direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic38, C38⋊Q8, C4.11D38, C38.1C23, C22.8D38, C76.11C22, Dic19.1C22, C19⋊1(C2×Q8), (C2×C76).4C2, (C2×C4).4D19, (C2×C38).8C22, C2.3(C22×D19), (C2×Dic19).3C2, SmallGroup(304,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic38
G = < a,b,c | a2=b76=1, c2=b38, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic38
►Regular action on 304 points
Generators in S304
(1 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 103)(8 104)(9 105)(10 106)(11 107)(12 108)(13 109)(14 110)(15 111)(16 112)(17 113)(18 114)(19 115)(20 116)(21 117)(22 118)(23 119)(24 120)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 127)(32 128)(33 129)(34 130)(35 131)(36 132)(37 133)(38 134)(39 135)(40 136)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 151)(56 152)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)(73 93)(74 94)(75 95)(76 96)(153 286)(154 287)(155 288)(156 289)(157 290)(158 291)(159 292)(160 293)(161 294)(162 295)(163 296)(164 297)(165 298)(166 299)(167 300)(168 301)(169 302)(170 303)(171 304)(172 229)(173 230)(174 231)(175 232)(176 233)(177 234)(178 235)(179 236)(180 237)(181 238)(182 239)(183 240)(184 241)(185 242)(186 243)(187 244)(188 245)(189 246)(190 247)(191 248)(192 249)(193 250)(194 251)(195 252)(196 253)(197 254)(198 255)(199 256)(200 257)(201 258)(202 259)(203 260)(204 261)(205 262)(206 263)(207 264)(208 265)(209 266)(210 267)(211 268)(212 269)(213 270)(214 271)(215 272)(216 273)(217 274)(218 275)(219 276)(220 277)(221 278)(222 279)(223 280)(224 281)(225 282)(226 283)(227 284)(228 285)
(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)(229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304)
(1 210 39 172)(2 209 40 171)(3 208 41 170)(4 207 42 169)(5 206 43 168)(6 205 44 167)(7 204 45 166)(8 203 46 165)(9 202 47 164)(10 201 48 163)(11 200 49 162)(12 199 50 161)(13 198 51 160)(14 197 52 159)(15 196 53 158)(16 195 54 157)(17 194 55 156)(18 193 56 155)(19 192 57 154)(20 191 58 153)(21 190 59 228)(22 189 60 227)(23 188 61 226)(24 187 62 225)(25 186 63 224)(26 185 64 223)(27 184 65 222)(28 183 66 221)(29 182 67 220)(30 181 68 219)(31 180 69 218)(32 179 70 217)(33 178 71 216)(34 177 72 215)(35 176 73 214)(36 175 74 213)(37 174 75 212)(38 173 76 211)(77 287 115 249)(78 286 116 248)(79 285 117 247)(80 284 118 246)(81 283 119 245)(82 282 120 244)(83 281 121 243)(84 280 122 242)(85 279 123 241)(86 278 124 240)(87 277 125 239)(88 276 126 238)(89 275 127 237)(90 274 128 236)(91 273 129 235)(92 272 130 234)(93 271 131 233)(94 270 132 232)(95 269 133 231)(96 268 134 230)(97 267 135 229)(98 266 136 304)(99 265 137 303)(100 264 138 302)(101 263 139 301)(102 262 140 300)(103 261 141 299)(104 260 142 298)(105 259 143 297)(106 258 144 296)(107 257 145 295)(108 256 146 294)(109 255 147 293)(110 254 148 292)(111 253 149 291)(112 252 150 290)(113 251 151 289)(114 250 152 288)
G:=sub<Sym(304)| (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,113)(18,114)(19,115)(20,116)(21,117)(22,118)(23,119)(24,120)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(153,286)(154,287)(155,288)(156,289)(157,290)(158,291)(159,292)(160,293)(161,294)(162,295)(163,296)(164,297)(165,298)(166,299)(167,300)(168,301)(169,302)(170,303)(171,304)(172,229)(173,230)(174,231)(175,232)(176,233)(177,234)(178,235)(179,236)(180,237)(181,238)(182,239)(183,240)(184,241)(185,242)(186,243)(187,244)(188,245)(189,246)(190,247)(191,248)(192,249)(193,250)(194,251)(195,252)(196,253)(197,254)(198,255)(199,256)(200,257)(201,258)(202,259)(203,260)(204,261)(205,262)(206,263)(207,264)(208,265)(209,266)(210,267)(211,268)(212,269)(213,270)(214,271)(215,272)(216,273)(217,274)(218,275)(219,276)(220,277)(221,278)(222,279)(