direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C2×C34, C51⋊4C23, C102⋊4C22, C6⋊(C2×C34), C3⋊(C22×C34), (C2×C6)⋊3C34, (C2×C102)⋊7C2, SmallGroup(408,44)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C2×C34 |
Generators and relations for S3×C2×C34
G = < a,b,c,d | a2=b34=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 108 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, S3, C6, C23, D6, C2×C6, C17, C22×S3, C34, C34, C51, C2×C34, C2×C34, S3×C17, C102, C22×C34, S3×C34, C2×C102, S3×C2×C34
Quotients: C1, C2, C22, S3, C23, D6, C17, C22×S3, C34, C2×C34, S3×C17, C22×C34, S3×C34, S3×C2×C34
►On 204 points
Generators in S204
(1 151)(2 152)(3 153)(4 154)(5 155)(6 156)(7 157)(8 158)(9 159)(10 160)(11 161)(12 162)(13 163)(14 164)(15 165)(16 166)(17 167)(18 168)(19 169)(20 170)(21 137)(22 138)(23 139)(24 140)(25 141)(26 142)(27 143)(28 144)(29 145)(30 146)(31 147)(32 148)(33 149)(34 150)(35 194)(36 195)(37 196)(38 197)(39 198)(40 199)(41 200)(42 201)(43 202)(44 203)(45 204)(46 171)(47 172)(48 173)(49 174)(50 175)(51 176)(52 177)(53 178)(54 179)(55 180)(56 181)(57 182)(58 183)(59 184)(60 185)(61 186)(62 187)(63 188)(64 189)(65 190)(66 191)(67 192)(68 193)(69 123)(70 124)(71 125)(72 126)(73 127)(74 128)(75 129)(76 130)(77 131)(78 132)(79 133)(80 134)(81 135)(82 136)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(101 121)(102 122)
(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 40 130)(2 41 131)(3 42 132)(4 43 133)(5 44 134)(6 45 135)(7 46 136)(8 47 103)(9 48 104)(10 49 105)(11 50 106)(12 51 107)(13 52 108)(14 53 109)(15 54 110)(16 55 111)(17 56 112)(18 57 113)(19 58 114)(20 59 115)(21 60 116)(22 61 117)(23 62 118)(24 63 119)(25 64 120)(26 65 121)(27 66 122)(28 67 123)(29 68 124)(30 35 125)(31 36 126)(32 37 127)(33 38 128)(34 39 129)(69 144 192)(70 145 193)(71 146 194)(72 147 195)(73 148 196)(74 149 197)(75 150 198)(76 151 199)(77 152 200)(78 153 201)(79 154 202)(80 155 203)(81 156 204)(82 157 171)(83 158 172)(84 159 173)(85 160 174)(86 161 175)(87 162 176)(88 163 177)(89 164 178)(90 165 179)(91 166 180)(92 167 181)(93 168 182)(94 169 183)(95 170 184)(96 137 185)(97 138 186)(98 139 187)(99 140 188)(100 141 189)(101 142 190)(102 143 191)
(1 168)(2 169)(3 170)(4 137)(5 138)(6 139)(7 140)(8 141)(9 142)(10 143)(11 144)(12 145)(13 146)(14 147)(15 148)(16 149)(17 150)(18 151)(19 152)(20 153)(21 154)(22 155)(23 156)(24 157)(25 158)(26 159)(27 160)(28 161)(29 162)(30 163)(31 164)(32 165)(33 166)(34 167)(35 88)(36 89)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 97)(45 98)(46 99)(47 100)(48 101)(49 102)(50 69)(51 70)(52 71)(53 72)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)(61 80)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(103 189)(104 190)(105 191)(106 192)(107 193)(108 194)(109 195)(110 196)(111 197)(112 198)(113 199)(114 200)(115 201)(116 202)(117 203)(118 204)(119 171)(120 172)(121 173)(122 174)(123 175)(124 176)(125 177)(126 178)(127 179)(128 180)(129 181)(130 182)(131 183)(132 184)(133 185)(134 186)(135 187)(136 188)
