direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C68, D6.C34, C204⋊6C2, C12⋊2C34, C34.14D6, Dic3⋊2C34, C102.19C22, C3⋊1(C2×C68), C51⋊9(C2×C4), C2.1(S3×C34), C6.2(C2×C34), (S3×C34).2C2, (Dic3×C17)⋊5C2, SmallGroup(408,21)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C68 |
Generators and relations for S3×C68
G = < a,b,c | a68=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C68
►On 204 points
Generators in S204
(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)(137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204)
(1 190 128)(2 191 129)(3 192 130)(4 193 131)(5 194 132)(6 195 133)(7 196 134)(8 197 135)(9 198 136)(10 199 69)(11 200 70)(12 201 71)(13 202 72)(14 203 73)(15 204 74)(16 137 75)(17 138 76)(18 139 77)(19 140 78)(20 141 79)(21 142 80)(22 143 81)(23 144 82)(24 145 83)(25 146 84)(26 147 85)(27 148 86)(28 149 87)(29 150 88)(30 151 89)(31 152 90)(32 153 91)(33 154 92)(34 155 93)(35 156 94)(36 157 95)(37 158 96)(38 159 97)(39 160 98)(40 161 99)(41 162 100)(42 163 101)(43 164 102)(44 165 103)(45 166 104)(46 167 105)(47 168 106)(48 169 107)(49 170 108)(50 171 109)(51 172 110)(52 173 111)(53 174 112)(54 175 113)(55 176 114)(56 177 115)(57 178 116)(58 179 117)(59 180 118)(60 181 119)(61 182 120)(62 183 121)(63 184 122)(64 185 123)(65 186 124)(66 187 125)(67 188 126)(68 189 127)
(69 199)(70 200)(71 201)(72 202)(73 203)(74 204)(75 137)(76 138)(77 139)(78 140)(79 141)(80 142)(81 143)(82 144)(83 145)(84 146)(85 147)(86 148)(87 149)(88 150)(89 151)(90 152)(91 153)(92 154)(93 155)(94 156)(95 157)(96 158)(97 159)(98 160)(99 161)(100 162)(101 163)(102 164)(103 165)(104 166)(105 167)(106 168)(107 169)(108 170)(109 171)(110 172)(111 173)(112 174)(113 175)(114 176)(115 177)(116 178)(117 179)(118 180)(119 181)(120 182)(121 183)(122 184)(123 185)(124 186)(125 187)(126 188)(127 189)(128 190)(129 191)(130 192)(131 193)(132 194)(133 195)(134 196)(135 197)(136 198)
G:=sub<Sym(204)| (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)(137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204), (1,190,128)(2,191,129)(3,192,130)(4,193,131)(5,194,132)(6,195,133)(7,196,134)(8,197,135)(9,198,136)(10,199,69)(11,200,70)(12,201,71)(13,202,72)(14,203,73)(15,204,74)(16,137,75)(17,138,76)(18,139,77)(19,140,78)(20,141,79)(21,142,80)(22,143,81)(23,144,82)(24,145,83)(25,146,84)(26,147,85)(27,148,86)(28,149,87)(29,150,88)(30,151,89)(31,152,90)(32,153,91)(33,154,92)(34,155,93)(35,156,94)(36,157,95)(37,158,96)(38,159,97)(39,160,98)(40,161,99)(41,162,100)(42,163,101)(43,164,102)(44,165,103)(45,166,104)(46,167,105)(47,168,106)(48,169,107)(49,170,108)(50,171,109)(51,172,110)(52,173,111)(53,174,112)(54,175,113)(55,176,114)(56,177,115)(57,178,116)(58,179,117)(59,180,118)(60,181,119)(61,182,120)(62,183,121)(63,184,122)(64,185,123)(65,186,124)(66,187,125)(67,188,126)(68,189,127), (69,199)(70,200)(71,201)(72,202)(73,203)(74,204)(75,137)(76,138)(77,139)(78,140)(79,141)(80,142)(81,143)(82,144)(83,145)(84,146)(85,147)(86,148)(87,149)(88,150)(89,151)(90,152)(91,153)(92,154)(93,155)(94,156)(95,157)(96,158)(97,159)(98,160)(99,161)(100,162)(101,163)(102,164)(103,165)(104,166)(105,167)(106,168)(107,169)(108,170)(109,171)(110,172)(111,173)(112,174)(113,175)(114,176)(115,177)(116,178)(117,179)(118,180)(119,181)(120,182)(121,183)(122,184)(123,185)(124,186)(125,187)(126,188)(127,189)(128,190)(129,191)(130,192)(131,193)(132,194)(133,195)(134,196)(135,197)(136,198)>;
