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## G = S3×C19order 114 = 2·3·19

### Direct product of C19 and S3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C19
 Chief series C1 — C3 — C57 — S3×C19
 Lower central C3 — S3×C19
 Upper central C1 — C19

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

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

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

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

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

57 conjugacy classes

 class 1 2 3 19A ··· 19R 38A ··· 38R 57A ··· 57R order 1 2 3 19 ··· 19 38 ··· 38 57 ··· 57 size 1 3 2 1 ··· 1 3 ··· 3 2 ··· 2

57 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C19 C38 S3 S3×C19 kernel S3×C19 C57 S3 C3 C19 C1 # reps 1 1 18 18 1 18

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

 121 0 0 121
,
 228 228 1 0
,
 1 0 228 228
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

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