direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C19⋊C8, C38⋊C8, C76.3C4, C4.14D38, C4.3Dic19, C76.14C22, C22.2Dic19, C19⋊2(C2×C8), (C2×C38).2C4, C38.6(C2×C4), (C2×C76).6C2, (C2×C4).5D19, C2.1(C2×Dic19), SmallGroup(304,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C2×C19⋊C8 |
Generators and relations for C2×C19⋊C8
G = < a,b,c | a2=b19=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×C19⋊C8
►Regular action on 304 points
Generators in S304
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 87)(12 88)(13 89)(14 90)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 97)(22 98)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 105)(30 106)(31 107)(32 108)(33 109)(34 110)(35 111)(36 112)(37 113)(38 114)(39 115)(40 116)(41 117)(42 118)(43 119)(44 120)(45 121)(46 122)(47 123)(48 124)(49 125)(50 126)(51 127)(52 128)(53 129)(54 130)(55 131)(56 132)(57 133)(58 134)(59 135)(60 136)(61 137)(62 138)(63 139)(64 140)(65 141)(66 142)(67 143)(68 144)(69 145)(70 146)(71 147)(72 148)(73 149)(74 150)(75 151)(76 152)(153 240)(154 241)(155 242)(156 243)(157 244)(158 245)(159 246)(160 247)(161 229)(162 230)(163 231)(164 232)(165 233)(166 234)(167 235)(168 236)(169 237)(170 238)(171 239)(172 250)(173 251)(174 252)(175 253)(176 254)(177 255)(178 256)(179 257)(180 258)(181 259)(182 260)(183 261)(184 262)(185 263)(186 264)(187 265)(188 266)(189 248)(190 249)(191 279)(192 280)(193 281)(194 282)(195 283)(196 284)(197 285)(198 267)(199 268)(200 269)(201 270)(202 271)(203 272)(204 273)(205 274)(206 275)(207 276)(208 277)(209 278)(210 286)(211 287)(212 288)(213 289)(214 290)(215 291)(216 292)(217 293)(218 294)(219 295)(220 296)(221 297)(222 298)(223 299)(224 300)(225 301)(226 302)(227 303)(228 304)
(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)
(1 300 58 249 20 276 39 237)(2 299 59 248 21 275 40 236)(3 298 60 266 22 274 41 235)(4 297 61 265 23 273 42 234)(5 296 62 264 24 272 43 233)(6 295 63 263 25 271 44 232)(7 294 64 262 26 270 45 231)(8 293 65 261 27 269 46 230)(9 292 66 260 28 268 47 229)(10 291 67 259 29 267 48 247)(11 290 68 258 30 285 49 246)(12 289 69 257 31 284 50 245)(13 288 70 256 32 283 51 244)(14 287 71 255 33 282 52 243)(15 286 72 254 34 281 53 242)(16 304 73 253 35 280 54 241)(17 303 74 252 36 279 55 240)(18 302 75 251 37 278 56 239)(19 301 76 250 38 277 57 238)(77 224 134 190 96 207 115 169)(78 223 135 189 97 206 116 168)(79 222 136 188 98 205 117 167)(80 221 137 187 99 204 118 166)(81 220 138 186 100 203 119 165)(82 219 139 185 101 202 120 164)(83 218 140 184 102 201 121 163)(84 217 141 183 103 200 122 162)(85 216 142 182 104 199 123 161)(86 215 143 181 105 198 124 160)(87 214 144 180 106 197 125 159)(88 213 145 179 107 196 126 158)(89 212 146 178 108 195 127 157)(90 211 147 177 109 194 128 156)(91 210 148 176 110 193 129 155)(92 228 149 175 111 192 130 154)(93 227 150 174 112 191 131 153)(94 226 151 173 113 209 132 171)(95 225 152 172 114 208 133 170)
G:=sub<Sym(304)| (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,109)(34,110)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(45,121)(46,122)(47,123)(48,124)(49,125)(50,126)(51,127)(52,128)(53,129)(54,130)(55,131)(56,132)(57,133)(58,134)(59,135)(60,136)(61,137)(62,138)(63,139)(64,140)(65,141)(66,142)(67,143)(68,144)(69,145)(70,146)(71,147)(72,148)(73,149)(74,150)(75,151)(76,152)(153,240)(154,241)(155,242)(156,243)(157,244)(158,245)(159,246)(160,247)(161,229)(162,230)(163,231)(164,232)(165,233)(166,234)(167,235)(168,236)(169,237)(170,238)(171,239)(172,250)(173,251)(174,252)(175,253)(176,254)(177,255)(178,256)(179,257)(180,258)(181,259)(182,260)(183,261)(184,262)(185,263)(186,264)(187,265)(188,266)(189,248)(190,249)(191,279)(192,280)(193,281)(194,282)(195,283)(196,284)(197,285)(198,267)(199,268)(200,269)(201,270)(202,271)(203,272)(204,273)(205,274)(206,275)(207,276)(208,277)(209,278)(210,286)(211,287)(212,288)(213,289)(214,290)(215,291)(216,292)(217,293)(218,294)(219,295)(220,296)(221,297)(222,298)(223,299)(224,300)(225,301)(226,302)(227,303)(228,304), (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), (1,300,58,249,20,276,39,237)(2,299,59,248,21,275,40,236)(3,298,60,266,22,274,41,235)(4,297,61,265,23,273,42,234)(5,296,62,264,24,272,43,233)(6,295,63,263,25,271,44,232)(7,294,64,262,26,270,45,231)(8,293,65,261,27,269,46,230)(9,292,66,260,28,268,47,229)(10,291,67,259,29,267,48,247)(11,290,68,258,30,285,49,246)(12,289,69,257,31,284,50,245)(13,288,70,256,32,283,51,244)(14,287,71,255,33,282,52,243)(15,286,72,254,34,281,53,242)(16,304,73,253,35,280,54,241)(17,303,74,252,36,279,55,240)(18,302,75,251,37,278,56,239)(19,301,76,250,38,277,57,238)(77,224,134,190,96,207,115,169)(78,223,135,189,97,206,116,168)(79,222,136,188,98,205,117,167)(80,221,137,187,99,204,118,166)(81,220,138,186,100,203,119,165)(82,219,139,185,101,202,120,164)(83,218,140,184,102,201,121,163)(84,217,141,183,103,200,122,162)(85,216,142,182,104,199,123,161)(86,215,143,181,105,198,124,160)(87,214,144,180,106,197,125,159)(88,213,145,179,107,196,126,158)(89,212,146,178,108,195,127,157)(90,211,147,177,109,194,128,156)(91,210,148,176,110,193,129,155)(92,228,149,175,111,192,130,154)(93,227,150,174,112,191,131,153)(94,226,151,173,113,209,132,171)(95,225,152,172,114,208,133,170)>;
G:=Group( (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,109)(34,110)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(45,121)(46,122)(47,123)(48,124)(49,125)(50,126)(51,127)(52,128)(53,129)(54,130)(55,131)(56,132)(57,133)(58,134)(59,135)(60,136)(61,137)(62,138)(63,139)(64,140)(65,141)(66,142)(67,143)(68,144)(69,145)(70,146)(71,147)(72,148)(73,149)(74,150)(75,151)(76,152)(153,240)(154,241)(155,242)(156,243)(157,244)(158,245)(159,246)(160,247)(161,229)(162,230)(163,231)(164,232)(165,233)(166,234)(167,235)(168,236)(169,237)(170,238)(171,239)(172,250)(173,251)(174,252)(175,253)(176,254)(177,255)(178,256)(179,257)(180,258)(181,259)(182,260)(183,261)(184,262)(185,263)(186,264)(187,265)(188,266)(189,248)(190,249)(191,279)(192,280)(193,281)(194,282)(195,283)(196,284)(197,285)(198,267)(199,268)(200,269)(201,270)(202,271)(203,272)(204,273)(205,274)(206,275)(207,276)(208,277)(209,278)(210,286)(211,287)(212,288)(213,289)(214,290)(215,291)(216,292)(217,293)(218,294)(219,295)(220,296)(221,297)(222,298)(223,299)(224,300)(225,301)(226,302)(227,303)(228,304), (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), (1,300,58,249,20,276,39,237)(2,299,59,248,21,275,40,236)(3,298,60,266,22,274,41,235)(4,297,61,265,23,273,42,234)(5,296,62,264,24,272,43,233)(6,295,63,263,25,271,44,232)(7,294,64,262,26,270,45,231)(8,293,65,261,27,269,46,230)(9,292,66,260,28,268,47,229)(10,291,67,259,29,267,48,247)(11,290,68,258,30,285,49,246)(12,289,69,257,31,284,50,245)(13,288,70,256,32,283,51,244)(14,287,71,255,33,282,52,243)(15,286,72,254,34,281,53,242)(16,304,73,253,35,280,54,241)(17,303,74,252,36,279,55,240)(18,302,75,251,37,278,56,239)(19,301,76,250,38,277,57,238)(77,224,134,190,96,207,115,169)(78,223,135,189,97,206,116,168)(79,222,136,188,98,205,117,167)(80,221,137,187,99,204,118,166)(81,220,138,186,100,203,119,165)(82,219,139,185,101,202,120,164)(83,218,140,184,102,201,121,163)(84,217,141,183,103,200,122,162)(85,216,142,182,104,199,123,161)(86,215,143,181,105,198,124,160)(87,214,144,180,106,197,125,159)(88,213,145,179,107,196,126,158)(89,212,146,178,108,195,127,157)(90,211,147,177,109,194,128,156)(91,210,148,176,110,193,129,155)(92,228,149,175,111,192,130,154)(93,227,150,174,112,191,131,153)(94,226,151,173,113,209,132,171)(95,225,152,172,114,208,133,170) );
