direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C59, C4⋊C118, C236⋊3C2, C22⋊C118, C118.6C22, (C2×C118)⋊1C2, C2.1(C2×C118), SmallGroup(472,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C59
G = < a,b,c | a59=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C59
►On 236 points
Generators in S236
(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)(178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236)
(1 165 188 115)(2 166 189 116)(3 167 190 117)(4 168 191 118)(5 169 192 60)(6 170 193 61)(7 171 194 62)(8 172 195 63)(9 173 196 64)(10 174 197 65)(11 175 198 66)(12 176 199 67)(13 177 200 68)(14 119 201 69)(15 120 202 70)(16 121 203 71)(17 122 204 72)(18 123 205 73)(19 124 206 74)(20 125 207 75)(21 126 208 76)(22 127 209 77)(23 128 210 78)(24 129 211 79)(25 130 212 80)(26 131 213 81)(27 132 214 82)(28 133 215 83)(29 134 216 84)(30 135 217 85)(31 136 218 86)(32 137 219 87)(33 138 220 88)(34 139 221 89)(35 140 222 90)(36 141 223 91)(37 142 224 92)(38 143 225 93)(39 144 226 94)(40 145 227 95)(41 146 228 96)(42 147 229 97)(43 148 230 98)(44 149 231 99)(45 150 232 100)(46 151 233 101)(47 152 234 102)(48 153 235 103)(49 154 236 104)(50 155 178 105)(51 156 179 106)(52 157 180 107)(53 158 181 108)(54 159 182 109)(55 160 183 110)(56 161 184 111)(57 162 185 112)(58 163 186 113)(59 164 187 114)
(60 169)(61 170)(62 171)(63 172)(64 173)(65 174)(66 175)(67 176)(68 177)(69 119)(70 120)(71 121)(72 122)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 129)(80 130)(81 131)(82 132)(83 133)(84 134)(85 135)(86 136)(87 137)(88 138)(89 139)(90 140)(91 141)(92 142)(93 143)(94 144)(95 145)(96 146)(97 147)(98 148)(99 149)(100 150)(101 151)(102 152)(103 153)(104 154)(105 155)(106 156)(107 157)(108 158)(109 159)(110 160)(111 161)(112 162)(113 163)(114 164)(115 165)(116 166)(117 167)(118 168)
G:=sub<Sym(236)| (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)(178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236), (1,165,188,115)(2,166,189,116)(3,167,190,117)(4,168,191,118)(5,169,192,60)(6,170,193,61)(7,171,194,62)(8,172,195,63)(9,173,196,64)(10,174,197,65)(11,175,198,66)(12,176,199,67)(13,177,200,68)(14,119,201,69)(15,120,202,70)(16,121,203,71)(17,122,204,72)(18,123,205,73)(19,124,206,74)(20,125,207,75)(21,126,208,76)(22,127,209,77)(23,128,210,78)(24,129,211,79)(25,130,212,80)(26,131,213,81)(27,132,214,82)(28,133,215,83)(29,134,216,84)(30,135,217,85)(31,136,218,86)(32,137,219,87)(33,138,220,88)(34,139,221,89)(35,140,222,90)(36,141,223,91)(37,142,224,92)(38,143,225,93)(39,144,226,94)(40,145,227,95)(41,146,228,96)(42,147,229,97)(43,148,230,98)(44,149,231,99)(45,150,232,100)(46,151,233,101)(47,152,234,102)(48,153,235,103)(49,154,236,104)(50,155,178,105)(51,156,179,106)(52,157,180,107)(53,158,181,108)(54,159,182,109)(55,160,183,110)(56,161,184,111)(57,162,185,112)(58,163,186,113)(59,164,187,114), (60,169)(61,170)(62,171)(63,172)(64,173)(65,174)(66,175)(67,176)(68,177)(69,119)(70,120)(71,121)(72,122)(73,123)(74,124)(75,125)(76,126)(77,127)(78,128)(79,129)(80,130)(81,131)(82,132)(83,133)(84,134)(85,135)(86,136)(87,137)(88,138)(89,139)(90,140)(91,141)(92,142)(93,143)(94,144)(95,145)(96,146)(97,147)(98,148)(99,149)(100,150)(101,151)(102,152)(103,153)(104,154)(105,155)(106,156)(107,157)(108,158)(109,159)(110,160)(111,161)(112,162)(113,163)(114,164)(115,165)(116,166)(117,167)(118,168)>;
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)(178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236), (1,165,188,115)(2,166,189,116)(3,167,190,117)(4,168,191,118)(5,169,192,60)(6,170,193,61)(7,171,194,62)(8,172,195,63)(9,173,196,64)(10,174,197,65)(11,175,198,66)(12,176,199,67)(13,177,200,68)(14,119,201,69)(15,120,202,70)(16,121,203,71)(17,122,204,72)(18,123,205,73)(19,124,206,74)(20,125,207,75)(21,126,208,76)(22,127,209,77)(23,128,210,78)(24,129,211,79)(25,130,212,80)(26,131,213,81)(27,132,214,82)(28,133,215,83)(29,134,216,84)(30,135,217,85)(31,136,218,86)(32,137,219,87)(33,138,220,88)(34,139,221,89)(35,140,222,90)(36,141,223,91)(37,142,224,92)(38,143,225,93)(39,144,226,94)(40,145,227,95)(41,146,228,96)(42,147,229,97)(43,148,230,98)(44,149,231,99)(45,150,232,100)(46,151,233,101)(47,152,234,102)(48,153,235,103)(49,154,236,104)(50,155,178,105)(51,156,179,106)(52,157,180,107)(53,158,181,108)(54,159,182,109)(55,160,183,110)(56,161,184,111)(57,162,185,112)(58,163,186,113)(59,164,187,114), (60,169)(61,170)(62,171)(63,172)(64,173)(65,174)(66,175)(67,176)(68,177)(69,119)(70,120)(71,121)(72,122)(73,123)(74,124)(75,125)(76,126)(77,127)(78,128)(79,129)(80,130)(81,131)(82,132)(83,133)(84,134)(85,135)(86,136)(87,137)(88,138)(89,139)(90,140)(91,141)(92,142)(93,143)(94,144)(95,145)(96,146)(97,147)(98,148)(99,149)(100,150)(101,151)(102,152)(103,153)(104,154)(105,155)(106,156)(107,157)(108,158)(109,159)(110,160)(111,161)(112,162)(113,163)(114,164)(115,165)(116,166)(117,167)(118,168) );
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),(178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236)], [(1,165,188,115),(2,166,189,116),(3,167,190,117),(4,168,191,118),(5,169,192,60),(6,170,193,61),(7,171,194,62),(8,172,195,63),(9,173,196,64),(10,174,197,65),(11,175,198,66),(12,176,199,67),(13,177,200,68),(14,119,201,69),(15,120,202,70),(16,121,203,71),(17,122,204,72),(18,123,205,73),(19,124,206,74),(20,125,207,75),(21,126,208,76),(22,127,209,77),(23,128,210,78),(24,129,211,79),(25,130,212,80),(26,131,213,81),(27,132,214,82),(28,133,215,83),(29,134,216,84),(30,135,217,85),(31,136,218,86),(32,137,219,87),(33,138,220,88),(34,139,221,89),(35,140,222,90),(36,141,223,91),(37,142,224,92),(38,143,225,93),(39,144,226,94),(40,145,227,95),(41,146,228,96),(42,147,229,97),(43,148,230,98),(44,149,231,99),(45,150,232,100),(46,151,233,101),(47,152,234,102),(48,153,235,103),(49,154,236,104),(50,155,178,105),(51,156,179,106),(52,157,180,107),(53,158,181,108),(54,159,182,109),(55,160,183,110),(56,161,184,111),(57,162,185,112),(58,163,186,113),(59,164,187,114)], [(60,169),(61,170),(62,171),(63,172),(64,173),(65,174),(66,175),(67,176),(68,177),(69,119),(70,120),(71,121),(72,122),(73,123),(74,124),(75,125),(76,126),(77,127),(78,128),(79,129),(80,130),(81,131),(82,132),(83,133),(84,134),(85,135),(86,136),(87,137),(88,138),(89,139),(90,140),(91,141),(92,142),(93,143),(94,144),(95,145),(96,146),(97,147),(98,148),(99,149),(100,150),(101,151),(102,152),(103,153),(104,154),(105,155),(106,156),(107,157),(108,158),(109,159),(110,160),(111,161),(112,162),(113,163),(114,164),(115,165),(116,166),(117,167),(118,168)]])
295 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 59A | ··· | 59BF | 118A | ··· | 118BF | 118BG | ··· | 118FR | 236A | ··· | 236BF |
| order | 1 | 2 | 2 | 2 | 4 | 59 | ··· | 59 | 118 | ··· | 118 | 118 | ··· | 118 | 236 | ··· | 236 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
295 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C59 | C118 | C118 | D4 | D4×C59 |
| kernel | D4×C59 | C236 | C2×C118 | D4 | C4 | C22 | C59 | C1 |
| # reps | 1 | 1 | 2 | 58 | 58 | 116 | 1 | 58 |
Matrix representation of D4×C59 ►in GL2(𝔽709) generated by
| 479 | 0 |
| 0 | 479 |
| 623 | 508 |
| 707 | 86 |
| 1 | 623 |
| 0 | 708 |
G:=sub<GL(2,GF(709))| [479,0,0,479],[623,707,508,86],[1,0,623,708] >;
D4×C59 in GAP, Magma, Sage, TeX
D_4\times C_{59} % in TeX
G:=Group("D4xC59"); // GroupNames label
G:=SmallGroup(472,9);
// by ID
G=gap.SmallGroup(472,9);
# by ID
G:=PCGroup([4,-2,-2,-59,-2,1905]);
// Polycyclic
G:=Group<a,b,c|a^59=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export