direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×D19, D57⋊C2, C3⋊1D38, C19⋊1D6, C57⋊C22, (S3×C19)⋊C2, (C3×D19)⋊C2, SmallGroup(228,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C57 — S3×D19 |
Generators and relations for S3×D19
G = < a,b,c,d | a3=b2=c19=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of S3×D19
►On 57 points
Generators in S57
(1 34 42)(2 35 43)(3 36 44)(4 37 45)(5 38 46)(6 20 47)(7 21 48)(8 22 49)(9 23 50)(10 24 51)(11 25 52)(12 26 53)(13 27 54)(14 28 55)(15 29 56)(16 30 57)(17 31 39)(18 32 40)(19 33 41)
(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 39)(32 40)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)
(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 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 28)(21 27)(22 26)(23 25)(29 38)(30 37)(31 36)(32 35)(33 34)(39 44)(40 43)(41 42)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)
G:=sub<Sym(57)| (1,34,42)(2,35,43)(3,36,44)(4,37,45)(5,38,46)(6,20,47)(7,21,48)(8,22,49)(9,23,50)(10,24,51)(11,25,52)(12,26,53)(13,27,54)(14,28,55)(15,29,56)(16,30,57)(17,31,39)(18,32,40)(19,33,41), (20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,28)(21,27)(22,26)(23,25)(29,38)(30,37)(31,36)(32,35)(33,34)(39,44)(40,43)(41,42)(45,57)(46,56)(47,55)(48,54)(49,53)(50,52)>;
G:=Group( (1,34,42)(2,35,43)(3,36,44)(4,37,45)(5,38,46)(6,20,47)(7,21,48)(8,22,49)(9,23,50)(10,24,51)(11,25,52)(12,26,53)(13,27,54)(14,28,55)(15,29,56)(16,30,57)(17,31,39)(18,32,40)(19,33,41), (20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,28)(21,27)(22,26)(23,25)(29,38)(30,37)(31,36)(32,35)(33,34)(39,44)(40,43)(41,42)(45,57)(46,56)(47,55)(48,54)(49,53)(50,52) );
G=PermutationGroup([[(1,34,42),(2,35,43),(3,36,44),(4,37,45),(5,38,46),(6,20,47),(7,21,48),(8,22,49),(9,23,50),(10,24,51),(11,25,52),(12,26,53),(13,27,54),(14,28,55),(15,29,56),(16,30,57),(17,31,39),(18,32,40),(19,33,41)], [(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,39),(32,40),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,28),(21,27),(22,26),(23,25),(29,38),(30,37),(31,36),(32,35),(33,34),(39,44),(40,43),(41,42),(45,57),(46,56),(47,55),(48,54),(49,53),(50,52)]])
S3×D19 is a maximal quotient of D57⋊C4 C57⋊D4 C3⋊D76 C19⋊D12 C57⋊Q8
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 19A | ··· | 19I | 38A | ··· | 38I | 57A | ··· | 57I |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 |
| size | 1 | 3 | 19 | 57 | 2 | 38 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D6 | D19 | D38 | S3×D19 |
| kernel | S3×D19 | S3×C19 | C3×D19 | D57 | D19 | C19 | S3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 9 | 9 | 9 |
Matrix representation of S3×D19 ►in GL4(𝔽229) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 227 | 11 |
| 0 | 0 | 83 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 146 | 228 |
| 78 | 1 | 0 | 0 |
| 120 | 210 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 58 | 123 | 0 | 0 |
| 224 | 171 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(229))| [1,0,0,0,0,1,0,0,0,0,227,83,0,0,11,1],[1,0,0,0,0,1,0,0,0,0,1,146,0,0,0,228],[78,120,0,0,1,210,0,0,0,0,1,0,0,0,0,1],[58,224,0,0,123,171,0,0,0,0,1,0,0,0,0,1] >;
S3×D19 in GAP, Magma, Sage, TeX
S_3\times D_{19} % in TeX
G:=Group("S3xD19"); // GroupNames label
G:=SmallGroup(228,8);
// by ID
G=gap.SmallGroup(228,8);
# by ID
G:=PCGroup([4,-2,-2,-3,-19,54,3459]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^19=d^2=1,b*a*b=a^-1,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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