(17 26 57)(18 27 58)(19 28 59)(20 29 60)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 39)(12 40)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 50)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 49)

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

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

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

S3xC2xC10 is a maximal subgroup of   D6:Dic5

60 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 5A5B5C5D6A6B6C10A···10L10M···10AB15A15B15C15D30A···30L
order122222223555566610···1010···101515151530···30
size11113333211112221···13···322222···2

60 irreducible representations

dim1111112222
type+++++
imageC1C2C2C5C10C10S3D6C5xS3S3xC10
kernelS3xC2xC10S3xC10C2xC30C22xS3D6C2xC6C2xC10C10C22C2
# reps161424413412

Matrix representation of S3xC2xC10 in GL4(F31) generated by

30000
03000
0010
0001
,
30000
0100
0020
0002
,
1000
0100
00030
00130
,
1000
0100
00130
00030
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,1,0,0,0,0,1],[30,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,30,30],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,30,30] >;

S3xC2xC10 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_{10}
% in TeX

G:=Group("S3xC2xC10");
// GroupNames label

G:=SmallGroup(120,45);
// by ID

G=gap.SmallGroup(120,45);
# by ID

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

G:=Group<a,b,c,d|a^2=b^10=c^3=d^2=1,a*b=b*a,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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