G:=sub<Sym(204)| (1,151)(2,152)(3,153)(4,154)(5,155)(6,156)(7,157)(8,158)(9,159)(10,160)(11,161)(12,162)(13,163)(14,164)(15,165)(16,166)(17,167)(18,168)(19,169)(20,170)(21,137)(22,138)(23,139)(24,140)(25,141)(26,142)(27,143)(28,144)(29,145)(30,146)(31,147)(32,148)(33,149)(34,150)(35,194)(36,195)(37,196)(38,197)(39,198)(40,199)(41,200)(42,201)(43,202)(44,203)(45,204)(46,171)(47,172)(48,173)(49,174)(50,175)(51,176)(52,177)(53,178)(54,179)(55,180)(56,181)(57,182)(58,183)(59,184)(60,185)(61,186)(62,187)(63,188)(64,189)(65,190)(66,191)(67,192)(68,193)(69,123)(70,124)(71,125)(72,126)(73,127)(74,128)(75,129)(76,130)(77,131)(78,132)(79,133)(80,134)(81,135)(82,136)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122), (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), (1,40,130)(2,41,131)(3,42,132)(4,43,133)(5,44,134)(6,45,135)(7,46,136)(8,47,103)(9,48,104)(10,49,105)(11,50,106)(12,51,107)(13,52,108)(14,53,109)(15,54,110)(16,55,111)(17,56,112)(18,57,113)(19,58,114)(20,59,115)(21,60,116)(22,61,117)(23,62,118)(24,63,119)(25,64,120)(26,65,121)(27,66,122)(28,67,123)(29,68,124)(30,35,125)(31,36,126)(32,37,127)(33,38,128)(34,39,129)(69,144,192)(70,145,193)(71,146,194)(72,147,195)(73,148,196)(74,149,197)(75,150,198)(76,151,199)(77,152,200)(78,153,201)(79,154,202)(80,155,203)(81,156,204)(82,157,171)(83,158,172)(84,159,173)(85,160,174)(86,161,175)(87,162,176)(88,163,177)(89,164,178)(90,165,179)(91,166,180)(92,167,181)(93,168,182)(94,169,183)(95,170,184)(96,137,185)(97,138,186)(98,139,187)(99,140,188)(100,141,189)(101,142,190)(102,143,191), (1,168)(2,169)(3,170)(4,137)(5,138)(6,139)(7,140)(8,141)(9,142)(10,143)(11,144)(12,145)(13,146)(14,147)(15,148)(16,149)(17,150)(18,151)(19,152)(20,153)(21,154)(22,155)(23,156)(24,157)(25,158)(26,159)(27,160)(28,161)(29,162)(30,163)(31,164)(32,165)(33,166)(34,167)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(46,99)(47,100)(48,101)(49,102)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(103,189)(104,190)(105,191)(106,192)(107,193)(108,194)(109,195)(110,196)(111,197)(112,198)(113,199)(114,200)(115,201)(116,202)(117,203)(118,204)(119,171)(120,172)(121,173)(122,174)(123,175)(124,176)(125,177)(126,178)(127,179)(128,180)(129,181)(130,182)(131,183)(132,184)(133,185)(134,186)(135,187)(136,188)>;
G:=Group( (1,151)(2,152)(3,153)(4,154)(5,155)(6,156)(7,157)(8,158)(9,159)(10,160)(11,161)(12,162)(13,163)(14,164)(15,165)(16,166)(17,167)(18,168)(19,169)(20,170)(21,137)(22,138)(23,139)(24,140)(25,141)(26,142)(27,143)(28,144)(29,145)(30,146)(31,147)(32,148)(33,149)(34,150)(35,194)(36,195)(37,196)(38,197)(39,198)(40,199)(41,200)(42,201)(43,202)(44,203)(45,204)(46,171)(47,172)(48,173)(49,174)(50,175)(51,176)(52,177)(53,178)(54,179)(55,180)(56,181)(57,182