direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C6×D17, C51⋊3C23, C102⋊3C22, C34⋊(C2×C6), C17⋊(C22×C6), (C2×C34)⋊5C6, (C2×C102)⋊5C2, SmallGroup(408,43)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C2×C6×D17 |
Generators and relations for C2×C6×D17
G = < a,b,c,d | a2=b6=c17=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 416 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C6, C6, C23, C2×C6, C2×C6, C17, C22×C6, D17, C34, C51, D34, C2×C34, C3×D17, C102, C22×D17, C6×D17, C2×C102, C2×C6×D17
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C22×C6, D17, D34, C3×D17, C22×D17, C6×D17, C2×C6×D17
►On 204 points
Generators in S204
(1 170)(2 154)(3 155)(4 156)(5 157)(6 158)(7 159)(8 160)(9 161)(10 162)(11 163)(12 164)(13 165)(14 166)(15 167)(16 168)(17 169)(18 180)(19 181)(20 182)(21 183)(22 184)(23 185)(24 186)(25 187)(26 171)(27 172)(28 173)(29 174)(30 175)(31 176)(32 177)(33 178)(34 179)(35 191)(36 192)(37 193)(38 194)(39 195)(40 196)(41 197)(42 198)(43 199)(44 200)(45 201)(46 202)(47 203)(48 204)(49 188)(50 189)(51 190)(52 119)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 121)(70 122)(71 123)(72 124)(73 125)(74 126)(75 127)(76 128)(77 129)(78 130)(79 131)(80 132)(81 133)(82 134)(83 135)(84 136)(85 120)(86 144)(87 145)(88 146)(89 147)(90 148)(91 149)(92 150)(93 151)(94 152)(95 153)(96 137)(97 138)(98 139)(99 140)(100 141)(101 142)(102 143)
(1 69 46 66 22 102)(2 70 47 67 23 86)(3 71 48 68 24 87)(4 72 49 52 25 88)(5 73 50 53 26 89)(6 74 51 54 27 90)(7 75 35 55 28 91)(8 76 36 56 29 92)(9 77 37 57 30 93)(10 78 38 58 31 94)(11 79 39 59 32 95)(12 80 40 60 33 96)(13 81 41 61 34 97)(14 82 42 62 18 98)(15 83 43 63 19 99)(16 84 44 64 20 100)(17 85 45 65 21 101)(103 171 147 157 125 189)(104 172 148 158 126 190)(105 173 149 159 127 191)(106 174 150 160 128 192)(107 175 151 161 129 193)(108 176 152 162 130 194)(109 177 153 163 131 195)(110 178 137 164 132 196)(111 179 138 165 133 197)(112 180 139 166 134 198)(113 181 140 167 135 199)(114 182 141 168 136 200)(115 183 142 169 120 201)(116 184 143 170 121 202)(117 185 144 154 122 203)(118 186 145 155 123 204)(119 187 146 156 124 188)
(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)
(1 115)(2 114)(3 113)(4 112)(5 111)(6 110)(7 109)(8 108)(9 107)(10 106)(11 105)(12 104)(13 103)(14 119)(15 118)(16 117)(17 116)(18 124)(19 123)(20 122)(21 121)(22 120)(23 136)(24 135)(25 134)(26 133)(27 132)(28 131)(29 130)(30 129)(31 128)(32 127)(33 126)(34 125)(35 153)(36 152)(37 151)(38 150)(39 149)(40 148)(41 147)(42 146)(43 145)(44 144)(45 143)(46 142)(47 141)(48 140)(49 139)(50 138)(51 137)(52 166)(53 165)(54 164)(55 163)(56 162)(57 161)(58 160)(59 159)(60 158)(61 157)(62 156)(63 155)(64 154)(65 170)(66 169)(67 168)(68 167)(69 183)(70 182)(71 181)(72 180)(73 179)(74 178)(75 177)(76 176)(77 175)(78 174)(79 173)(80 172)(81 171)(82 187)(83 186)(84 185)(85 184)(86 200)(87 199)(88 198)(89 197)(90 196)(91 195)(92 194)(93 193)(94 192)(95 191)(96 190)(97 189)(98 188)(99 204)(100 203)(101 202)(102 201)
