direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic37, C74⋊2C4, C2.2D74, C22.D37, C74.4C22, C37⋊3(C2×C4), (C2×C74).C2, SmallGroup(296,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C37 — C2×Dic37 |
Generators and relations for C2×Dic37
G = < a,b,c | a2=b74=1, c2=b37, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic37
►Regular action on 296 points
Generators in S296
(1 138)(2 139)(3 140)(4 141)(5 142)(6 143)(7 144)(8 145)(9 146)(10 147)(11 148)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 85)(23 86)(24 87)(25 88)(26 89)(27 90)(28 91)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 99)(37 100)(38 101)(39 102)(40 103)(41 104)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 121)(59 122)(60 123)(61 124)(62 125)(63 126)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)(73 136)(74 137)(149 223)(150 224)(151 225)(152 226)(153 227)(154 228)(155 229)(156 230)(157 231)(158 232)(159 233)(160 234)(161 235)(162 236)(163 237)(164 238)(165 239)(166 240)(167 241)(168 242)(169 243)(170 244)(171 245)(172 246)(173 247)(174 248)(175 249)(176 250)(177 251)(178 252)(179 253)(180 254)(181 255)(182 256)(183 257)(184 258)(185 259)(186 260)(187 261)(188 262)(189 263)(190 264)(191 265)(192 266)(193 267)(194 268)(195 269)(196 270)(197 271)(198 272)(199 273)(200 274)(201 275)(202 276)(203 277)(204 278)(205 279)(206 280)(207 281)(208 282)(209 283)(210 284)(211 285)(212 286)(213 287)(214 288)(215 289)(216 290)(217 291)(218 292)(219 293)(220 294)(221 295)(222 296)
(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)(223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296)
(1 186 38 149)(2 185 39 222)(3 184 40 221)(4 183 41 220)(5 182 42 219)(6 181 43 218)(7 180 44 217)(8 179 45 216)(9 178 46 215)(10 177 47 214)(11 176 48 213)(12 175 49 212)(13 174 50 211)(14 173 51 210)(15 172 52 209)(16 171 53 208)(17 170 54 207)(18 169 55 206)(19 168 56 205)(20 167 57 204)(21 166 58 203)(22 165 59 202)(23 164 60 201)(24 163 61 200)(25 162 62 199)(26 161 63 198)(27 160 64 197)(28 159 65 196)(29 158 66 195)(30 157 67 194)(31 156 68 193)(32 155 69 192)(33 154 70 191)(34 153 71 190)(35 152 72 189)(36 151 73 188)(37 150 74 187)(75 249 112 286)(76 248 113 285)(77 247 114 284)(78 246 115 283)(79 245 116 282)(80 244 117 281)(81 243 118 280)(82 242 119 279)(83 241 120 278)(84 240 121 277)(85 239 122 276)(86 238 123 275)(87 237 124 274)(88 236 125 273)(89 235 126 272)(90 234 127 271)(91 233 128 270)(92 232 129 269)(93 231 130 268)(94 230 131 267)(95 229 132 266)(96 228 133 265)(97 227 134 264)(98 226 135 263)(99 225 136 262)(100 224 137 261)(101 223 138 260)(102 296 139 259)(103 295 140 258)(104 294 141 257)(105 293 142 256)(106 292 143 255)(107 291 144 254)(108 290 145 253)(109 289 146 252)(110 288 147 251)(111 287 148 250)
G:=sub<Sym(296)| (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,145)(9,146)(10,147)(11,148)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135)(73,136)(74,137)(149,223)(150,224)(151,225)(152,226)(153,227)(154,228)(155,229)(156,230)(157,231)(158,232)(159,233)(160,234)(161,235)(162,236)(163,237)(164,238)(165,239)(166,240)(167,241)(168,242)(169,243)(170,244)(171,245)(172,246)(173,247)(174,248)(175,249)(176,250)(177,251)(178,252)(179,253)(180,254)(181,255)(182,256)(183,257)(184,258)(185,259)(186,260)(187,261)(188,262)(189,263)(190,264)(191,265)(192,266)(193,267)(194,268)(195,269)(196,270)(197,271)(198,272)(199,273)(200,274)(201,275)(202,276)(203,277)(204,278)(205,279)(206,280)(207,281)(208,282)(209,283)(210,284)(211,285)(212,286)(213,287)(214,288)(215,289)(216,290)(217,291)(218,292)(219,293)(220,294)(221