)(58,183)(59,184)(60,185)(61,186)(62,187)(63,188)(64,189)(65,190)(66,191)(67,192)(68,193)(69,123)(70,124)(71,125)(72,126)(73,127)(74,128)(75,129)(76,130)(77,131)(78,132)(79,133)(80,134)(81,135)(82,136)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122), (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), (1,40,130)(2,41,131)(3,42,132)(4,43,133)(5,44,134)(6,45,135)(7,46,136)(8,47,103)(9,48,104)(10,49,105)(11,50,106)(12,51,107)(13,52,108)(14,53,109)(15,54,110)(16,55,111)(17,56,112)(18,57,113)(19,58,114)(20,59,115)(21,60,116)(22,61,117)(23,62,118)(24,63,119)(25,64,120)(26,65,121)(27,66,122)(28,67,123)(29,68,124)(30,35,125)(31,36,126)(32,37,127)(33,38,128)(34,39,129)(69,144,192)(70,145,193)(71,146,194)(72,147,195)(73,148,196)(74,149,197)(75,150,198)(76,151,199)(77,152,200)(78,153,201)(79,154,202)(80,155,203)(81,156,204)(82,157,171)(83,158,172)(84,159,173)(85,160,174)(86,161,175)(87,162,176)(88,163,177)(89,164,178)(90,165,179)(91,166,180)(92,167,181)(93,168,182)(94,169,183)(95,170,184)(96,137,185)(97,138,186)(98,139,187)(99,140,188)(100,141,189)(101,142,190)(102,143,191), (1,168)(2,169)(3,170)(4,137)(5,138)(6,139)(7,140)(8,141)(9,142)(10,143)(11,144)(12,145)(13,146)(14,147)(15,148)(16,149)(17,150)(18,151)(19,152)(20,153)(21,154)(22,155)(23,156)(24,157)(25,158)(26,159)(27,160)(28,161)(29,162)(30,163)(31,164)(32,165)(33,166)(34,167)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(46,99)(47,100)(48,101)(49,102)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(103,189)(104,190)(105,191)(106,192)(107,193)(108,194)(109,195)(110,196)(111,197)(112,198)(113,199)(114,200)(115,201)(116,202)(117,203)(118,204)(119,171)(120,172)(121,173)(122,174)(123,175)(124,176)(125,177)(126,178)(127,179)(128,180)(129,181)(130,182)(131,183)(132,184)(133,185)(134,186)(135,187)(136,188) );
G=PermutationGroup([[(1,151),(2,152),(3,153),(4,154),(5,155),(6,156),(7,157),(8,158),(9,159),(10,160),(11,161),(12,162),(13,163),(14,164),(15,165),(16,166),(17,167),(18,168),(19,169),(20,170),(21,137),(22,138),(23,139),(24,140),(25,141),(26,142),(27,143),(28,144),(29,145),(30,146),(31,147),(32,148),(33,149),(34,150),(35,194),(36,195),(37,196),(38,197),(39,198),(40,199),(41,200),(42,201),(43,202),(44,203),(45,204),(46,171),(47,172),(48,173),(49,174),(50,175),(51,176),(52,177),(53,178),(54,179),(55,180),(56,181),(57,182),(58,183),(59,184),(60,185),(61,186),(62,187),(63,188),(64,189),(65,190),(66,191),(67,192),(68,193),(69,123),(70,124),(71,125),(72,126),(73,127),(74,128),(75,129),(76,130),(77,131),(78,132),(79,133),(80,134),(81,135),(82,136),(83,103),(84,104),(85,105),(86,106),(87,107),(88,108),(89,109),(90,110),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(101,121),(102,122)], [(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)], [(1,40,130),(2,41,131),(3,42,132),(4,43,133),(5,44,134),(6,45,135),(7,46,136),(8,47,103),(9,48,104),(10,49,105