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G = S3×C19order 114 = 2·3·19

Direct product of C19 and S3

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

Aliases: S3×C19, C3⋊C38, C573C2, SmallGroup(114,3)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C19
C1C3C57 — S3×C19
C3 — S3×C19
C1C19

Generators and relations for S3×C19
 G = < a,b,c | a19=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C38

Smallest permutation representation of S3×C19
On 57 points
Generators in S57
(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)
(1 28 42)(2 29 43)(3 30 44)(4 31 45)(5 32 46)(6 33 47)(7 34 48)(8 35 49)(9 36 50)(10 37 51)(11 38 52)(12 20 53)(13 21 54)(14 22 55)(15 23 56)(16 24 57)(17 25 39)(18 26 40)(19 27 41)
(20 53)(21 54)(22 55)(23 56)(24 57)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)

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

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

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

57 conjugacy classes

class 1  2  3 19A···19R38A···38R57A···57R
order12319···1938···3857···57
size1321···13···32···2

57 irreducible representations

dim111122
type+++
imageC1C2C19C38S3S3×C19
kernelS3×C19C57S3C3C19C1
# reps111818118

Matrix representation of S3×C19 in GL2(𝔽229) generated by

1210
0121
,
228228
10
,
10
228228
G:=sub<GL(2,GF(229))| [121,0,0,121],[228,1,228,0],[1,228,0,228] >;

S3×C19 in GAP, Magma, Sage, TeX

S_3\times C_{19}
% in TeX

G:=Group("S3xC19");
// GroupNames label

G:=SmallGroup(114,3);
// by ID

G=gap.SmallGroup(114,3);
# by ID

G:=PCGroup([3,-2,-19,-3,686]);
// Polycyclic

G:=Group<a,b,c|a^19=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×C19 in TeX

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