,295)(222,296), (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)(223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296), (1,186,38,149)(2,185,39,222)(3,184,40,221)(4,183,41,220)(5,182,42,219)(6,181,43,218)(7,180,44,217)(8,179,45,216)(9,178,46,215)(10,177,47,214)(11,176,48,213)(12,175,49,212)(13,174,50,211)(14,173,51,210)(15,172,52,209)(16,171,53,208)(17,170,54,207)(18,169,55,206)(19,168,56,205)(20,167,57,204)(21,166,58,203)(22,165,59,202)(23,164,60,201)(24,163,61,200)(25,162,62,199)(26,161,63,198)(27,160,64,197)(28,159,65,196)(29,158,66,195)(30,157,67,194)(31,156,68,193)(32,155,69,192)(33,154,70,191)(34,153,71,190)(35,152,72,189)(36,151,73,188)(37,150,74,187)(75,249,112,286)(76,248,113,285)(77,247,114,284)(78,246,115,283)(79,245,116,282)(80,244,117,281)(81,243,118,280)(82,242,119,279)(83,241,120,278)(84,240,121,277)(85,239,122,276)(86,238,123,275)(87,237,124,274)(88,236,125,273)(89,235,126,272)(90,234,127,271)(91,233,128,270)(92,232,129,269)(93,231,130,268)(94,230,131,267)(95,229,132,266)(96,228,133,265)(97,227,134,264)(98,226,135,263)(99,225,136,262)(100,224,137,261)(101,223,138,260)(102,296,139,259)(103,295,140,258)(104,294,141,257)(105,293,142,256)(106,292,143,255)(107,291,144,254)(108,290,145,253)(109,289,146,252)(110,288,147,251)(111,287,148,250)>;
G:=Group( (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,145)(9,146)(10,147)(11,148)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135)(73,136)(74,137)(149,223)(150,224)(151,225)(152,226)(153,227)(154,228)(155,229)(156,230)(157,231)(158,232)(159,233)(160,234)(161,235)(162,236)(163,237)(164,238)(165,239)(166,240)(167,241)(168,242)(169,243)(170,244)(171,245)(172,246)(173,247)(174,248)(175,249)(176,250)(177,251)(178,252)(179,253)(180,254)(181,255)(182,256)(183,257)(184,258)(185,259)(186,260)(187,261)(188,262)(189,263)(190,264)(191,265)(192,266)(193,267)(194,268)(195,269)(196,270)(197,271)(198,272)(199,273)(200,274)(201,275)(202,276)(203,277)(204,278)(205,279)(206,280)(207,281)(208,282)(209,283)(210,284)(211,285)(212,286)(213,287)(214,288)(215,289)(216,290)(217,291)(218,292)(219,293)(220,294)(221,295)(222,296), (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)(223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296), (1,186,38,149)(2,185,39,222)(3,184,40,221)(4,183,41,220)(5,182,42,219)(6,181,43,218)(7,180,44,217)(8,179,45,216)(9,178,46,215)(10,177,47,214)(11,176,48,213)(12,175,49,212)(13,174,50,211)(14,173,51,210)(15,172,52,209)(16,171,53,208)(17,170,54,207)(18,169,55,206)(19,168,56,205)(20,167,57,204)(21,166,58,203)(22,165,59,202)(23,164,60,201)(24,163,61,200)(25,162,62,199)(26,161,63,198)(27,160,64,197)(28,159,65,196)(29,158,66,195)(30,157,67,194)(31,156,68,193)(32,155,69,192)(33,154,70,191)(34,153,71,190)(35,152,72,189)(36,151,73,188)(37,150,74,187)(75,249,112,286)(76,248,113,285)(77,247,114,284)(78,246,115,283)(79,245,116,282)(80,244,117,281)(81,243,118,280)(82,242,119,279)(83,241,120,278)(84,240,121,277)(85,239,122,276)(86,238,123,275)(87,237,124,274)(88,236,125,273)(89,235,126,272)(90,234,127,271)(91,233,128,270)(92,232,129,269)(93,231,130,268)(94,230,131,267)(95,229,132,266)(96,228,133,265)(97,227,134,264)(98,226,135,263)(99,225,136,262)(100,224,137,261)(101,223,138,260)(102,296,139,259)(103,295,140,258)(104,294,141,257)(105,293,142,256)(106,292,143,255)(107,291,144,254)(108,290,145,253)(109,289,146,252)(110,288,147,251)(111,287,148,250) );
