direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C49, C3⋊C98, C147⋊3C2, C21.C14, C7.(S3×C7), (S3×C7).C7, SmallGroup(294,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C49 |
Generators and relations for S3×C49
G = < a,b,c | a49=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C49
►On 147 points
Generators in S147
(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)(99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147)
(1 112 80)(2 113 81)(3 114 82)(4 115 83)(5 116 84)(6 117 85)(7 118 86)(8 119 87)(9 120 88)(10 121 89)(11 122 90)(12 123 91)(13 124 92)(14 125 93)(15 126 94)(16 127 95)(17 128 96)(18 129 97)(19 130 98)(20 131 50)(21 132 51)(22 133 52)(23 134 53)(24 135 54)(25 136 55)(26 137 56)(27 138 57)(28 139 58)(29 140 59)(30 141 60)(31 142 61)(32 143 62)(33 144 63)(34 145 64)(35 146 65)(36 147 66)(37 99 67)(38 100 68)(39 101 69)(40 102 70)(41 103 71)(42 104 72)(43 105 73)(44 106 74)(45 107 75)(46 108 76)(47 109 77)(48 110 78)(49 111 79)
(50 131)(51 132)(52 133)(53 134)(54 135)(55 136)(56 137)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 145)(65 146)(66 147)(67 99)(68 100)(69 101)(70 102)(71 103)(72 104)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)(81 113)(82 114)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(89 121)(90 122)(91 123)(92 124)(93 125)(94 126)(95 127)(96 128)(97 129)(98 130)
G:=sub<Sym(147)| (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)(99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147), (1,112,80)(2,113,81)(3,114,82)(4,115,83)(5,116,84)(6,117,85)(7,118,86)(8,119,87)(9,120,88)(10,121,89)(11,122,90)(12,123,91)(13,124,92)(14,125,93)(15,126,94)(16,127,95)(17,128,96)(18,129,97)(19,130,98)(20,131,50)(21,132,51)(22,133,52)(23,134,53)(24,135,54)(25,136,55)(26,137,56)(27,138,57)(28,139,58)(29,140,59)(30,141,60)(31,142,61)(32,143,62)(33,144,63)(34,145,64)(35,146,65)(36,147,66)(37,99,67)(38,100,68)(39,101,69)(40,102,70)(41,103,71)(42,104,72)(43,105,73)(44,106,74)(45,107,75)(46,108,76)(47,109,77)(48,110,78)(49,111,79), (50,131)(51,132)(52,133)(53,134)(54,135)(55,136)(56,137)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128)(97,129)(98,130)>;
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)(99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147), (1,112,80)(2,113,81)(3,114,82)(4,115,83)(5,116,84)(6,117,85)(7,118,86)(8,119,87)(9,120,88)(10,121,89)(11,122,90)(12,123,91)(13,124,92)(14,125,93)(15,126,94)(16,127,95)(17,128,96)(18,129,97)(19,130,98)(20,131,50)(21,132,51)(22,133,52)(23,134,53)(24,135,54)(25,136,55)(26,137,56)(27,138,57)(28,139,58)(29,140,59)(30,141,60)(31,142,61)(32,143,62)(33,144,63)(34,145,64)(35,146,65)(36,147,66)(37,99,67)(38,100,68)(39,101,69)(40,102,70)(41,103,71)(42,104,72)(43,105,73)(44,106,74)(45,107,75)(46,108,76)(47,109,77)(48,110,78)(49,111,79), (50,131)(51,132)(52,133)(53,134)(54,135)(55,136)(56,137)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128)(97,129)(98,130) );
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),(99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147)], [(1,112,80),(2,113,81),(3,114,82),(4,115,83),(5,116,84),(6,117,85),(7,118,86),(8,119,87),(9,120,88),(10,121,89),(11,122,90),(12,123,91),(13,124,92),(14,125,93),(15,126,94),(16,127,95),(17,128,96),(18,129,97),(19,130,98),(20,131,50),(21,132,51),(22,133,52),(23,134,53),(24,135,54),(25,136,55),(26,137,56),(27,138,57),(28,139,58),(29,140,59),(30,141,60),(31,142,61),(32,143,62),(33,144,63),(34,145,64),(35,146,65),(36,147,66),(37,99,67),(38,100,68),(39,101,69),(40,102,70),(41,103,71),(42,104,72),(43,105,73),(44,106,74),(45,107,75),(46,108,76),(47,109,77),(48,110,78),(49,111,79)], [(50,131),(51,132),(52,133),(53,134),(54,135),(55,136),(56,137),(57,138),(58,139),(59,140),(60,141),(61,142),(62,143),(63,144),(64,145),(65,146),(66,147),(67,99),(68,100),(69,101),(70,102),(71,103),(72,104),(73,105),(74,106),(75,107),(76,108),(77,109),(78,110),(79,111),(80,112),(81,113),(82,114),(83,115),(84,116),(85,117),(86,118),(87,119),(88,120),(89,121),(90,122),(91,123),(92,124),(93,125),(94,126),(95,127),(96,128),(97,129),(98,130)]])
147 conjugacy classes
| class | 1 | 2 | 3 | 7A | ··· | 7F | 14A | ··· | 14F | 21A | ··· | 21F | 49A | ··· | 49AP | 98A | ··· | 98AP | 147A | ··· | 147AP |
| order | 1 | 2 | 3 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 49 | ··· | 49 | 98 | ··· | 98 | 147 | ··· | 147 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
147 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | ||||||
| image | C1 | C2 | C7 | C14 | C49 | C98 | S3 | S3×C7 | S3×C49 |
| kernel | S3×C49 | C147 | S3×C7 | C21 | S3 | C3 | C49 | C7 | C1 |
| # reps | 1 | 1 | 6 | 6 | 42 | 42 | 1 | 6 | 42 |
Matrix representation of S3×C49 ►in GL2(𝔽883) generated by
| 52 | 0 |
| 0 | 52 |
| 882 | 882 |
| 1 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(883))| [52,0,0,52],[882,1,882,0],[0,1,1,0] >;
S3×C49 in GAP, Magma, Sage, TeX
S_3\times C_{49} % in TeX
G:=Group("S3xC49"); // GroupNames label
G:=SmallGroup(294,3);
// by ID
G=gap.SmallGroup(294,3);
# by ID
G:=PCGroup([4,-2,-7,-7,-3,61,3139]);
// Polycyclic
G:=Group<a,b,c|a^49=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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