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)(137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204), (1,190,128)(2,191,129)(3,192,130)(4,193,131)(5,194,132)(6,195,133)(7,196,134)(8,197,135)(9,198,136)(10,199,69)(11,200,70)(12,201,71)(13,202,72)(14,203,73)(15,204,74)(16,137,75)(17,138,76)(18,139,77)(19,140,78)(20,141,79)(21,142,80)(22,143,81)(23,144,82)(24,145,83)(25,146,84)(26,147,85)(27,148,86)(28,149,87)(29,150,88)(30,151,89)(31,152,90)(32,153,91)(33,154,92)(34,155,93)(35,156,94)(36,157,95)(37,158,96)(38,159,97)(39,160,98)(40,161,99)(41,162,100)(42,163,101)(43,164,102)(44,165,103)(45,166,104)(46,167,105)(47,168,106)(48,169,107)(49,170,108)(50,171,109)(51,172,110)(52,173,111)(53,174,112)(54,175,113)(55,176,114)(56,177,115)(57,178,116)(58,179,117)(59,180,118)(60,181,119)(61,182,120)(62,183,121)(63,184,122)(64,185,123)(65,186,124)(66,187,125)(67,188,126)(68,189,127), (69,199)(70,200)(71,201)(72,202)(73,203)(74,204)(75,137)(76,138)(77,139)(78,140)(79,141)(80,142)(81,143)(82,144)(83,145)(84,146)(85,147)(86,148)(87,149)(88,150)(89,151)(90,152)(91,153)(92,154)(93,155)(94,156)(95,157)(96,158)(97,159)(98,160)(99,161)(100,162)(101,163)(102,164)(103,165)(104,166)(105,167)(106,168)(107,169)(108,170)(109,171)(110,172)(111,173)(112,174)(113,175)(114,176)(115,177)(116,178)(117,179)(118,180)(119,181)(120,182)(121,183)(122,184)(123,185)(124,186)(125,187)(126,188)(127,189)(128,190)(129,191)(130,192)(131,193)(132,194)(133,195)(134,196)(135,197)(136,198) );
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),(137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204)], [(1,190,128),(2,191,129),(3,192,130),(4,193,131),(5,194,132),(6,195,133),(7,196,134),(8,197,135),(9,198,136),(10,199,69),(11,200,70),(12,201,71),(13,202,72),(14,203,73),(15,204,74),(16,137,75),(17,138,76),(18,139,77),(19,140,78),(20,141,79),(21,142,80),(22,143,81),(23,144,82),(24,145,83),(25,146,84),(26,147,85),(27,148,86),(28,149,87),(29,150,88),(30,151,89),(31,152,90),(32,153,91),(33,154,92),(34,155,93),(35,156,94),(36,157,95),(37,158,96),(38,159,97),(39,160,98),(40,161,99),(41,162,100),(42,163,101),(43,164,102),(44,165,103),(45,166,104),(46,167,105),(47,168,106),(48,169,107),(49,170,108),(50,171,109),(51,172,110),(52,173,111),(53,174,112),(54,175,113),(55,176,114),(56,177,115),(57,178,116),(58,179,117),(59,180,118),(60,181,119),(61,182,120),(62,183,121),(63,184,122),(64,185,123),(65,186,124),(66,187,125),(67,188,126),(68,189,127)], [(69,199),(70,200),(71,201),(72,202),(73,203),(74,204),(75,137),(76,138),(77,139),(78,140),(79,141),(80,142),(81,143),(82,144),(83,145),(84,146),(85,147),(86,148),(87,149),(88,150),(89,151),(90,152),(91,153),(92,154),(93,155),(94,156),(95,157),(96,158),(97,159),(98,160),(99,161),(100,162),(101,163),(102,164),(103,165),(104,166),(105,167),(106,168),(107,169),(108,170),(109,171),(110,172),(111,173),(112,174),(113,175),(114,176),(115,177),(116,178),(117,179),(118,180),(119,181),(120,182),(121,183),(122,184),(123,185),(124,186),(125,187),(126,188),(127,189),(128,190),(129,191),(130,192),(131,193),(132,194),(133,195),(134,196),(135,197),(136,198)]])
204 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 12A | 12B | 17A | ··· | 17P | 34A | ··· | 34P | 34Q | ··· | 34AV | 51A | ··· | 51P | 68A | ··· | 68AF | 68AG | ··· | 68BL | 102A | ··· | 102P | 204A | ··· | 204AF |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 12 | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 68 | ··· | 68 | 102 | ··· | 102 | 204 | ··· | 204 |
| size | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 3 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
204 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C17 | C34 | C34 | C34 | C68 | S3 | D6 | C4×S3 | S3×C17 | S3×C34 | S3×C68 |
| kernel | S3×C68 | Dic3×C17 | C204 | S3×C34 | S3×C17 | C4×S3 | Dic3 | C12 | D6 | S3 | C68 | C34 | C17 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 16 | 16 | 16 | 16 | 64 | 1 | 1 | 2 | 16 | 16 | 32 |
Matrix representation of S3×C68 ►in GL3(𝔽409) generated by
| 143 | 0 | 0 |
| 0 | 340 | 0 |
| 0 | 0 | 340 |
| 1 | 0 | 0 |
| 0 | 0 | 408 |
| 0 | 1 | 408 |
| 408 | 0 | 0 |
| 0 | 1 | 408 |
| 0 | 0 | 408 |
G:=sub<GL(3,GF(409))| [143,0,0,0,340,0,0,0,340],[1,0,0,0,0,1,0,408,408],[408,0,0,0,1,0,0,408,408] >;
S3×C68 in GAP, Magma, Sage, TeX
S_3\times C_{68} % in TeX
G:=Group("S3xC68"); // GroupNames label
G:=SmallGroup(408,21);
// by ID
G=gap.SmallGroup(408,21);
# by ID
G:=PCGroup([5,-2,-2,-17,-2,-3,346,6804]);
// Polycyclic
G:=Group<a,b,c|a^68=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export