G=PermutationGroup([[(1,77),(2,78),(3,79),(4,80),(5,81),(6,82),(7,83),(8,84),(9,85),(10,86),(11,87),(12,88),(13,89),(14,90),(15,91),(16,92),(17,93),(18,94),(19,95),(20,96),(21,97),(22,98),(23,99),(24,100),(25,101),(26,102),(27,103),(28,104),(29,105),(30,106),(31,107),(32,108),(33,109),(34,110),(35,111),(36,112),(37,113),(38,114),(39,115),(40,116),(41,117),(42,118),(43,119),(44,120),(45,121),(46,122),(47,123),(48,124),(49,125),(50,126),(51,127),(52,128),(53,129),(54,130),(55,131),(56,132),(57,133),(58,134),(59,135),(60,136),(61,137),(62,138),(63,139),(64,140),(65,141),(66,142),(67,143),(68,144),(69,145),(70,146),(71,147),(72,148),(73,149),(74,150),(75,151),(76,152),(153,240),(154,241),(155,242),(156,243),(157,244),(158,245),(159,246),(160,247),(161,229),(162,230),(163,231),(164,232),(165,233),(166,234),(167,235),(168,236),(169,237),(170,238),(171,239),(172,250),(173,251),(174,252),(175,253),(176,254),(177,255),(178,256),(179,257),(180,258),(181,259),(182,260),(183,261),(184,262),(185,263),(186,264),(187,265),(188,266),(189,248),(190,249),(191,279),(192,280),(193,281),(194,282),(195,283),(196,284),(197,285),(198,267),(199,268),(200,269),(201,270),(202,271),(203,272),(204,273),(205,274),(206,275),(207,276),(208,277),(209,278),(210,286),(211,287),(212,288),(213,289),(214,290),(215,291),(216,292),(217,293),(218,294),(219,295),(220,296),(221,297),(222,298),(223,299),(224,300),(225,301),(226,302),(227,303),(228,304)], [(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)], [(1,300,58,249,20,276,39,237),(2,299,59,248,21,275,40,236),(3,298,60,266,22,274,41,235),(4,297,61,265,23,273,42,234),(5,296,62,264,24,272,43,233),(6,295,63,263,25,271,44,232),(7,294,64,262,26,270,45,231),(8,293,65,261,27,269,46,230),(9,292,66,260,28,268,47,229),(10,291,67,259,29,267,48,247),(11,290,68,258,30,285,49,246),(12,289,69,257,31,284,50,245),(13,288,70,256,32,283,51,244),(14,287,71,255,33,282,52,243),(15,286,72,254,34,281,53,242),(16,304,73,253,35,280,54,241),(17,303,74,252,36,279,55,240),(18,302,75,251,37,278,56,239),(19,301,76,250,38,277,57,238),(77,224,134,190,96,207,115,169),(78,223,135,189,97,206,116,168),(79,222,136,188,98,205,117,167),(80,221,137,187,99,204,118,166),(81,220,138,186,100,203,119,165),(82,219,139,185,101,202,120,164),(83,218,140,184,102,201,121,163),(84,217,141,183,103,200,122,162),(85,216,142,182,104,199,123,161),(86,215,143,181,105,198,124,160),(87,214,144,180,106,197,125,159),(88,213,145,179,107,196,126,158),(89,212,146,178,108,195,127,157),(90,211,147,177,109,194,128,156),(91,210,148,176,110,193,129,155),(92,228,149,175,111,192,130,154),(93,227,150,174,112,191,131,153),(94,226,151,173,113,209,132,171),(95,225,152,172,114,208,133,170)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 19A | ··· | 19I | 38A | ··· | 38AA | 76A | ··· | 76AJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 19 | ··· | 19 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D19 | Dic19 | D38 | Dic19 | C19⋊C8 |
| kernel | C2×C19⋊C8 | C19⋊C8 | C2×C76 | C76 | C2×C38 | C38 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 9 | 9 | 9 | 9 | 36 |
Matrix representation of C2×C19⋊C8 ►in GL3(𝔽457) generated by
| 1 | 0 | 0 |
| 0 | 456 | 0 |
| 0 | 0 | 456 |
| 1 | 0 | 0 |
| 0 | 392 | 456 |
| 0 | 1 | 0 |
| 207 | 0 | 0 |
| 0 | 39 | 83 |
| 0 | 290 | 418 |
G:=sub<GL(3,GF(457))| [1,0,0,0,456,0,0,0,456],[1,0,0,0,392,1,0,456,0],[207,0,0,0,39,290,0,83,418] >;
C2×C19⋊C8 in GAP, Magma, Sage, TeX
C_2\times C_{19}\rtimes C_8 % in TeX
G:=Group("C2xC19:C8"); // GroupNames label
G:=SmallGroup(304,8);
// by ID
G=gap.SmallGroup(304,8);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,20,42,7204]);
// Polycyclic
G:=Group<a,b,c|a^2=b^19=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export