),(11,50,106),(12,51,107),(13,52,108),(14,53,109),(15,54,110),(16,55,111),(17,56,112),(18,57,113),(19,58,114),(20,59,115),(21,60,116),(22,61,117),(23,62,118),(24,63,119),(25,64,120),(26,65,121),(27,66,122),(28,67,123),(29,68,124),(30,35,125),(31,36,126),(32,37,127),(33,38,128),(34,39,129),(69,144,192),(70,145,193),(71,146,194),(72,147,195),(73,148,196),(74,149,197),(75,150,198),(76,151,199),(77,152,200),(78,153,201),(79,154,202),(80,155,203),(81,156,204),(82,157,171),(83,158,172),(84,159,173),(85,160,174),(86,161,175),(87,162,176),(88,163,177),(89,164,178),(90,165,179),(91,166,180),(92,167,181),(93,168,182),(94,169,183),(95,170,184),(96,137,185),(97,138,186),(98,139,187),(99,140,188),(100,141,189),(101,142,190),(102,143,191)], [(1,168),(2,169),(3,170),(4,137),(5,138),(6,139),(7,140),(8,141),(9,142),(10,143),(11,144),(12,145),(13,146),(14,147),(15,148),(16,149),(17,150),(18,151),(19,152),(20,153),(21,154),(22,155),(23,156),(24,157),(25,158),(26,159),(27,160),(28,161),(29,162),(30,163),(31,164),(32,165),(33,166),(34,167),(35,88),(36,89),(37,90),(38,91),(39,92),(40,93),(41,94),(42,95),(43,96),(44,97),(45,98),(46,99),(47,100),(48,101),(49,102),(50,69),(51,70),(52,71),(53,72),(54,73),(55,74),(56,75),(57,76),(58,77),(59,78),(60,79),(61,80),(62,81),(63,82),(64,83),(65,84),(66,85),(67,86),(68,87),(103,189),(104,190),(105,191),(106,192),(107,193),(108,194),(109,195),(110,196),(111,197),(112,198),(113,199),(114,200),(115,201),(116,202),(117,203),(118,204),(119,171),(120,172),(121,173),(122,174),(123,175),(124,176),(125,177),(126,178),(127,179),(128,180),(129,181),(130,182),(131,183),(132,184),(133,185),(134,186),(135,187),(136,188)]])
204 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 6A | 6B | 6C | 17A | ··· | 17P | 34A | ··· | 34AV | 34AW | ··· | 34DH | 51A | ··· | 51P | 102A | ··· | 102AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 6 | 6 | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
204 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C17 | C34 | C34 | S3 | D6 | S3×C17 | S3×C34 |
| kernel | S3×C2×C34 | S3×C34 | C2×C102 | C22×S3 | D6 | C2×C6 | C2×C34 | C34 | C22 | C2 |
| # reps | 1 | 6 | 1 | 16 | 96 | 16 | 1 | 3 | 16 | 48 |
Matrix representation of S3×C2×C34 ►in GL3(𝔽103) generated by
| 102 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 95 | 0 |
| 0 | 0 | 95 |
| 1 | 0 | 0 |
| 0 | 0 | 102 |
| 0 | 1 | 102 |
| 1 | 0 | 0 |
| 0 | 0 | 102 |
| 0 | 102 | 0 |
G:=sub<GL(3,GF(103))| [102,0,0,0,1,0,0,0,1],[1,0,0,0,95,0,0,0,95],[1,0,0,0,0,1,0,102,102],[1,0,0,0,0,102,0,102,0] >;
S3×C2×C34 in GAP, Magma, Sage, TeX
S_3\times C_2\times C_{34} % in TeX
G:=Group("S3xC2xC34"); // GroupNames label
G:=SmallGroup(408,44);
// by ID
G=gap.SmallGroup(408,44);
# by ID
G:=PCGroup([5,-2,-2,-2,-17,-3,6804]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^34=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations