direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C43, C3⋊C86, C129⋊3C2, SmallGroup(258,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C43 |
Generators and relations for S3×C43
G = < a,b,c | a43=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C43
►On 129 points
Generators in S129
(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 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129)
(1 112 51)(2 113 52)(3 114 53)(4 115 54)(5 116 55)(6 117 56)(7 118 57)(8 119 58)(9 120 59)(10 121 60)(11 122 61)(12 123 62)(13 124 63)(14 125 64)(15 126 65)(16 127 66)(17 128 67)(18 129 68)(19 87 69)(20 88 70)(21 89 71)(22 90 72)(23 91 73)(24 92 74)(25 93 75)(26 94 76)(27 95 77)(28 96 78)(29 97 79)(30 98 80)(31 99 81)(32 100 82)(33 101 83)(34 102 84)(35 103 85)(36 104 86)(37 105 44)(38 106 45)(39 107 46)(40 108 47)(41 109 48)(42 110 49)(43 111 50)
(44 105)(45 106)(46 107)(47 108)(48 109)(49 110)(50 111)(51 112)(52 113)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 121)(61 122)(62 123)(63 124)(64 125)(65 126)(66 127)(67 128)(68 129)(69 87)(70 88)(71 89)(72 90)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)
G:=sub<Sym(129)| (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)(87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129), (1,112,51)(2,113,52)(3,114,53)(4,115,54)(5,116,55)(6,117,56)(7,118,57)(8,119,58)(9,120,59)(10,121,60)(11,122,61)(12,123,62)(13,124,63)(14,125,64)(15,126,65)(16,127,66)(17,128,67)(18,129,68)(19,87,69)(20,88,70)(21,89,71)(22,90,72)(23,91,73)(24,92,74)(25,93,75)(26,94,76)(27,95,77)(28,96,78)(29,97,79)(30,98,80)(31,99,81)(32,100,82)(33,101,83)(34,102,84)(35,103,85)(36,104,86)(37,105,44)(38,106,45)(39,107,46)(40,108,47)(41,109,48)(42,110,49)(43,111,50), (44,105)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,121)(61,122)(62,123)(63,124)(64,125)(65,126)(66,127)(67,128)(68,129)(69,87)(70,88)(71,89)(72,90)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)>;
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)(87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129), (1,112,51)(2,113,52)(3,114,53)(4,115,54)(5,116,55)(6,117,56)(7,118,57)(8,119,58)(9,120,59)(10,121,60)(11,122,61)(12,123,62)(13,124,63)(14,125,64)(15,126,65)(16,127,66)(17,128,67)(18,129,68)(19,87,69)(20,88,70)(21,89,71)(22,90,72)(23,91,73)(24,92,74)(25,93,75)(26,94,76)(27,95,77)(28,96,78)(29,97,79)(30,98,80)(31,99,81)(32,100,82)(33,101,83)(34,102,84)(35,103,85)(36,104,86)(37,105,44)(38,106,45)(39,107,46)(40,108,47)(41,109,48)(42,110,49)(43,111,50), (44,105)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,121)(61,122)(62,123)(63,124)(64,125)(65,126)(66,127)(67,128)(68,129)(69,87)(70,88)(71,89)(72,90)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104) );
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),(87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129)], [(1,112,51),(2,113,52),(3,114,53),(4,115,54),(5,116,55),(6,117,56),(7,118,57),(8,119,58),(9,120,59),(10,121,60),(11,122,61),(12,123,62),(13,124,63),(14,125,64),(15,126,65),(16,127,66),(17,128,67),(18,129,68),(19,87,69),(20,88,70),(21,89,71),(22,90,72),(23,91,73),(24,92,74),(25,93,75),(26,94,76),(27,95,77),(28,96,78),(29,97,79),(30,98,80),(31,99,81),(32,100,82),(33,101,83),(34,102,84),(35,103,85),(36,104,86),(37,105,44),(38,106,45),(39,107,46),(40,108,47),(41,109,48),(42,110,49),(43,111,50)], [(44,105),(45,106),(46,107),(47,108),(48,109),(49,110),(50,111),(51,112),(52,113),(53,114),(54,115),(55,116),(56,117),(57,118),(58,119),(59,120),(60,121),(61,122),(62,123),(63,124),(64,125),(65,126),(66,127),(67,128),(68,129),(69,87),(70,88),(71,89),(72,90),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102),(85,103),(86,104)]])
129 conjugacy classes
| class | 1 | 2 | 3 | 43A | ··· | 43AP | 86A | ··· | 86AP | 129A | ··· | 129AP |
| order | 1 | 2 | 3 | 43 | ··· | 43 | 86 | ··· | 86 | 129 | ··· | 129 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
129 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C43 | C86 | S3 | S3×C43 |
| kernel | S3×C43 | C129 | S3 | C3 | C43 | C1 |
| # reps | 1 | 1 | 42 | 42 | 1 | 42 |
Matrix representation of S3×C43 ►in GL2(𝔽1033) generated by
| 335 | 0 |
| 0 | 335 |
| 1032 | 1032 |
| 1 | 0 |
| 1 | 0 |
| 1032 | 1032 |
G:=sub<GL(2,GF(1033))| [335,0,0,335],[1032,1,1032,0],[1,1032,0,1032] >;
S3×C43 in GAP, Magma, Sage, TeX
S_3\times C_{43} % in TeX
G:=Group("S3xC43"); // GroupNames label
G:=SmallGroup(258,3);
// by ID
G=gap.SmallGroup(258,3);
# by ID
G:=PCGroup([3,-2,-43,-3,1550]);
// Polycyclic
G:=Group<a,b,c|a^43=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export