direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C55, C3⋊C110, C165⋊7C2, C15⋊3C22, C33⋊7C10, SmallGroup(330,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C55 |
Generators and relations for S3×C55
G = < a,b,c | a55=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C55
►On 165 points
Generators in S165
(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)(111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165)
(1 137 66)(2 138 67)(3 139 68)(4 140 69)(5 141 70)(6 142 71)(7 143 72)(8 144 73)(9 145 74)(10 146 75)(11 147 76)(12 148 77)(13 149 78)(14 150 79)(15 151 80)(16 152 81)(17 153 82)(18 154 83)(19 155 84)(20 156 85)(21 157 86)(22 158 87)(23 159 88)(24 160 89)(25 161 90)(26 162 91)(27 163 92)(28 164 93)(29 165 94)(30 111 95)(31 112 96)(32 113 97)(33 114 98)(34 115 99)(35 116 100)(36 117 101)(37 118 102)(38 119 103)(39 120 104)(40 121 105)(41 122 106)(42 123 107)(43 124 108)(44 125 109)(45 126 110)(46 127 56)(47 128 57)(48 129 58)(49 130 59)(50 131 60)(51 132 61)(52 133 62)(53 134 63)(54 135 64)(55 136 65)
(56 127)(57 128)(58 129)(59 130)(60 131)(61 132)(62 133)(63 134)(64 135)(65 136)(66 137)(67 138)(68 139)(69 140)(70 141)(71 142)(72 143)(73 144)(74 145)(75 146)(76 147)(77 148)(78 149)(79 150)(80 151)(81 152)(82 153)(83 154)(84 155)(85 156)(86 157)(87 158)(88 159)(89 160)(90 161)(91 162)(92 163)(93 164)(94 165)(95 111)(96 112)(97 113)(98 114)(99 115)(100 116)(101 117)(102 118)(103 119)(104 120)(105 121)(106 122)(107 123)(108 124)(109 125)(110 126)
G:=sub<Sym(165)| (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)(111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165), (1,137,66)(2,138,67)(3,139,68)(4,140,69)(5,141,70)(6,142,71)(7,143,72)(8,144,73)(9,145,74)(10,146,75)(11,147,76)(12,148,77)(13,149,78)(14,150,79)(15,151,80)(16,152,81)(17,153,82)(18,154,83)(19,155,84)(20,156,85)(21,157,86)(22,158,87)(23,159,88)(24,160,89)(25,161,90)(26,162,91)(27,163,92)(28,164,93)(29,165,94)(30,111,95)(31,112,96)(32,113,97)(33,114,98)(34,115,99)(35,116,100)(36,117,101)(37,118,102)(38,119,103)(39,120,104)(40,121,105)(41,122,106)(42,123,107)(43,124,108)(44,125,109)(45,126,110)(46,127,56)(47,128,57)(48,129,58)(49,130,59)(50,131,60)(51,132,61)(52,133,62)(53,134,63)(54,135,64)(55,136,65), (56,127)(57,128)(58,129)(59,130)(60,131)(61,132)(62,133)(63,134)(64,135)(65,136)(66,137)(67,138)(68,139)(69,140)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,161)(91,162)(92,163)(93,164)(94,165)(95,111)(96,112)(97,113)(98,114)(99,115)(100,116)(101,117)(102,118)(103,119)(104,120)(105,121)(106,122)(107,123)(108,124)(109,125)(110,126)>;
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)(111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165), (1,137,66)(2,138,67)(3,139,68)(4,140,69)(5,141,70)(6,142,71)(7,143,72)(8,144,73)(9,145,74)(10,146,75)(11,147,76)(12,148,77)(13,149,78)(14,150,79)(15,151,80)(16,152,81)(17,153,82)(18,154,83)(19,155,84)(20,156,85)(21,157,86)(22,158,87)(23,159,88)(24,160,89)(25,161,90)(26,162,91)(27,163,92)(28,164,93)(29,165,94)(30,111,95)(31,112,96)(32,113,97)(33,114,98)(34,115,99)(35,116,100)(36,117,101)(37,118,102)(38,119,103)(39,120,104)(40,121,105)(41,122,106)(42,123,107)(43,124,108)(44,125,109)(45,126,110)(46,127,56)(47,128,57)(48,129,58)(49,130,59)(50,131,60)(51,132,61)(52,133,62)(53,134,63)(54,135,64)(55,136,65), (56,127)(57,128)(58,129)(59,130)(60,131)(61,132)(62,133)(63,134)(64,135)(65,136)(66,137)(67,138)(68,139)(69,140)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,161)(91,162)(92,163)(93,164)(94,165)(95,111)(96,112)(97,113)(98,114)(99,115)(100,116)(101,117)(102,118)(103,119)(104,120)(105,121)(106,122)(107,123)(108,124)(109,125)(110,126) );
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),(111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165)], [(1,137,66),(2,138,67),(3,139,68),(4,140,69),(5,141,70),(6,142,71),(7,143,72),(8,144,73),(9,145,74),(10,146,75),(11,147,76),(12,148,77),(13,149,78),(14,150,79),(15,151,80),(16,152,81),(17,153,82),(18,154,83),(19,155,84),(20,156,85),(21,157,86),(22,158,87),(23,159,88),(24,160,89),(25,161,90),(26,162,91),(27,163,92),(28,164,93),(29,165,94),(30,111,95),(31,112,96),(32,113,97),(33,114,98),(34,115,99),(35,116,100),(36,117,101),(37,118,102),(38,119,103),(39,120,104),(40,121,105),(41,122,106),(42,123,107),(43,124,108),(44,125,109),(45,126,110),(46,127,56),(47,128,57),(48,129,58),(49,130,59),(50,131,60),(51,132,61),(52,133,62),(53,134,63),(54,135,64),(55,136,65)], [(56,127),(57,128),(58,129),(59,130),(60,131),(61,132),(62,133),(63,134),(64,135),(65,136),(66,137),(67,138),(68,139),(69,140),(70,141),(71,142),(72,143),(73,144),(74,145),(75,146),(76,147),(77,148),(78,149),(79,150),(80,151),(81,152),(82,153),(83,154),(84,155),(85,156),(86,157),(87,158),(88,159),(89,160),(90,161),(91,162),(92,163),(93,164),(94,165),(95,111),(96,112),(97,113),(98,114),(99,115),(100,116),(101,117),(102,118),(103,119),(104,120),(105,121),(106,122),(107,123),(108,124),(109,125),(110,126)]])
165 conjugacy classes
| class | 1 | 2 | 3 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 11A | ··· | 11J | 15A | 15B | 15C | 15D | 22A | ··· | 22J | 33A | ··· | 33J | 55A | ··· | 55AN | 110A | ··· | 110AN | 165A | ··· | 165AN |
| order | 1 | 2 | 3 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 11 | ··· | 11 | 15 | 15 | 15 | 15 | 22 | ··· | 22 | 33 | ··· | 33 | 55 | ··· | 55 | 110 | ··· | 110 | 165 | ··· | 165 |
| size | 1 | 3 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
165 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C5 | C10 | C11 | C22 | C55 | C110 | S3 | C5×S3 | S3×C11 | S3×C55 |
| kernel | S3×C55 | C165 | S3×C11 | C33 | C5×S3 | C15 | S3 | C3 | C55 | C11 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 10 | 10 | 40 | 40 | 1 | 4 | 10 | 40 |
Matrix representation of S3×C55 ►in GL3(𝔽331) generated by
| 124 | 0 | 0 |
| 0 | 180 | 0 |
| 0 | 0 | 180 |
| 1 | 0 | 0 |
| 0 | 0 | 330 |
| 0 | 1 | 330 |
| 330 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(331))| [124,0,0,0,180,0,0,0,180],[1,0,0,0,0,1,0,330,330],[330,0,0,0,0,1,0,1,0] >;
S3×C55 in GAP, Magma, Sage, TeX
S_3\times C_{55} % in TeX
G:=Group("S3xC55"); // GroupNames label
G:=SmallGroup(330,8);
// by ID
G=gap.SmallGroup(330,8);
# by ID
G:=PCGroup([4,-2,-5,-11,-3,3523]);
// Polycyclic
G:=Group<a,b,c|a^55=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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