direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C23, C3⋊C46, C69⋊3C2, SmallGroup(138,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C23 |
Generators and relations for S3×C23
G = < a,b,c | a23=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C23
►On 69 points
Generators in S69
(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 61 62 63 64 65 66 67 68 69)
(1 27 66)(2 28 67)(3 29 68)(4 30 69)(5 31 47)(6 32 48)(7 33 49)(8 34 50)(9 35 51)(10 36 52)(11 37 53)(12 38 54)(13 39 55)(14 40 56)(15 41 57)(16 42 58)(17 43 59)(18 44 60)(19 45 61)(20 46 62)(21 24 63)(22 25 64)(23 26 65)
(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)
G:=sub<Sym(69)| (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)(47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69), (1,27,66)(2,28,67)(3,29,68)(4,30,69)(5,31,47)(6,32,48)(7,33,49)(8,34,50)(9,35,51)(10,36,52)(11,37,53)(12,38,54)(13,39,55)(14,40,56)(15,41,57)(16,42,58)(17,43,59)(18,44,60)(19,45,61)(20,46,62)(21,24,63)(22,25,64)(23,26,65), (24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)>;
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)(47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69), (1,27,66)(2,28,67)(3,29,68)(4,30,69)(5,31,47)(6,32,48)(7,33,49)(8,34,50)(9,35,51)(10,36,52)(11,37,53)(12,38,54)(13,39,55)(14,40,56)(15,41,57)(16,42,58)(17,43,59)(18,44,60)(19,45,61)(20,46,62)(21,24,63)(22,25,64)(23,26,65), (24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62) );
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),(47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69)], [(1,27,66),(2,28,67),(3,29,68),(4,30,69),(5,31,47),(6,32,48),(7,33,49),(8,34,50),(9,35,51),(10,36,52),(11,37,53),(12,38,54),(13,39,55),(14,40,56),(15,41,57),(16,42,58),(17,43,59),(18,44,60),(19,45,61),(20,46,62),(21,24,63),(22,25,64),(23,26,65)], [(24,63),(25,64),(26,65),(27,66),(28,67),(29,68),(30,69),(31,47),(32,48),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62)]])
69 conjugacy classes
| class | 1 | 2 | 3 | 23A | ··· | 23V | 46A | ··· | 46V | 69A | ··· | 69V |
| order | 1 | 2 | 3 | 23 | ··· | 23 | 46 | ··· | 46 | 69 | ··· | 69 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
69 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C23 | C46 | S3 | S3×C23 |
| kernel | S3×C23 | C69 | S3 | C3 | C23 | C1 |
| # reps | 1 | 1 | 22 | 22 | 1 | 22 |
Matrix representation of S3×C23 ►in GL2(𝔽47) generated by
| 3 | 0 |
| 0 | 3 |
| 0 | 43 |
| 12 | 46 |
| 1 | 43 |
| 0 | 46 |
G:=sub<GL(2,GF(47))| [3,0,0,3],[0,12,43,46],[1,0,43,46] >;
S3×C23 in GAP, Magma, Sage, TeX
S_3\times C_{23} % in TeX
G:=Group("S3xC23"); // GroupNames label
G:=SmallGroup(138,1);
// by ID
G=gap.SmallGroup(138,1);
# by ID
G:=PCGroup([3,-2,-23,-3,830]);
// Polycyclic
G:=Group<a,b,c|a^23=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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