223,280)(224,281)(225,282)(226,283)(227,284)(228,285), (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)(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304), (1,210,39,172)(2,209,40,171)(3,208,41,170)(4,207,42,169)(5,206,43,168)(6,205,44,167)(7,204,45,166)(8,203,46,165)(9,202,47,164)(10,201,48,163)(11,200,49,162)(12,199,50,161)(13,198,51,160)(14,197,52,159)(15,196,53,158)(16,195,54,157)(17,194,55,156)(18,193,56,155)(19,192,57,154)(20,191,58,153)(21,190,59,228)(22,189,60,227)(23,188,61,226)(24,187,62,225)(25,186,63,224)(26,185,64,223)(27,184,65,222)(28,183,66,221)(29,182,67,220)(30,181,68,219)(31,180,69,218)(32,179,70,217)(33,178,71,216)(34,177,72,215)(35,176,73,214)(36,175,74,213)(37,174,75,212)(38,173,76,211)(77,287,115,249)(78,286,116,248)(79,285,117,247)(80,284,118,246)(81,283,119,245)(82,282,120,244)(83,281,121,243)(84,280,122,242)(85,279,123,241)(86,278,124,240)(87,277,125,239)(88,276,126,238)(89,275,127,237)(90,274,128,236)(91,273,129,235)(92,272,130,234)(93,271,131,233)(94,270,132,232)(95,269,133,231)(96,268,134,230)(97,267,135,229)(98,266,136,304)(99,265,137,303)(100,264,138,302)(101,263,139,301)(102,262,140,300)(103,261,141,299)(104,260,142,298)(105,259,143,297)(106,258,144,296)(107,257,145,295)(108,256,146,294)(109,255,147,293)(110,254,148,292)(111,253,149,291)(112,252,150,290)(113,251,151,289)(114,250,152,288)>;
G:=Group( (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,113)(18,114)(19,115)(20,116)(21,117)(22,118)(23,119)(24,120)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(153,286)(154,287)(155,288)(156,289)(157,290)(158,291)(159,292)(160,293)(161,294)(162,295)(163,296)(164,297)(165,298)(166,299)(167,300)(168,301)(169,302)(170,303)(171,304)(172,229)(173,230)(174,231)(175,232)(176,233)(177,234)(178,235)(179,236)(180,237)(181,238)(182,239)(183,240)(184,241)(185,242)(186,243)(187,244)(188,245)(189,246)(190,247)(191,248)(192,249)(193,250)(194,251)(195,252)(196,253)(197,254)(198,255)(199,256)(200,257)(201,258)(202,259)(203,260)(204,261)(205,262)(206,263)(207,264)(208,265)(209,266)(210,267)(211,268)(212,269)(213,270)(214,271)(215,272)(216,273)(217,274)(218,275)(219,276)(220,277)(221,278)(222,279)(223,280)(224,281)(225,282)(226,283)(227,284)(228,285), (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)(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304), (1,210,39,172)(2,209,40,171)(3,208,41,170)(4,207,42,169)(5,206,43,168)(6,205,44,167)(7,204,45,166)(8,203,46,165)(9,202,47,164)(10,201,48,163)(11,200,49,162)(12,199,50,161)(13,198,51,160)(14,197,52,159)(15,196,53,158)(16,195,54,157)(17,194,55,156)(18,193,56,155)(19,192,57,154)(20,191,58,153)(21,190,59,228)(22,189,60,227)(23,188,61,226)(24,187,62,225)(25,186,63,224)(26,185,64,223)(27,184,65,222)(28,183,66,221)(29,182,67,220)(30,181,68,219)(31,180,69,218)(32,179,70,217)(33,178,71,216)(34,177,72,215)(35,176,73,214)(36,175,74,213)(37,174,75,212)(38,173,76,211)(77,287,115,249)(78,286,116,248)(79,285,117,247)(80,284,118,246)(81,283,119,245)(82,282,120,244)(83,281,121,243)(84,280,122,242)(85,279,123,241)(86,278,124,240)(87,277,125,239)(88,276,126,238)(89,275,127,237)(90,274,128,236)(91,273,129,235)(92,272,130,234)(93,271,131,233)(94,270,132,232)(95,269,133,231)(96,268,134,230)(97,267,135,229)(98,266,136,304)(99,265,137,303)(100,264,138,302)(101,263,139,301)(102,262,140,300)(103,261,141,299)(104,260,142,298)(105,259,143,297)(106,258,144,296)(107,257,145,295)(108,256,146,294)(109,255,147,293)(110,254,148,292)(111,253,149,291)(112,252,150,290)(113,251,151,289)(114,250,152,288) );