G:=sub<Sym(204)| (1,170)(2,154)(3,155)(4,156)(5,157)(6,158)(7,159)(8,160)(9,161)(10,162)(11,163)(12,164)(13,165)(14,166)(15,167)(16,168)(17,169)(18,180)(19,181)(20,182)(21,183)(22,184)(23,185)(24,186)(25,187)(26,171)(27,172)(28,173)(29,174)(30,175)(31,176)(32,177)(33,178)(34,179)(35,191)(36,192)(37,193)(38,194)(39,195)(40,196)(41,197)(42,198)(43,199)(44,200)(45,201)(46,202)(47,203)(48,204)(49,188)(50,189)(51,190)(52,119)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,121)(70,122)(71,123)(72,124)(73,125)(74,126)(75,127)(76,128)(77,129)(78,130)(79,131)(80,132)(81,133)(82,134)(83,135)(84,136)(85,120)(86,144)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,137)(97,138)(98,139)(99,140)(100,141)(101,142)(102,143), (1,69,46,66,22,102)(2,70,47,67,23,86)(3,71,48,68,24,87)(4,72,49,52,25,88)(5,73,50,53,26,89)(6,74,51,54,27,90)(7,75,35,55,28,91)(8,76,36,56,29,92)(9,77,37,57,30,93)(10,78,38,58,31,94)(11,79,39,59,32,95)(12,80,40,60,33,96)(13,81,41,61,34,97)(14,82,42,62,18,98)(15,83,43,63,19,99)(16,84,44,64,20,100)(17,85,45,65,21,101)(103,171,147,157,125,189)(104,172,148,158,126,190)(105,173,149,159,127,191)(106,174,150,160,128,192)(107,175,151,161,129,193)(108,176,152,162,130,194)(109,177,153,163,131,195)(110,178,137,164,132,196)(111,179,138,165,133,197)(112,180,139,166,134,198)(113,181,140,167,135,199)(114,182,141,168,136,200)(115,183,142,169,120,201)(116,184,143,170,121,202)(117,185,144,154,122,203)(118,186,145,155,123,204)(119,187,146,156,124,188), (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), (1,115)(2,114)(3,113)(4,112)(5,111)(6,110)(7,109)(8,108)(9,107)(10,106)(11,105)(12,104)(13,103)(14,119)(15,118)(16,117)(17,116)(18,124)(19,123)(20,122)(21,121)(22,120)(23,136)(24,135)(25,134)(26,133)(27,132)(28,131)(29,130)(30,129)(31,128)(32,127)(33,126)(34,125)(35,153)(36,152)(37,151)(38,150)(39,149)(40,148)(41,147)(42,146)(43,145)(44,144)(45,143)(46,142)(47,141)(48,140)(49,139)(50,138)(51,137)(52,166)(53,165)(54,164)(55,163)(56,162)(57,161)(58,160)(59,159)(60,158)(61,157)(62,156)(63,155)(64,154)(65,170)(66,169)(67,168)(68,167)(69,183)(70,182)(71,181)(72,180)(73,179)(74,178)(75,177)(76,176)(77,175)(78,174)(79,173)(80,172)(81,171)(82,187)(83,186)(84,185)(85,184)(86,200)(87,199)(88,198)(89,197)(90,196)(91,195)(92,194)(93,193)(94,192)(95,191)(96,190)(97,189)(98,188)(99,204)(100,203)(101,202)(102,201)>;
G:=Group( (1,170)(2,154)(3,155)(4,156)(5,157)(6,158)(7,159)(8,160)(9,161)(10,162)(11,163)(12,164)(13,165)(14,166)(15,167)(16,168)(17,169)(18,180)(19,181)(20,182)(21,183)(22,184)(23,185)(24,186)(25,187)(26,171)(27,172)(28,173)(29,174)(30,175)(31,176)(32,177)(33,178)(34,179)(35,191)(36,192)(37,193)(38,194)(39,195)(40,196)(41,197)(42,198)(43,199)(44,200)(45,201)(46,202)(47,203)(48,204)(49,188)(50,189)(51,190)(52,119)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,121)(70,122)(71,123)(72,124)(73,125)(74,126)(75,127)(76,128)(77,129)(78,130)(79,131)(80,132)(81,133)(82,134)(83,135)(84,136)(85,120)(86,144)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,137)(97,138)(98,139)(99,140)(100,141)(101,142)(102,143), (1,69,46,66,22,102)(2,70,47,67,23,86)(3,71,48,68,24,87)(4,72,49,52,25,88)(5,73,50,53,26,89)(6,74,51,54,27,90)(7,75,35,55,28,91)(8,76,36,56,29,92)(9,77,37,57,30,93)(10,78,38,58,31,94)(11,79,39,59,32,95)(12,80,40,60,33,96)(13,81,41,61,34,97)(14,82,42,62,18,98)(15,83,43,63,19,99)(16,84,44,64,20,100)(17,85,45,65,21,101)(103,171,147,157,125,189)(104,172,148,158,126,190)(105,173,149,159,127,191)(106,174,150,160,128,192)(107,175,151,161,129,193)(108,176,152,162,130,194)(109,177,153,163,131,195)(110,178,137,164,132,196)(111,179,138,165,133,197)(112,180,139,166,134,198)(113,181,140,167,135,199)(114,182,141,168,136,200)(115,183,142,169,120,201)(116,184,143,170,121,202)(117,185,144,154,122,203)(118,186,145,155,123,204)(119,187,146,156,124,188), (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), (1,115)(2,114)(3,113)(4,112)(5,111)(6,110)(7,109)(8,108)(9,107)(10,106)(11,105)(12,104)(13,103)(14,119)(15,118)(16,117)(17,116)(18,124)(19,123)(20,122)(21,121)(22,120)(23,136)(24,135)(25,134)(26,133)(27,132)(28,131)(29,130)(30,129)(31,128)(32,127)(33,126)(34,125)(35,153)(36,152)(37,151)(38,150)(39,149)(40,148)(41,147)(42,146)(43,145)(44,144)(45,143)(46,142)(47,141)(48,140)(49,139)(50,138)(51,137)(52,166)(53,165)(54,164)(55,163)(56,162)(57,161)(58,160)(59,159)(60,158)(61,157)(62,156)(63,155)(64,154)(65,170)(66,169)(67,168)(68,167)(69,183)(70,182)(71,181)(72,180)(73,179)(74,178)(75,177)(76,176)(77,175)(78,174)(79,173)(80,172)(81,171)(82,187)(83,186)(84,185)(85,184)(86,200)(87,199)(88,198)(89,197)(90,196)(91,195)(92,194)(93,193)(94,192)(95,191)(96,190)(97,189)(98,188)(99,204)(100,203)(101,202)(102,201) );
G=PermutationGroup([[(1,170),(2,154),(3,155),(4,156),(5,157),(6,158),(7,159),(8,160),(9,161),(10,162),(11,163),(12,164),(13,165),(14,166),(15,167),(16,168),(17,169),(18,180),(19,181),(20,182),(21,183),(22,184),(23,185),(24,186),(25,187),(26,171),(27,172),(28,173),(29,174),(30,175),(31,176),(32,177),(33,178),(34,179),(35,191),(36,192),(37,193),(38,194),(39,195),(40,196),(41,197),(42,198),(43,199),(44,200),(45,201),(46,202),(47,203),(48,204),(49,188),(50,189),(51,190),(52,119),(53,103),(54,104),(55,105),(56,106),(57,107),(58,108),(59,109),(60,110),(61,111),(62,112),(63,113),(64,114),(65,115),(66,116),(67,117),(68,118),(69,121),(70,122),(71,123),(72,124),(73,125),(74,126),(75,127),(76,128),(77,129),(78,130),(79,131),(80,132),(81,133),(82,134),(83,135),(84,136),(85,120),(86,144),(87,145),(88,146),(89,147),(90,148),(91,149),(92,150),(93,151),(94,152),(95,153),(96,137),(97,138),(98,139),(99,140),(100,141),(101,142),(102,143)], [(1,69,46,66,22,102),(2,70,47,67,23,86),(3,71,48,68,24,87),(4,72,49,52,25,88),(5,73,50,53,26,89),(6,74,51,54,27,90),(7,75,35,55,28,91),(8,76,36,56,29,92),(9,77,37,57,30,93),(10,78,38,58,31,94),(11,79,39,59,32,95),(12,80,40,60,33,96),(13,81,41,61,34,97),(14,82,42,62,18,98),(15,83,43,63,19,99),(16,84,44,64,20,100),(17,85,45,65,21,101),(103,171,147,157,125,189),(104,172,148,158,126,190),(105,173,149,159,127,191),(106,174,150,160,128,192),(107,175,151,161,129,193),(108,176,152,162,130,194),(109,177,153,163,131,195),(110,178,137,164,132,196),(111,179,138,165,133,197),(112,180,139,166,134,198),(113,181,140,167,135,199),(114,182,141,168,136,200),(115,183,142,169,120,201),(116,184,143,170,121,202),(117,185,144,154,122,203),(118,186,145,155,123,204),(119,187,146,156,124,188)], [(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)], [(1,115),(2,114),(3,113),(4,112),(5,111),(6,110),(7,109),(8,108),(9,107),(10,106),(11,105),(12,104),(13,103),(14,119),(15,118),(16,117),(17,116),(18,124),(19,123),(20,122),(21,121),(22,120),(23,136),(24,135),(25,134),(26,133),(27,132),(28,131),(29,130),(30,129),(31,128),(32,127),(33,126),(34,125),(35,153),(36,152),(37,151),(38,150),(39,149),(40,148),(41,147),(42,146),(43,145),(44,144),(45,143),(46,142),(47,141),(48,140),(49,139),(50,138),(51,137),(52,166),(53,165),(54,164),(55,163),(56,162),(57,161),(58,160),(59,159),(60,158),(61,157),(62,156),(63,155),(64,154),(65,170),(66,169),(67,168),(68,167),(69,183),(70,182),(71,181),(72,180),(73,179),(74,178),(75,177),(76,176),(77,175),(78,174),(79,173),(80,172),(81,171),(82,187),(83,186),(84,185),(85,184),(86,200),(87,199),(88,198),(89,197),(90,196),(91,195),(92,194),(93,193),(94,192),(95,191),(96,190),(97,189),(98,188),(99,204),(100,203),(101,202),(102,201)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 6A | ··· | 6F | 6G | ··· | 6N | 17A | ··· | 17H | 34A | ··· | 34X | 51A | ··· | 51P | 102A | ··· | 102AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | 1 | 1 | 17 | 17 | 17 | 17 | 1 | 1 | 1 | ··· | 1 | 17 | ··· | 17 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D17 | D34 | C3×D17 | C6×D17 |
| kernel | C2×C6×D17 | C6×D17 | C2×C102 | C22×D17 | D34 | C2×C34 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 6 | 1 | 2 | 12 | 2 | 8 | 24 | 16 | 48 |
Matrix representation of C2×C6×D17 ►in GL4(𝔽103) generated by
| 1 | 0 | 0 | 0 |
| 0 | 102 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 57 | 0 | 0 | 0 |
| 0 | 102 | 0 | 0 |
| 0 | 0 | 102 | 0 |
| 0 | 0 | 0 | 102 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 20 | 1 |
| 0 | 0 | 36 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 102 | 0 | 0 |
| 0 | 0 | 81 | 7 |
| 0 | 0 | 34 | 22 |
G:=sub<GL(4,GF(103))| [1,0,0,0,0,102,0,0,0,0,1,0,0,0,0,1],[57,0,0,0,0,102,0,0,0,0,102,0,0,0,0,102],[1,0,0,0,0,1,0,0,0,0,20,36,0,0,1,7],[1,0,0,0,0,102,0,0,0,0,81,34,0,0,7,22] >;
C2×C6×D17 in GAP, Magma, Sage, TeX
C_2\times C_6\times D_{17} % in TeX
G:=Group("C2xC6xD17"); // GroupNames label
G:=SmallGroup(408,43);
// by ID
G=gap.SmallGroup(408,43);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-17,9604]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^17=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