direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: Q16×C19, C8.C38, Q8.C38, C152.3C2, C38.16D4, C76.19C22, C4.3(C2×C38), C2.5(D4×C19), (Q8×C19).2C2, SmallGroup(304,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16×C19
G = < a,b,c | a19=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q16×C19
►Regular action on 304 points
Generators in S304
(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 94 46 251 71 241 272 119)(2 95 47 252 72 242 273 120)(3 77 48 253 73 243 274 121)(4 78 49 254 74 244 275 122)(5 79 50 255 75 245 276 123)(6 80 51 256 76 246 277 124)(7 81 52 257 58 247 278 125)(8 82 53 258 59 229 279 126)(9 83 54 259 60 230 280 127)(10 84 55 260 61 231 281 128)(11 85 56 261 62 232 282 129)(12 86 57 262 63 233 283 130)(13 87 39 263 64 234 284 131)(14 88 40 264 65 235 285 132)(15 89 41 265 66 236 267 133)(16 90 42 266 67 237 268 115)(17 91 43 248 68 238 269 116)(18 92 44 249 69 239 270 117)(19 93 45 250 70 240 271 118)(20 201 171 189 228 113 303 145)(21 202 153 190 210 114 304 146)(22 203 154 172 211 96 286 147)(23 204 155 173 212 97 287 148)(24 205 156 174 213 98 288 149)(25 206 157 175 214 99 289 150)(26 207 158 176 215 100 290 151)(27 208 159 177 216 101 291 152)(28 209 160 178 217 102 292 134)(29 191 161 179 218 103 293 135)(30 192 162 180 219 104 294 136)(31 193 163 181 220 105 295 137)(32 194 164 182 221 106 296 138)(33 195 165 183 222 107 297 139)(34 196 166 184 223 108 298 140)(35 197 167 185 224 109 299 141)(36 198 168 186 225 110 300 142)(37 199 169 187 226 111 301 143)(38 200 170 188 227 112 302 144)
(1 190 71 146)(2 172 72 147)(3 173 73 148)(4 174 74 149)(5 175 75 150)(6 176 76 151)(7 177 58 152)(8 178 59 134)(9 179 60 135)(10 180 61 136)(11 181 62 137)(12 182 63 138)(13 183 64 139)(14 184 65 140)(15 185 66 141)(16 186 67 142)(17 187 68 143)(18 188 69 144)(19 189 70 145)(20 118 228 250)(21 119 210 251)(22 120 211 252)(23 121 212 253)(24 122 213 254)(25 123 214 255)(26 124 215 256)(27 125 216 257)(28 126 217 258)(29 127 218 259)(30 128 219 260)(31 129 220 261)(32 130 221 262)(33 131 222 263)(34 132 223 264)(35 133 224 265)(36 115 225 266)(37 116 226 248)(38 117 227 249)(39 195 284 107)(40 196 285 108)(41 197 267 109)(42 198 268 110)(43 199 269 111)(44 200 270 112)(45 201 271 113)(46 202 272 114)(47 203 273 96)(48 204 274 97)(49 205 275 98)(50 206 276 99)(51 207 277 100)(52 208 278 101)(53 209 279 102)(54 191 280 103)(55 192 281 104)(56 193 282 105)(57 194 283 106)(77 155 243 287)(78 156 244 288)(79 157 245 289)(80 158 246 290)(81 159 247 291)(82 160 229 292)(83 161 230 293)(84 162 231 294)(85 163 232 295)(86 164 233 296)(87 165 234 297)(88 166 235 298)(89 167 236 299)(90 168 237 300)(91 169 238 301)(92 170 239 302)(93 171 240 303)(94 153 241 304)(95 154 242 286)
G:=sub<Sym(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), (1,94,46,251,71,241,272,119)(2,95,47,252,72,242,273,120)(3,77,48,253,73,243,274,121)(4,78,49,254,74,244,275,122)(5,79,50,255,75,245,276,123)(6,80,51,256,76,246,277,124)(7,81,52,257,58,247,278,125)(8,82,53,258,59,229,279,126)(9,83,54,259,60,230,280,127)(10,84,55,260,61,231,281,128)(11,85,56,261,62,232,282,129)(12,86,57,262,63,233,283,130)(13,87,39,263,64,234,284,131)(14,88,40,264,65,235,285,132)(15,89,41,265,66,236,267,133)(16,90,42,266,67,237,268,115)(17,91,43,248,68,238,269,116)(18,92,44,249,69,239,270,117)(19,93,45,250,70,240,271,118)(20,201,171,189,228,113,303,145)(21,202,153,190,210,114,304,146)(22,203,154,172,211,96,286,147)(23,204,155,173,212,97,287,148)(24,205,156,174,213,98,288,149)(25,206,157,175,214,99,289,150)(26,207,158,176,215,100,290,151)(27,208,159,177,216,101,291,152)(28,209,160,178,217,102,292,134)(29,191,161,179,218,103,293,135)(30,192,162,180,219,104,294,136)(31,193,163,181,220,105,295,137)(32,194,164,182,221,106,296,138)(33,195,165,183,222,107,297,139)(34,196,166,184,223,108,298,140)(35,197,167,185,224,109,299,141)(36,198,168,186,225,110,300,142)(37,199,169,187,226,111,301,143)(38,200,170,188,227,112,302,144), (1,190,71,146)(2,172,72,147)(3,173,73,148)(4,174,74,149)(5,175,75,150)(6,176,76,151)(7,177,58,152)(8,178,59,134)(9,179,60,135)(10,180,61,136)(11,181,62,137)(12,182,63,138)(13,183,64,139)(14,184,65,140)(15,185,66,141)(16,186,67,142)(17,187,68,143)(18,188,69,144)(19,189,70,145)(20,118,228,250)(21,119,210,251)(22,120,211,252)(23,121,212,253)(24,122,213,254)(25,123,214,255)(26,124,215,256)(27,125,216,257)(28,126,217,258)(29,127,218,259)(30,128,219,260)(31,129,220,261)(32,130,221,262)(33,131,222,263)(34,132,223,264)(35,133,224,265)(36,115,225,266)(37,116,226,248)(38,117,227,249)(39,195,284,107)(40,196,285,108)(41,197,267,109)(42,198,268,110)(43,199,269,111)(44,200,270,112)(45,201,271,113)(46,202,272,114)(47,203,273,96)(48,204,274,97)(49,205,275,98)(50,206,276,99)(51,207,277,100)(52,208,278,101)(53,209,279,102)(54,191,280,103)(55,192,281,104)(56,193,282,105)(57,194,283,106)(77,155,243,287)(78,156,244,288)(79,157,245,289)(80,158,246,290)(81,159,247,291)(82,160,229,292)(83,161,230,293)(84,162,231,294)(85,163,232,295)(86,164,233,296)(87,165,234,297)(88,166,235,298)(89,167,236,299)(90,168,237,300)(91,169,238,301)(92,170,239,302)(93,171,240,303)(94,153,241,304)(95,154,242,286)>;
G:=Group( (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), (1,94,46,251,71,241,272,119)(2,95,47,252,72,242,273,120)(3,77,48,253,73,243,274,121)(4,78,49,254,74,244,275,122)(5,79,50,255,75,245,276,123)(6,80,51,256,76,246,277,124)(7,81,52,257,58,247,278,125)(8,82,53,258,59,229,279,126)(9,83,54,259,60,230,280,127)(10,84,55,260,61,231,281,128)(11,85,56,261,62,232,282,129)(12,86,57,262,63,233,283,130)(13,87,39,263,64,234,284,131)(14,88,40,264,65,235,285,132)(15,89,41,265,66,236,267,133)(16,90,42,266,67,237,268,115)(17,91,43,248,68,238,269,116)(18,92,44,249,69,239,270,117)(19,93,45,250,70,240,271,118)(20,201,171,189,228,113,303,145)(21,202,153,190,210,114,304,146)(22,203,154,172,211,96,286,147)(23,204,155,173,212,97,287,148)(24,205,156,174,213,98,288,149)(25,206,157,175,214,99,289,150)(26,207,158,176,215,100,290,151)(27,208,159,177,216,101,291,152)(28,209,160,178,217,102,292,134)(29,191,161,179,218,103,293,135)(30,192,162,180,219,104,294,136)(31,193,163,181,220,105,295,137)(32,194,164,182,221,106,296,138)(33,195,165,183,222,107,297,139)(34,196,166,184,223,108,298,140)(35,197,167,185,224,109,299,141)(36,198,168,186,225,110,300,142)(37,199,169,187,226,111,301,143)(38,200,170,188,227,112,302,144), (1,190,71,146)(2,172,72,147)(3,173,73,148)(4,174,74,149)(5,175,75,150)(6,176,76,151)(7,177,58,152)(8,178,59,134)(9,179,60,135)(10,180,61,136)(11,181,62,137)(12,182,63,138)(13,183,64,139)(14,184,65,140)(15,185,66,141)(16,186,67,142)(17,187,68,143)(18,188,69,144)(19,189,70,145)(20,118,228,250)(21,119,210,251)(22,120,211,252)(23,121,212,253)(24,122,213,254)(25,123,214,255)(26,124,215,256)(27,125,216,257)(28,126,217,258)(29,127,218,259)(30,128,219,260)(31,129,220,261)(32,130,221,262)(33,131,222,263)(34,132,223,264)(35,133,224,265)(36,115,225,266)(37,116,226,248)(38,117,227,249)(39,195,284,107)(40,196,285,108)(41,197,267,109)(42,198,268,110)(43,199,269,111)(44,200,270,112)(45,201,271,113)(46,202,272,114)(47,203,273,96)(48,204,274,97)(49,205,275,98)(50,206,276,99)(51,207,277,100)(52,208,278,101)(53,209,279,102)(54,191,280,103)(55,192,281,104)(56,193,282,105)(57,194,283,106)(77,155,243,287)(78,156,244,288)(79,157,245,289)(80,158,246,290)(81,159,247,291)(82,160,229,292)(83,161,230,293)(84,162,231,294)(85,163,232,295)(86,164,233,296)(87,165,234,297)(88,166,235,298)(89,167,236,299)(90,168,237,300)(91,169,238,301)(92,170,239,302)(93,171,240,303)(94,153,241,304)(95,154,242,286) );
G=PermutationGroup([[(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)], [(1,94,46,251,71,241,272,119),(2,95,47,252,72,242,273,120),(3,77,48,253,73,243,274,121),(4,78,49,254,74,244,275,122),(5,79,50,255,75,245,276,123),(6,80,51,256,76,246,277,124),(7,81,52,257,58,247,278,125),(8,82,53,258,59,229,279,126),(9,83,54,259,60,230,280,127),(10,84,55,260,61,231,281,128),(11,85,56,261,62,232,282,129),(12,86,57,262,63,233,283,130),(13,87,39,263,64,234,284,131),(14,88,40,264,65,235,285,132),(15,89,41,265,66,236,267,133),(16,90,42,266,67,237,268,115),(17,91,43,248,68,238,269,116),(18,92,44,249,69,239,270,117),(19,93,45,250,70,240,271,118),(20,201,171,189,228,113,303,145),(21,202,153,190,210,114,304,146),(22,203,154,172,211,96,286,147),(23,204,155,173,212,97,287,148),(24,205,156,174,213,98,288,149),(25,206,157,175,214,99,289,150),(26,207,158,176,215,100,290,151),(27,208,159,177,216,101,291,152),(28,209,160,178,217,102,292,134),(29,191,161,179,218,103,293,135),(30,192,162,180,219,104,294,136),(31,193,163,181,220,105,295,137),(32,194,164,182,221,106,296,138),(33,195,165,183,222,107,297,139),(34,196,166,184,223,108,298,140),(35,197,167,185,224,109,299,141),(36,198,168,186,225,110,300,142),(37,199,169,187,226,111,301,143),(38,200,170,188,227,112,302,144)], [(1,190,71,146),(2,172,72,147),(3,173,73,148),(4,174,74,149),(5,175,75,150),(6,176,76,151),(7,177,58,152),(8,178,59,134),(9,179,60,135),(10,180,61,136),(11,181,62,137),(12,182,63,138),(13,183,64,139),(14,184,65,140),(15,185,66,141),(16,186,67,142),(17,187,68,143),(18,188,69,144),(19,189,70,145),(20,118,228,250),(21,119,210,251),(22,120,211,252),(23,121,212,253),(24,122,213,254),(25,123,214,255),(26,124,215,256),(27,125,216,257),(28,126,217,258),(29,127,218,259),(30,128,219,260),(31,129,220,261),(32,130,221,262),(33,131,222,263),(34,132,223,264),(35,133,224,265),(36,115,225,266),(37,116,226,248),(38,117,227,249),(39,195,284,107),(40,196,285,108),(41,197,267,109),(42,198,268,110),(43,199,269,111),(44,200,270,112),(45,201,271,113),(46,202,272,114),(47,203,273,96),(48,204,274,97),(49,205,275,98),(50,206,276,99),(51,207,277,100),(52,208,278,101),(53,209,279,102),(54,191,280,103),(55,192,281,104),(56,193,282,105),(57,194,283,106),(77,155,243,287),(78,156,244,288),(79,157,245,289),(80,158,246,290),(81,159,247,291),(82,160,229,292),(83,161,230,293),(84,162,231,294),(85,163,232,295),(86,164,233,296),(87,165,234,297),(88,166,235,298),(89,167,236,299),(90,168,237,300),(91,169,238,301),(92,170,239,302),(93,171,240,303),(94,153,241,304),(95,154,242,286)]])
133 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 19A | ··· | 19R | 38A | ··· | 38R | 76A | ··· | 76R | 76S | ··· | 76BB | 152A | ··· | 152AJ |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 | 76 | ··· | 76 | 152 | ··· | 152 |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C19 | C38 | C38 | D4 | Q16 | D4×C19 | Q16×C19 |
| kernel | Q16×C19 | C152 | Q8×C19 | Q16 | C8 | Q8 | C38 | C19 | C2 | C1 |
| # reps | 1 | 1 | 2 | 18 | 18 | 36 | 1 | 2 | 18 | 36 |
Matrix representation of Q16×C19 ►in GL2(𝔽457) generated by
| 68 | 0 |
| 0 | 68 |
| 40 | 417 |
| 40 | 40 |
| 310 | 82 |
| 82 | 147 |
G:=sub<GL(2,GF(457))| [68,0,0,68],[40,40,417,40],[310,82,82,147] >;
Q16×C19 in GAP, Magma, Sage, TeX
Q_{16}\times C_{19} % in TeX
G:=Group("Q16xC19"); // GroupNames label
G:=SmallGroup(304,26);
// by ID
G=gap.SmallGroup(304,26);
# by ID
G:=PCGroup([5,-2,-2,-19,-2,-2,760,781,766,4563,2288,58]);
// Polycyclic
G:=Group<a,b,c|a^19=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export