direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×Dic19, C19⋊C42, C76⋊2C4, C22.3D38, (C2×C76).7C2, C38.3(C2×C4), (C2×C4).6D19, C2.2(C4×D19), (C2×C38).3C22, C2.2(C2×Dic19), (C2×Dic19).4C2, SmallGroup(304,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C4×Dic19 |
Generators and relations for C4×Dic19
G = < a,b,c | a4=b38=1, c2=b19, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4×Dic19
►Regular action on 304 points
Generators in S304
(1 64 274 237)(2 65 275 238)(3 66 276 239)(4 67 277 240)(5 68 278 241)(6 69 279 242)(7 70 280 243)(8 71 281 244)(9 72 282 245)(10 73 283 246)(11 74 284 247)(12 75 285 248)(13 76 286 249)(14 39 287 250)(15 40 288 251)(16 41 289 252)(17 42 290 253)(18 43 291 254)(19 44 292 255)(20 45 293 256)(21 46 294 257)(22 47 295 258)(23 48 296 259)(24 49 297 260)(25 50 298 261)(26 51 299 262)(27 52 300 263)(28 53 301 264)(29 54 302 265)(30 55 303 266)(31 56 304 229)(32 57 267 230)(33 58 268 231)(34 59 269 232)(35 60 270 233)(36 61 271 234)(37 62 272 235)(38 63 273 236)(77 192 129 181)(78 193 130 182)(79 194 131 183)(80 195 132 184)(81 196 133 185)(82 197 134 186)(83 198 135 187)(84 199 136 188)(85 200 137 189)(86 201 138 190)(87 202 139 153)(88 203 140 154)(89 204 141 155)(90 205 142 156)(91 206 143 157)(92 207 144 158)(93 208 145 159)(94 209 146 160)(95 210 147 161)(96 211 148 162)(97 212 149 163)(98 213 150 164)(99 214 151 165)(100 215 152 166)(101 216 115 167)(102 217 116 168)(103 218 117 169)(104 219 118 170)(105 220 119 171)(106 221 120 172)(107 222 121 173)(108 223 122 174)(109 224 123 175)(110 225 124 176)(111 226 125 177)(112 227 126 178)(113 228 127 179)(114 191 128 180)
(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)
(1 221 20 202)(2 220 21 201)(3 219 22 200)(4 218 23 199)(5 217 24 198)(6 216 25 197)(7 215 26 196)(8 214 27 195)(9 213 28 194)(10 212 29 193)(11 211 30 192)(12 210 31 191)(13 209 32 228)(14 208 33 227)(15 207 34 226)(16 206 35 225)(17 205 36 224)(18 204 37 223)(19 203 38 222)(39 145 58 126)(40 144 59 125)(41 143 60 124)(42 142 61 123)(43 141 62 122)(44 140 63 121)(45 139 64 120)(46 138 65 119)(47 137 66 118)(48 136 67 117)(49 135 68 116)(50 134 69 115)(51 133 70 152)(52 132 71 151)(53 131 72 150)(54 130 73 149)(55 129 74 148)(56 128 75 147)(57 127 76 146)(77 247 96 266)(78 246 97 265)(79 245 98 264)(80 244 99 263)(81 243 100 262)(82 242 101 261)(83 241 102 260)(84 240 103 259)(85 239 104 258)(86 238 105 257)(87 237 106 256)(88 236 107 255)(89 235 108 254)(90 234 109 253)(91 233 110 252)(92 232 111 251)(93 231 112 250)(94 230 113 249)(95 229 114 248)(153 274 172 293)(154 273 173 292)(155 272 174 291)(156 271 175 290)(157 270 176 289)(158 269 177 288)(159 268 178 287)(160 267 179 286)(161 304 180 285)(162 303 181 284)(163 302 182 283)(164 301 183 282)(165 300 184 281)(166 299 185 280)(167 298 186 279)(168 297 187 278)(169 296 188 277)(170 295 189 276)(171 294 190 275)