G=PermutationGroup([[(1,138),(2,139),(3,140),(4,141),(5,142),(6,143),(7,144),(8,145),(9,146),(10,147),(11,148),(12,75),(13,76),(14,77),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,85),(23,86),(24,87),(25,88),(26,89),(27,90),(28,91),(29,92),(30,93),(31,94),(32,95),(33,96),(34,97),(35,98),(36,99),(37,100),(38,101),(39,102),(40,103),(41,104),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,112),(50,113),(51,114),(52,115),(53,116),(54,117),(55,118),(56,119),(57,120),(58,121),(59,122),(60,123),(61,124),(62,125),(63,126),(64,127),(65,128),(66,129),(67,130),(68,131),(69,132),(70,133),(71,134),(72,135),(73,136),(74,137),(149,223),(150,224),(151,225),(152,226),(153,227),(154,228),(155,229),(156,230),(157,231),(158,232),(159,233),(160,234),(161,235),(162,236),(163,237),(164,238),(165,239),(166,240),(167,241),(168,242),(169,243),(170,244),(171,245),(172,246),(173,247),(174,248),(175,249),(176,250),(177,251),(178,252),(179,253),(180,254),(181,255),(182,256),(183,257),(184,258),(185,259),(186,260),(187,261),(188,262),(189,263),(190,264),(191,265),(192,266),(193,267),(194,268),(195,269),(196,270),(197,271),(198,272),(199,273),(200,274),(201,275),(202,276),(203,277),(204,278),(205,279),(206,280),(207,281),(208,282),(209,283),(210,284),(211,285),(212,286),(213,287),(214,288),(215,289),(216,290),(217,291),(218,292),(219,293),(220,294),(221,295),(222,296)], [(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),(223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296)], [(1,186,38,149),(2,185,39,222),(3,184,40,221),(4,183,41,220),(5,182,42,219),(6,181,43,218),(7,180,44,217),(8,179,45,216),(9,178,46,215),(10,177,47,214),(11,176,48,213),(12,175,49,212),(13,174,50,211),(14,173,51,210),(15,172,52,209),(16,171,53,208),(17,170,54,207),(18,169,55,206),(19,168,56,205),(20,167,57,204),(21,166,58,203),(22,165,59,202),(23,164,60,201),(24,163,61,200),(25,162,62,199),(26,161,63,198),(27,160,64,197),(28,159,65,196),(29,158,66,195),(30,157,67,194),(31,156,68,193),(32,155,69,192),(33,154,70,191),(34,153,71,190),(35,152,72,189),(36,151,73,188),(37,150,74,187),(75,249,112,286),(76,248,113,285),(77,247,114,284),(78,246,115,283),(79,245,116,282),(80,244,117,281),(81,243,118,280),(82,242,119,279),(83,241,120,278),(84,240,121,277),(85,239,122,276),(86,238,123,275),(87,237,124,274),(88,236,125,273),(89,235,126,272),(90,234,127,271),(91,233,128,270),(92,232,129,269),(93,231,130,268),(94,230,131,267),(95,229,132,266),(96,228,133,265),(97,227,134,264),(98,226,135,263),(99,225,136,262),(100,224,137,261),(101,223,138,260),(102,296,139,259),(103,295,140,258),(104,294,141,257),(105,293,142,256),(106,292,143,255),(107,291,144,254),(108,290,145,253),(109,289,146,252),(110,288,147,251),(111,287,148,250)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 37A | ··· | 37R | 74A | ··· | 74BB |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 37 | ··· | 37 | 74 | ··· | 74 |
| size | 1 | 1 | 1 | 1 | 37 | 37 | 37 | 37 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | D37 | Dic37 | D74 |
| kernel | C2×Dic37 | Dic37 | C2×C74 | C74 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 18 | 36 | 18 |
Matrix representation of C2×Dic37 ►in GL3(𝔽149) generated by
| 148 | 0 | 0 |
| 0 | 148 | 0 |
| 0 | 0 | 148 |
| 148 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 148 | 51 |
| 105 | 0 | 0 |
| 0 | 96 | 56 |
| 0 | 35 | 53 |
G:=sub<GL(3,GF(149))| [148,0,0,0,148,0,0,0,148],[148,0,0,0,0,148,0,1,51],[105,0,0,0,96,35,0,56,53] >;
C2×Dic37 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{37} % in TeX
G:=Group("C2xDic37"); // GroupNames label
G:=SmallGroup(296,7);
// by ID
G=gap.SmallGroup(296,7);
# by ID
G:=PCGroup([4,-2,-2,-2,-37,16,4611]);
// Polycyclic
G:=Group<a,b,c|a^2=b^74=1,c^2=b^37,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export