direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D59, C236⋊2C2, D118.C2, C2.1D118, Dic59⋊2C2, C118.2C22, C59⋊1(C2×C4), SmallGroup(472,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C59 — C4×D59 |
Generators and relations for C4×D59
G = < a,b,c | a4=b59=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D59
►On 236 points
Generators in S236
(1 220 76 162)(2 221 77 163)(3 222 78 164)(4 223 79 165)(5 224 80 166)(6 225 81 167)(7 226 82 168)(8 227 83 169)(9 228 84 170)(10 229 85 171)(11 230 86 172)(12 231 87 173)(13 232 88 174)(14 233 89 175)(15 234 90 176)(16 235 91 177)(17 236 92 119)(18 178 93 120)(19 179 94 121)(20 180 95 122)(21 181 96 123)(22 182 97 124)(23 183 98 125)(24 184 99 126)(25 185 100 127)(26 186 101 128)(27 187 102 129)(28 188 103 130)(29 189 104 131)(30 190 105 132)(31 191 106 133)(32 192 107 134)(33 193 108 135)(34 194 109 136)(35 195 110 137)(36 196 111 138)(37 197 112 139)(38 198 113 140)(39 199 114 141)(40 200 115 142)(41 201 116 143)(42 202 117 144)(43 203 118 145)(44 204 60 146)(45 205 61 147)(46 206 62 148)(47 207 63 149)(48 208 64 150)(49 209 65 151)(50 210 66 152)(51 211 67 153)(52 212 68 154)(53 213 69 155)(54 214 70 156)(55 215 71 157)(56 216 72 158)(57 217 73 159)(58 218 74 160)(59 219 75 161)
(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 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 52)(9 51)(10 50)(11 49)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(60 91)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(92 118)(93 117)(94 116)(95 115)(96 114)(97 113)(98 112)(99 111)(100 110)(101 109)(102 108)(103 107)(104 106)(119 145)(120 144)(121 143)(122 142)(123 141)(124 140)(125 139)(126 138)(127 137)(128 136)(129 135)(130 134)(131 133)(146 177)(147 176)(148 175)(149 174)(150 173)(151 172)(152 171)(153 170)(154 169)(155 168)(156 167)(157 166)(158 165)(159 164)(160 163)(161 162)(178 202)(179 201)(180 200)(181 199)(182 198)(183 197)(184 196)(185 195)(186 194)(187 193)(188 192)(189 191)(203 236)(204 235)(205 234)(206 233)(207 232)(208 231)(209 230)(210 229)(211 228)(212 227)(213 226)(214 225)(215 224)(216 223)(217 222)(218 221)(219 220)
G:=sub<Sym(236)| (1,220,76,162)(2,221,77,163)(3,222,78,164)(4,223,79,165)(5,224,80,166)(6,225,81,167)(7,226,82,168)(8,227,83,169)(9,228,84,170)(10,229,85,171)(11,230,86,172)(12,231,87,173)(13,232,88,174)(14,233,89,175)(15,234,90,176)(16,235,91,177)(17,236,92,119)(18,178,93,120)(19,179,94,121)(20,180,95,122)(21,181,96,123)(22,182,97,124)(23,183,98,125)(24,184,99,126)(25,185,100,127)(26,186,101,128)(27,187,102,129)(28,188,103,130)(29,189,104,131)(30,190,105,132)(31,191,106,133)(32,192,107,134)(33,193,108,135)(34,194,109,136)(35,195,110,137)(36,196,111,138)(37,197,112,139)(38,198,113,140)(39,199,114,141)(40,200,115,142)(41,201,116,143)(42,202,117,144)(43,203,118,145)(44,204,60,146)(45,205,61,147)(46,206,62,148)(47,207,63,149)(48,208,64,150)(49,209,65,151)(50,210,66,152)(51,211,67,153)(52,212,68,154)(53,213,69,155)(54,214,70,156)(55,215,71,157)(56,216,72,158)(57,217,73,159)(58,218,74,160)(59,219,75,161), (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), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,50)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(92,118)(93,117)(94,116)(95,115)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)(119,145)(120,144)(121,143)(122,142)(123,141)(124,140)(125,139)(126,138)(127,137)(128,136)(129,135)(130,134)(131,133)(146,177)(147,176)(148,175)(149,174)(150,173)(151,172)(152,171)(153,170)(154,169)(155,168)(156,167)(157,166)(158,165)(159,164)(160,163)(161,162)(178,202)(179,201)(180,200)(181,199)(182,198)(183,197)(184,196)(185,195)(186,194)(187,193)(188,192)(189,191)(203,236)(204,235)(205,234)(206,233)(207,232)(208,231)(209,230)(210,229)(211,228)(212,227)(213,226)(214,225)(215,224)(216,223)(217,222)(218,221)(219,220)>;
G:=Group( (1,220,76,162)(2,221,77,163)(3,222,78,164)(4,223,79,165)(5,224,80,166)(6,225,81,167)(7,226,82,168)(8,227,83,169)(9,228,84,170)(10,229,85,171)(11,230,86,172)(12,231,87,173)(13,232,88,174)(14,233,89,175)(15,234,90,176)(16,235,91,177)(17,236,92,119)(18,178,93,120)(19,179,94,121)(20,180,95,122)(21,181,96,123)(22,182,97,124)(23,183,98,125)(24,184,99,126)(25,185,100,127)(26,186,101,128)(27,187,102,129)(28,188,103,130)(29,189,104,131)(30,190,105,132)(31,191,106,133)(32,192,107,134)(33,193,108,135)(34,194,109,136)(35,195,110,137)(36,196,111,138)(37,197,112,139)(38,198,113,140)(39,199,114,141)(40,200,115,142)(41,201,116,143)(42,202,117,144)(43,203,118,145)(44,204,60,146)(45,205,61,147)(46,206,62,148)(47,207,63,149)(48,208,64,150)(49,209,65,151)(50,210,66,152)(51,211,67,153)(52,212,68,154)(53,213,69,155)(54,214,70,156)(55,215,71,157)(56,216,72,158)(57,217,73,159)(58,218,74,160)(59,219,75,161), (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), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,50)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(92,118)(93,117)(94,116)(95,115)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)(119,145)(120,144)(121,143)(122,142)(123,141)(124,140)(125,139)(126,138)(127,137)(128,136)(129,135)(130,134)(131,133)(146,177)(147,176)(148,175)(149,174)(150,173)(151,172)(152,171)(153,170)(154,169)(155,168)(156,167)(157,166)(158,165)(159,164)(160,163)(161,162)(178,202)(179,201)(180,200)(181,199)(182,198)(183,197)(184,196)(185,195)(186,194)(187,193)(188,192)(189,191)(203,236)(204,235)(205,234)(206,233)(207,232)(208,231)(209,230)(210,229)(211,228)(212,227)(213,226)(214,225)(215,224)(216,223)(217,222)(218,221)(219,220) );
G=PermutationGroup([[(1,220,76,162),(2,221,77,163),(3,222,78,164),(4,223,79,165),(5,224,80,166),(6,225,81,167),(7,226,82,168),(8,227,83,169),(9,228,84,170),(10,229,85,171),(11,230,86,172),(12,231,87,173),(13,232,88,174),(14,233,89,175),(15,234,90,176),(16,235,91,177),(17,236,92,119),(18,178,93,120),(19,179,94,121),(20,180,95,122),(21,181,96,123),(22,182,97,124),(23,183,98,125),(24,184,99,126),(25,185,100,127),(26,186,101,128),(27,187,102,129),(28,188,103,130),(29,189,104,131),(30,190,105,132),(31,191,106,133),(32,192,107,134),(33,193,108,135),(34,194,109,136),(35,195,110,137),(36,196,111,138),(37,197,112,139),(38,198,113,140),(39,199,114,141),(40,200,115,142),(41,201,116,143),(42,202,117,144),(43,203,118,145),(44,204,60,146),(45,205,61,147),(46,206,62,148),(47,207,63,149),(48,208,64,150),(49,209,65,151),(50,210,66,152),(51,211,67,153),(52,212,68,154),(53,213,69,155),(54,214,70,156),(55,215,71,157),(56,216,72,158),(57,217,73,159),(58,218,74,160),(59,219,75,161)], [(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)], [(1,59),(2,58),(3,57),(4,56),(5,55),(6,54),(7,53),(8,52),(9,51),(10,50),(11,49),(12,48),(13,47),(14,46),(15,45),(16,44),(17,43),(18,42),(19,41),(20,40),(21,39),(22,38),(23,37),(24,36),(25,35),(26,34),(27,33),(28,32),(29,31),(60,91),(61,90),(62,89),(63,88),(64,87),(65,86),(66,85),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,77),(75,76),(92,118),(93,117),(94,116),(95,115),(96,114),(97,113),(98,112),(99,111),(100,110),(101,109),(102,108),(103,107),(104,106),(119,145),(120,144),(121,143),(122,142),(123,141),(124,140),(125,139),(126,138),(127,137),(128,136),(129,135),(130,134),(131,133),(146,177),(147,176),(148,175),(149,174),(150,173),(151,172),(152,171),(153,170),(154,169),(155,168),(156,167),(157,166),(158,165),(159,164),(160,163),(161,162),(178,202),(179,201),(180,200),(181,199),(182,198),(183,197),(184,196),(185,195),(186,194),(187,193),(188,192),(189,191),(203,236),(204,235),(205,234),(206,233),(207,232),(208,231),(209,230),(210,229),(211,228),(212,227),(213,226),(214,225),(215,224),(216,223),(217,222),(218,221),(219,220)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 59A | ··· | 59AC | 118A | ··· | 118AC | 236A | ··· | 236BF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 59 | ··· | 59 | 118 | ··· | 118 | 236 | ··· | 236 |
| size | 1 | 1 | 59 | 59 | 1 | 1 | 59 | 59 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D59 | D118 | C4×D59 |
| kernel | C4×D59 | Dic59 | C236 | D118 | D59 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 29 | 29 | 58 |
Matrix representation of C4×D59 ►in GL3(𝔽709) generated by
| 613 | 0 | 0 |
| 0 | 708 | 0 |
| 0 | 0 | 708 |
| 1 | 0 | 0 |
| 0 | 27 | 1 |
| 0 | 554 | 572 |
| 708 | 0 | 0 |
| 0 | 572 | 708 |
| 0 | 334 | 137 |
G:=sub<GL(3,GF(709))| [613,0,0,0,708,0,0,0,708],[1,0,0,0,27,554,0,1,572],[708,0,0,0,572,334,0,708,137] >;
C4×D59 in GAP, Magma, Sage, TeX
C_4\times D_{59} % in TeX
G:=Group("C4xD59"); // GroupNames label
G:=SmallGroup(472,4);
// by ID
G=gap.SmallGroup(472,4);
# by ID
G:=PCGroup([4,-2,-2,-2,-59,21,7427]);
// Polycyclic
G:=Group<a,b,c|a^4=b^59=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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