G:=sub<Sym(304)| (1,64,274,237)(2,65,275,238)(3,66,276,239)(4,67,277,240)(5,68,278,241)(6,69,279,242)(7,70,280,243)(8,71,281,244)(9,72,282,245)(10,73,283,246)(11,74,284,247)(12,75,285,248)(13,76,286,249)(14,39,287,250)(15,40,288,251)(16,41,289,252)(17,42,290,253)(18,43,291,254)(19,44,292,255)(20,45,293,256)(21,46,294,257)(22,47,295,258)(23,48,296,259)(24,49,297,260)(25,50,298,261)(26,51,299,262)(27,52,300,263)(28,53,301,264)(29,54,302,265)(30,55,303,266)(31,56,304,229)(32,57,267,230)(33,58,268,231)(34,59,269,232)(35,60,270,233)(36,61,271,234)(37,62,272,235)(38,63,273,236)(77,192,129,181)(78,193,130,182)(79,194,131,183)(80,195,132,184)(81,196,133,185)(82,197,134,186)(83,198,135,187)(84,199,136,188)(85,200,137,189)(86,201,138,190)(87,202,139,153)(88,203,140,154)(89,204,141,155)(90,205,142,156)(91,206,143,157)(92,207,144,158)(93,208,145,159)(94,209,146,160)(95,210,147,161)(96,211,148,162)(97,212,149,163)(98,213,150,164)(99,214,151,165)(100,215,152,166)(101,216,115,167)(102,217,116,168)(103,218,117,169)(104,219,118,170)(105,220,119,171)(106,221,120,172)(107,222,121,173)(108,223,122,174)(109,224,123,175)(110,225,124,176)(111,226,125,177)(112,227,126,178)(113,228,127,179)(114,191,128,180), (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), (1,221,20,202)(2,220,21,201)(3,219,22,200)(4,218,23,199)(5,217,24,198)(6,216,25,197)(7,215,26,196)(8,214,27,195)(9,213,28,194)(10,212,29,193)(11,211,30,192)(12,210,31,191)(13,209,32,228)(14,208,33,227)(15,207,34,226)(16,206,35,225)(17,205,36,224)(18,204,37,223)(19,203,38,222)(39,145,58,126)(40,144,59,125)(41,143,60,124)(42,142,61,123)(43,141,62,122)(44,140,63,121)(45,139,64,120)(46,138,65,119)(47,137,66,118)(48,136,67,117)(49,135,68,116)(50,134,69,115)(51,133,70,152)(52,132,71,151)(53,131,72,150)(54,130,73,149)(55,129,74,148)(56,128,75,147)(57,127,76,146)(77,247,96,266)(78,246,97,265)(79,245,98,264)(80,244,99,263)(81,243,100,262)(82,242,101,261)(83,241,102,260)(84,240,103,259)(85,239,104,258)(86,238,105,257)(87,237,106,256)(88,236,107,255)(89,235,108,254)(90,234,109,253)(91,233,110,252)(92,232,111,251)(93,231,112,250)(94,230,113,249)(95,229,114,248)(153,274,172,293)(154,273,173,292)(155,272,174,291)(156,271,175,290)(157,270,176,289)(158,269,177,288)(159,268,178,287)(160,267,179,286)(161,304,180,285)(162,303,181,284)(163,302,182,283)(164,301,183,282)(165,300,184,281)(166,299,185,280)(167,298,186,279)(168,297,187,278)(169,296,188,277)(170,295,189,276)(171,294,190,275)>;
G:=Group( (1,64,274,237)(2,65,275,238)(3,66,276,239)(4,67,277,240)(5,68,278,241)(6,69,279,242)(7,70,280,243)(8,71,281,244)(9,72,282,245)(10,73,283,246)(11,74,284,247)(12,75,285,248)(13,76,286,249)(14,39,287,250)(15,40,288,251)(16,41,289,252)(17,42,290,253)(18,43,291,254)(19,44,292,255)(20,45,293,256)(21,46,294,257)(22,47,295,258)(23,48,296,259)(24,49,297,260)(25,50,298,261)(26,51,299,262)(27,52,300,263)(28,53,301,264)(29,54,302,265)(30,55,303,266)(31,56,304,229)(32,57,267,230)(33,58,268,231)(34,59,269,232)(35,60,270,233)(36,61,271,234)(37,62,272,235)(38,63,273,236)(77,192,129,181)(78,193,130,182)(79,194,131,183)(80,195,132,184)(81,196,133,185)(82,197,134,186)(83,198,135,187)(84,199,136,188)(85,200,137,189)(86,201,138,190)(87,202,139,153)(88,203,140,154)(89,204,141,155)(90,205,142,156)(91,206,143,157)(92,207,144,158)(93,208,145,159)(94,209,146,160)(95,210,147,161)(96,211,148,162)(97,212,149,163)(98,213,150,164)(99,214,151,165)(100,215,152,166)(101,216,115,167)(102,217,116,168)(103,218,117,169)(104,219,118,170)(105,220,119,171)(106,221,120,172)(107,222,121,173)(108,223,122,174)(109,224,123,175)(110,225,124,176)(111,226,125,177)(112,227,126,178)(113,228,127,179)(114,191,128,180), (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), (1,221,20,202)(2,220,21,201)(3,219,22,200)(4,218,23,199)(5,217,24,198)(6,216,25,197)(7,215,26,196)(8,214,27,195)(9,213,28,194)(10,212,29,193)(11,211,30,192)(12,210,31,191)(13,209,32,228)(14,208,33,227)(15,207,34,226)(16,206,35,225)(17,205,36,224)(18,204,37,223)(19,203,38,222)(39,145,58,126)(40,144,59,125)(41,143,60,124)(42,142,61,123)(43,141,62,122)(44,140,63,121)(45,139,64,120)(46,138,65,119)(47,137,66,118)(48,136,67,117)(49,135,68,116)(50,134,69,115)(51,133,70,152)(52,132,71,151)(53,131,72,150)(54,130,73,149)(55,129,74,148)(56,128,75,147)(57,127,76,146)(77,247,96,266)(78,246,97,265)(79,245,98,264)(80,244,99,263)(81,243,100,262)(82,242,101,261)(83,241,102,260)(84,240,103,259)(85,239,104,258)(86,238,105,257)(87,237,106,256)(88,236,107,255)(89,235,108,254)(90,234,109,253)(91,233,110,252)(92,232,111,251)(93,231,112,250)(94,230,113,249)(95,229,114,248)(153,274,172,293)(154,273,173,292)(155,272,174,291)(156,271,175,290)(157,270,176,289)(158,269,177,288)(159,268,178,287)(160,267,179,286)(161,304,180,285)(162,303,181,284)(163,302,182,283)(164,301,183,282)(165,300,184,281)(166,299,185,280)(167,298,186,279)(168,297,187,278)(169,296,188,277)(170,295,189,276)(171,294,190,275) );
G=PermutationGroup([[(1,64,274,237),(2,65,275,238),(3,66,276,239),(4,67,277,240),(5,68,278,241),(6,69,279,242),(7,70,280,243),(8,71,281,244),(9,72,282,245),(10,73,283,246),(11,74,284,247),(12,75,285,248),(13,76,286,249),(14,39,287,250),(15,40,288,251),(16,41,289,252),(17,42,290,253),(18,43,291,254),(19,44,292,255),(20,45,293,256),(21,46,294,257),(22,47,295,258),(23,48,296,259),(24,49,297,260),(25,50,298,261),(26,51,299,262),(27,52,300,263),(28,53,301,264),(29,54,302,265),(30,55,303,266),(31,56,304,229),(32,57,267,230),(33,58,268,231),(34,59,269,232),(35,60,270,233),(36,61,271,234),(37,62,272,235),(38,63,273,236),(77,192,129,181),(78,193,130,182),(79,194,131,183),(80,195,132,184),(81,196,133,185),(82,197,134,186),(83,198,135,187),(84,199,136,188),(85,200,137,189),(86,201,138,190),(87,202,139,153),(88,203,140,154),(89,204,141,155),(90,205,142,156),(91,206,143,157),(92,207,144,158),(93,208,145,159),(94,209,146,160),(95,210,147,161),(96,211,148,162),(97,212,149,163),(98,213,150,164),(99,214,151,165),(100,215,152,166),(101,216,115,167),(102,217,116,168),(103,218,117,169),(104,219,118,170),(105,220,119,171),(106,221,120,172),(107,222,121,173),(108,223,122,174),(109,224,123,175),(110,225,124,176),(111,226,125,177),(112,227,126,178),(113,228,127,179),(114,191,128,180)], [(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)], [(1,221,20,202),(2,220,21,201),(3,219,22,200),(4,218,23,199),(5,217,24,198),(6,216,25,197),(7,215,26,196),(8,214,27,195),(9,213,28,194),(10,212,29,193),(11,211,30,192),(12,210,31,191),(13,209,32,228),(14,208,33,227),(15,207,34,226),(16,206,35,225),(17,205,36,224),(18,204,37,223),(19,203,38,222),(39,145,58,126),(40,144,59,125),(41,143,60,124),(42,142,61,123),(43,141,62,122),(44,140,63,121),(45,139,64,120),(46,138,65,119),(47,137,66,118),(48,136,67,117),(49,135,68,116),(50,134,69,115),(51,133,70,152),(52,132,71,151),(53,131,72,150),(54,130,73,149),(55,129,74,148),(56,128,75,147),(57,127,76,146),(77,247,96,266),(78,246,97,265),(79,245,98,264),(80,244,99,263),(81,243,100,262),(82,242,101,261),(83,241,102,260),(84,240,103,259),(85,239,104,258),(86,238,105,257),(87,237,106,256),(88,236,107,255),(89,235,108,254),(90,234,109,253),(91,233,110,252),(92,232,111,251),(93,231,112,250),(94,230,113,249),(95,229,114,248),(153,274,172,293),(154,273,173,292),(155,272,174,291),(156,271,175,290),(157,270,176,289),(158,269,177,288),(159,268,178,287),(160,267,179,286),(161,304,180,285),(162,303,181,284),(163,302,182,283),(164,301,183,282),(165,300,184,281),(166,299,185,280),(167,298,186,279),(168,297,187,278),(169,296,188,277),(170,295,189,276),(171,294,190,275)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 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 | 1 | 1 | 1 | 1 | 19 | ··· | 19 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C4 | C4 | D19 | Dic19 | D38 | C4×D19 |
| kernel | C4×Dic19 | C2×Dic19 | C2×C76 | Dic19 | C76 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 9 | 18 | 9 | 36 |
Matrix representation of C4×Dic19 ►in GL3(𝔽229) generated by
| 107 | 0 | 0 |
| 0 | 122 | 0 |
| 0 | 0 | 122 |
| 228 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 228 | 102 |
| 122 | 0 | 0 |
| 0 | 167 | 156 |
| 0 | 15 | 62 |
G:=sub<GL(3,GF(229))| [107,0,0,0,122,0,0,0,122],[228,0,0,0,0,228,0,1,102],[122,0,0,0,167,15,0,156,62] >;
C4×Dic19 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_{19} % in TeX
G:=Group("C4xDic19"); // GroupNames label
G:=SmallGroup(304,10);
// by ID
G=gap.SmallGroup(304,10);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,20,46,7204]);
// Polycyclic
G:=Group<a,b,c|a^4=b^38=1,c^2=b^19,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export