direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C19, C3⋊C38, C57⋊3C2, SmallGroup(114,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×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
Export