direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C29, C3⋊C58, C87⋊3C2, SmallGroup(174,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C29 |
Generators and relations for S3×C29
G = < a,b,c | a29=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C29
►On 87 points
Generators in S87
(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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87)
(1 69 42)(2 70 43)(3 71 44)(4 72 45)(5 73 46)(6 74 47)(7 75 48)(8 76 49)(9 77 50)(10 78 51)(11 79 52)(12 80 53)(13 81 54)(14 82 55)(15 83 56)(16 84 57)(17 85 58)(18 86 30)(19 87 31)(20 59 32)(21 60 33)(22 61 34)(23 62 35)(24 63 36)(25 64 37)(26 65 38)(27 66 39)(28 67 40)(29 68 41)
(30 86)(31 87)(32 59)(33 60)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 81)(55 82)(56 83)(57 84)(58 85)
G:=sub<Sym(87)| (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)(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87), (1,69,42)(2,70,43)(3,71,44)(4,72,45)(5,73,46)(6,74,47)(7,75,48)(8,76,49)(9,77,50)(10,78,51)(11,79,52)(12,80,53)(13,81,54)(14,82,55)(15,83,56)(16,84,57)(17,85,58)(18,86,30)(19,87,31)(20,59,32)(21,60,33)(22,61,34)(23,62,35)(24,63,36)(25,64,37)(26,65,38)(27,66,39)(28,67,40)(29,68,41), (30,86)(31,87)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81)(55,82)(56,83)(57,84)(58,85)>;
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)(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87), (1,69,42)(2,70,43)(3,71,44)(4,72,45)(5,73,46)(6,74,47)(7,75,48)(8,76,49)(9,77,50)(10,78,51)(11,79,52)(12,80,53)(13,81,54)(14,82,55)(15,83,56)(16,84,57)(17,85,58)(18,86,30)(19,87,31)(20,59,32)(21,60,33)(22,61,34)(23,62,35)(24,63,36)(25,64,37)(26,65,38)(27,66,39)(28,67,40)(29,68,41), (30,86)(31,87)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81)(55,82)(56,83)(57,84)(58,85) );
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),(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87)], [(1,69,42),(2,70,43),(3,71,44),(4,72,45),(5,73,46),(6,74,47),(7,75,48),(8,76,49),(9,77,50),(10,78,51),(11,79,52),(12,80,53),(13,81,54),(14,82,55),(15,83,56),(16,84,57),(17,85,58),(18,86,30),(19,87,31),(20,59,32),(21,60,33),(22,61,34),(23,62,35),(24,63,36),(25,64,37),(26,65,38),(27,66,39),(28,67,40),(29,68,41)], [(30,86),(31,87),(32,59),(33,60),(34,61),(35,62),(36,63),(37,64),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(46,73),(47,74),(48,75),(49,76),(50,77),(51,78),(52,79),(53,80),(54,81),(55,82),(56,83),(57,84),(58,85)]])
87 conjugacy classes
| class | 1 | 2 | 3 | 29A | ··· | 29AB | 58A | ··· | 58AB | 87A | ··· | 87AB |
| order | 1 | 2 | 3 | 29 | ··· | 29 | 58 | ··· | 58 | 87 | ··· | 87 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
87 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C29 | C58 | S3 | S3×C29 |
| kernel | S3×C29 | C87 | S3 | C3 | C29 | C1 |
| # reps | 1 | 1 | 28 | 28 | 1 | 28 |
Matrix representation of S3×C29 ►in GL2(𝔽349) generated by
| 263 | 0 |
| 0 | 263 |
| 0 | 348 |
| 1 | 348 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(349))| [263,0,0,263],[0,1,348,348],[0,1,1,0] >;
S3×C29 in GAP, Magma, Sage, TeX
S_3\times C_{29} % in TeX
G:=Group("S3xC29"); // GroupNames label
G:=SmallGroup(174,1);
// by ID
G=gap.SmallGroup(174,1);
# by ID
G:=PCGroup([3,-2,-29,-3,1046]);
// Polycyclic
G:=Group<a,b,c|a^29=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export