G=PermutationGroup([[(1,97),(2,98),(3,99),(4,100),(5,101),(6,102),(7,103),(8,104),(9,105),(10,106),(11,107),(12,108),(13,109),(14,110),(15,111),(16,112),(17,113),(18,114),(19,115),(20,116),(21,117),(22,118),(23,119),(24,120),(25,121),(26,122),(27,123),(28,124),(29,125),(30,126),(31,127),(32,128),(33,129),(34,130),(35,131),(36,132),(37,133),(38,134),(39,135),(40,136),(41,137),(42,138),(43,139),(44,140),(45,141),(46,142),(47,143),(48,144),(49,145),(50,146),(51,147),(52,148),(53,149),(54,150),(55,151),(56,152),(57,77),(58,78),(59,79),(60,80),(61,81),(62,82),(63,83),(64,84),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(71,91),(72,92),(73,93),(74,94),(75,95),(76,96),(153,286),(154,287),(155,288),(156,289),(157,290),(158,291),(159,292),(160,293),(161,294),(162,295),(163,296),(164,297),(165,298),(166,299),(167,300),(168,301),(169,302),(170,303),(171,304),(172,229),(173,230),(174,231),(175,232),(176,233),(177,234),(178,235),(179,236),(180,237),(181,238),(182,239),(183,240),(184,241),(185,242),(186,243),(187,244),(188,245),(189,246),(190,247),(191,248),(192,249),(193,250),(194,251),(195,252),(196,253),(197,254),(198,255),(199,256),(200,257),(201,258),(202,259),(203,260),(204,261),(205,262),(206,263),(207,264),(208,265),(209,266),(210,267),(211,268),(212,269),(213,270),(214,271),(215,272),(216,273),(217,274),(218,275),(219,276),(220,277),(221,278),(222,279),(223,280),(224,281),(225,282),(226,283),(227,284),(228,285)], [(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),(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304)], [(1,210,39,172),(2,209,40,171),(3,208,41,170),(4,207,42,169),(5,206,43,168),(6,205,44,167),(7,204,45,166),(8,203,46,165),(9,202,47,164),(10,201,48,163),(11,200,49,162),(12,199,50,161),(13,198,51,160),(14,197,52,159),(15,196,53,158),(16,195,54,157),(17,194,55,156),(18,193,56,155),(19,192,57,154),(20,191,58,153),(21,190,59,228),(22,189,60,227),(23,188,61,226),(24,187,62,225),(25,186,63,224),(26,185,64,223),(27,184,65,222),(28,183,66,221),(29,182,67,220),(30,181,68,219),(31,180,69,218),(32,179,70,217),(33,178,71,216),(34,177,72,215),(35,176,73,214),(36,175,74,213),(37,174,75,212),(38,173,76,211),(77,287,115,249),(78,286,116,248),(79,285,117,247),(80,284,118,246),(81,283,119,245),(82,282,120,244),(83,281,121,243),(84,280,122,242),(85,279,123,241),(86,278,124,240),(87,277,125,239),(88,276,126,238),(89,275,127,237),(90,274,128,236),(91,273,129,235),(92,272,130,234),(93,271,131,233),(94,270,132,232),(95,269,133,231),(96,268,134,230),(97,267,135,229),(98,266,136,304),(99,265,137,303),(100,264,138,302),(101,263,139,301),(102,262,140,300),(103,261,141,299),(104,260,142,298),(105,259,143,297),(106,258,144,296),(107,257,145,295),(108,256,146,294),(109,255,147,293),(110,254,148,292),(111,253,149,291),(112,252,150,290),(113,251,151,289),(114,250,152,288)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 19A | ··· | 19I | 38A | ··· | 38AA | 76A | ··· | 76AJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 38 | 38 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | C2 | Q8 | D19 | D38 | D38 | Dic38 |
| kernel | C2×Dic38 | Dic38 | C2×Dic19 | C2×C76 | C38 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 9 | 18 | 9 | 36 |
Matrix representation of C2×Dic38 ►in GL3(𝔽229) generated by
| 228 | 0 | 0 |
| 0 | 228 | 0 |
| 0 | 0 | 228 |
| 228 | 0 | 0 |
| 0 | 80 | 33 |
| 0 | 85 | 224 |
| 228 | 0 | 0 |
| 0 | 73 | 146 |
| 0 | 227 | 156 |
G:=sub<GL(3,GF(229))| [228,0,0,0,228,0,0,0,228],[228,0,0,0,80,85,0,33,224],[228,0,0,0,73,227,0,146,156] >;
C2×Dic38 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{38} % in TeX
G:=Group("C2xDic38"); // GroupNames label
G:=SmallGroup(304,27);
// by ID
G=gap.SmallGroup(304,27);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,40,182,42,7204]);
// Polycyclic
G:=Group<a,b,c|a^2=b^76=1,c^2=b^38,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export