metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C19⋊2Q16, Q8.D19, C4.4D38, C38.10D4, C76.4C22, Dic38.2C2, C19⋊C8.C2, (Q8×C19).1C2, C2.7(C19⋊D4), SmallGroup(304,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C19⋊Q16
G = < a,b,c | a19=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C19⋊Q16
►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)
(1 267 49 248 38 286 60 229)(2 285 50 266 20 304 61 247)(3 284 51 265 21 303 62 246)(4 283 52 264 22 302 63 245)(5 282 53 263 23 301 64 244)(6 281 54 262 24 300 65 243)(7 280 55 261 25 299 66 242)(8 279 56 260 26 298 67 241)(9 278 57 259 27 297 68 240)(10 277 39 258 28 296 69 239)(11 276 40 257 29 295 70 238)(12 275 41 256 30 294 71 237)(13 274 42 255 31 293 72 236)(14 273 43 254 32 292 73 235)(15 272 44 253 33 291 74 234)(16 271 45 252 34 290 75 233)(17 270 46 251 35 289 76 232)(18 269 47 250 36 288 58 231)(19 268 48 249 37 287 59 230)(77 204 138 166 105 223 124 185)(78 203 139 165 106 222 125 184)(79 202 140 164 107 221 126 183)(80 201 141 163 108 220 127 182)(81 200 142 162 109 219 128 181)(82 199 143 161 110 218 129 180)(83 198 144 160 111 217 130 179)(84 197 145 159 112 216 131 178)(85 196 146 158 113 215 132 177)(86 195 147 157 114 214 133 176)(87 194 148 156 96 213 115 175)(88 193 149 155 97 212 116 174)(89 192 150 154 98 211 117 173)(90 191 151 153 99 210 118 172)(91 209 152 171 100 228 119 190)(92 208 134 170 101 227 120 189)(93 207 135 169 102 226 121 188)(94 206 136 168 103 225 122 187)(95 205 137 167 104 224 123 186)
(1 99 38 90)(2 100 20 91)(3 101 21 92)(4 102 22 93)(5 103 23 94)(6 104 24 95)(7 105 25 77)(8 106 26 78)(9 107 27 79)(10 108 28 80)(11 109 29 81)(12 110 30 82)(13 111 31 83)(14 112 32 84)(15 113 33 85)(16 114 34 86)(17 96 35 87)(18 97 36 88)(19 98 37 89)(39 141 69 127)(40 142 70 128)(41 143 71 129)(42 144 72 130)(43 145 73 131)(44 146 74 132)(45 147 75 133)(46 148 76 115)(47 149 58 116)(48 150 59 117)(49 151 60 118)(50 152 61 119)(51 134 62 120)(52 135 63 121)(53 136 64 122)(54 137 65 123)(55 138 66 124)(56 139 67 125)(57 140 68 126)(153 286 172 267)(154 287 173 268)(155 288 174 269)(156 289 175 270)(157 290 176 271)(158 291 177 272)(159 292 178 273)(160 293 179 274)(161 294 180 275)(162 295 181 276)(163 296 182 277)(164 297 183 278)(165 298 184 279)(166 299 185 280)(167 300 186 281)(168 301 187 282)(169 302 188 283)(170 303 189 284)(171 304 190 285)(191 229 210 248)(192 230 211 249)(193 231 212 250)(194 232 213 251)(195 233 214 252)(196 234 215 253)(197 235 216 254)(198 236 217 255)(199 237 218 256)(200 238 219 257)(201 239 220 258)(202 240 221 259)(203 241 222 260)(204 242 223 261)(205 243 224 262)(206 244 225 263)(207 245 226 264)(208 246 227 265)(209 247 228 266)
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), (1,267,49,248,38,286,60,229)(2,285,50,266,20,304,61,247)(3,284,51,265,21,303,62,246)(4,283,52,264,22,302,63,245)(5,282,53,263,23,301,64,244)(6,281,54,262,24,300,65,243)(7,280,55,261,25,299,66,242)(8,279,56,260,26,298,67,241)(9,278,57,259,27,297,68,240)(10,277,39,258,28,296,69,239)(11,276,40,257,29,295,70,238)(12,275,41,256,30,294,71,237)(13,274,42,255,31,293,72,236)(14,273,43,254,32,292,73,235)(15,272,44,253,33,291,74,234)(16,271,45,252,34,290,75,233)(17,270,46,251,35,289,76,232)(18,269,47,250,36,288,58,231)(19,268,48,249,37,287,59,230)(77,204,138,166,105,223,124,185)(78,203,139,165,106,222,125,184)(79,202,140,164,107,221,126,183)(80,201,141,163,108,220,127,182)(81,200,142,162,109,219,128,181)(82,199,143,161,110,218,129,180)(83,198,144,160,111,217,130,179)(84,197,145,159,112,216,131,178)(85,196,146,158,113,215,132,177)(86,195,147,157,114,214,133,176)(87,194,148,156,96,213,115,175)(88,193,149,155,97,212,116,174)(89,192,150,154,98,211,117,173)(90,191,151,153,99,210,118,172)(91,209,152,171,100,228,119,190)(92,208,134,170,101,227,120,189)(93,207,135,169,102,226,121,188)(94,206,136,168,103,225,122,187)(95,205,137,167,104,224,123,186), (1,99,38,90)(2,100,20,91)(3,101,21,92)(4,102,22,93)(5,103,23,94)(6,104,24,95)(7,105,25,77)(8,106,26,78)(9,107,27,79)(10,108,28,80)(11,109,29,81)(12,110,30,82)(13,111,31,83)(14,112,32,84)(15,113,33,85)(16,114,34,86)(17,96,35,87)(18,97,36,88)(19,98,37,89)(39,141,69,127)(40,142,70,128)(41,143,71,129)(42,144,72,130)(43,145,73,131)(44,146,74,132)(45,147,75,133)(46,148,76,115)(47,149,58,116)(48,150,59,117)(49,151,60,118)(50,152,61,119)(51,134,62,120)(52,135,63,121)(53,136,64,122)(54,137,65,123)(55,138,66,124)(56,139,67,125)(57,140,68,126)(153,286,172,267)(154,287,173,268)(155,288,174,269)(156,289,175,270)(157,290,176,271)(158,291,177,272)(159,292,178,273)(160,293,179,274)(161,294,180,275)(162,295,181,276)(163,296,182,277)(164,297,183,278)(165,298,184,279)(166,299,185,280)(167,300,186,281)(168,301,187,282)(169,302,188,283)(170,303,189,284)(171,304,190,285)(191,229,210,248)(192,230,211,249)(193,231,212,250)(194,232,213,251)(195,233,214,252)(196,234,215,253)(197,235,216,254)(198,236,217,255)(199,237,218,256)(200,238,219,257)(201,239,220,258)(202,240,221,259)(203,241,222,260)(204,242,223,261)(205,243,224,262)(206,244,225,263)(207,245,226,264)(208,246,227,265)(209,247,228,266)>;
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), (1,267,49,248,38,286,60,229)(2,285,50,266,20,304,61,247)(3,284,51,265,21,303,62,246)(4,283,52,264,22,302,63,245)(5,282,53,263,23,301,64,244)(6,281,54,262,24,300,65,243)(7,280,55,261,25,299,66,242)(8,279,56,260,26,298,67,241)(9,278,57,259,27,297,68,240)(10,277,39,258,28,296,69,239)(11,276,40,257,29,295,70,238)(12,275,41,256,30,294,71,237)(13,274,42,255,31,293,72,236)(14,273,43,254,32,292,73,235)(15,272,44,253,33,291,74,234)(16,271,45,252,34,290,75,233)(17,270,46,251,35,289,76,232)(18,269,47,250,36,288,58,231)(19,268,48,249,37,287,59,230)(77,204,138,166,105,223,124,185)(78,203,139,165,106,222,125,184)(79,202,140,164,107,221,126,183)(80,201,141,163,108,220,127,182)(81,200,142,162,109,219,128,181)(82,199,143,161,110,218,129,180)(83,198,144,160,111,217,130,179)(84,197,145,159,112,216,131,178)(85,196,146,158,113,215,132,177)(86,195,147,157,114,214,133,176)(87,194,148,156,96,213,115,175)(88,193,149,155,97,212,116,174)(89,192,150,154,98,211,117,173)(90,191,151,153,99,210,118,172)(91,209,152,171,100,228,119,190)(92,208,134,170,101,227,120,189)(93,207,135,169,102,226,121,188)(94,206,136,168,103,225,122,187)(95,205,137,167,104,224,123,186), (1,99,38,90)(2,100,20,91)(3,101,21,92)(4,102,22,93)(5,103,23,94)(6,104,24,95)(7,105,25,77)(8,106,26,78)(9,107,27,79)(10,108,28,80)(11,109,29,81)(12,110,30,82)(13,111,31,83)(14,112,32,84)(15,113,33,85)(16,114,34,86)(17,96,35,87)(18,97,36,88)(19,98,37,89)(39,141,69,127)(40,142,70,128)(41,143,71,129)(42,144,72,130)(43,145,73,131)(44,146,74,132)(45,147,75,133)(46,148,76,115)(47,149,58,116)(48,150,59,117)(49,151,60,118)(50,152,61,119)(51,134,62,120)(52,135,63,121)(53,136,64,122)(54,137,65,123)(55,138,66,124)(56,139,67,125)(57,140,68,126)(153,286,172,267)(154,287,173,268)(155,288,174,269)(156,289,175,270)(157,290,176,271)(158,291,177,272)(159,292,178,273)(160,293,179,274)(161,294,180,275)(162,295,181,276)(163,296,182,277)(164,297,183,278)(165,298,184,279)(166,299,185,280)(167,300,186,281)(168,301,187,282)(169,302,188,283)(170,303,189,284)(171,304,190,285)(191,229,210,248)(192,230,211,249)(193,231,212,250)(194,232,213,251)(195,233,214,252)(196,234,215,253)(197,235,216,254)(198,236,217,255)(199,237,218,256)(200,238,219,257)(201,239,220,258)(202,240,221,259)(203,241,222,260)(204,242,223,261)(205,243,224,262)(206,244,225,263)(207,245,226,264)(208,246,227,265)(209,247,228,266) );
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)], [(1,267,49,248,38,286,60,229),(2,285,50,266,20,304,61,247),(3,284,51,265,21,303,62,246),(4,283,52,264,22,302,63,245),(5,282,53,263,23,301,64,244),(6,281,54,262,24,300,65,243),(7,280,55,261,25,299,66,242),(8,279,56,260,26,298,67,241),(9,278,57,259,27,297,68,240),(10,277,39,258,28,296,69,239),(11,276,40,257,29,295,70,238),(12,275,41,256,30,294,71,237),(13,274,42,255,31,293,72,236),(14,273,43,254,32,292,73,235),(15,272,44,253,33,291,74,234),(16,271,45,252,34,290,75,233),(17,270,46,251,35,289,76,232),(18,269,47,250,36,288,58,231),(19,268,48,249,37,287,59,230),(77,204,138,166,105,223,124,185),(78,203,139,165,106,222,125,184),(79,202,140,164,107,221,126,183),(80,201,141,163,108,220,127,182),(81,200,142,162,109,219,128,181),(82,199,143,161,110,218,129,180),(83,198,144,160,111,217,130,179),(84,197,145,159,112,216,131,178),(85,196,146,158,113,215,132,177),(86,195,147,157,114,214,133,176),(87,194,148,156,96,213,115,175),(88,193,149,155,97,212,116,174),(89,192,150,154,98,211,117,173),(90,191,151,153,99,210,118,172),(91,209,152,171,100,228,119,190),(92,208,134,170,101,227,120,189),(93,207,135,169,102,226,121,188),(94,206,136,168,103,225,122,187),(95,205,137,167,104,224,123,186)], [(1,99,38,90),(2,100,20,91),(3,101,21,92),(4,102,22,93),(5,103,23,94),(6,104,24,95),(7,105,25,77),(8,106,26,78),(9,107,27,79),(10,108,28,80),(11,109,29,81),(12,110,30,82),(13,111,31,83),(14,112,32,84),(15,113,33,85),(16,114,34,86),(17,96,35,87),(18,97,36,88),(19,98,37,89),(39,141,69,127),(40,142,70,128),(41,143,71,129),(42,144,72,130),(43,145,73,131),(44,146,74,132),(45,147,75,133),(46,148,76,115),(47,149,58,116),(48,150,59,117),(49,151,60,118),(50,152,61,119),(51,134,62,120),(52,135,63,121),(53,136,64,122),(54,137,65,123),(55,138,66,124),(56,139,67,125),(57,140,68,126),(153,286,172,267),(154,287,173,268),(155,288,174,269),(156,289,175,270),(157,290,176,271),(158,291,177,272),(159,292,178,273),(160,293,179,274),(161,294,180,275),(162,295,181,276),(163,296,182,277),(164,297,183,278),(165,298,184,279),(166,299,185,280),(167,300,186,281),(168,301,187,282),(169,302,188,283),(170,303,189,284),(171,304,190,285),(191,229,210,248),(192,230,211,249),(193,231,212,250),(194,232,213,251),(195,233,214,252),(196,234,215,253),(197,235,216,254),(198,236,217,255),(199,237,218,256),(200,238,219,257),(201,239,220,258),(202,240,221,259),(203,241,222,260),(204,242,223,261),(205,243,224,262),(206,244,225,263),(207,245,226,264),(208,246,227,265),(209,247,228,266)]])
52 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 19A | ··· | 19I | 38A | ··· | 38I | 76A | ··· | 76AA |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 2 | 4 | 76 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | D4 | Q16 | D19 | D38 | C19⋊D4 | C19⋊Q16 |
| kernel | C19⋊Q16 | C19⋊C8 | Dic38 | Q8×C19 | C38 | C19 | Q8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 9 | 9 | 18 | 9 |
Matrix representation of C19⋊Q16 ►in GL4(𝔽457) generated by
| 246 | 1 | 0 | 0 |
| 269 | 146 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 71 | 399 | 0 | 0 |
| 276 | 386 | 0 | 0 |
| 0 | 0 | 242 | 80 |
| 0 | 0 | 197 | 135 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 405 | 317 |
| 0 | 0 | 3 | 52 |
G:=sub<GL(4,GF(457))| [246,269,0,0,1,146,0,0,0,0,1,0,0,0,0,1],[71,276,0,0,399,386,0,0,0,0,242,197,0,0,80,135],[1,0,0,0,0,1,0,0,0,0,405,3,0,0,317,52] >;
C19⋊Q16 in GAP, Magma, Sage, TeX
C_{19}\rtimes Q_{16} % in TeX
G:=Group("C19:Q16"); // GroupNames label
G:=SmallGroup(304,17);
// by ID
G=gap.SmallGroup(304,17);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,40,61,46,182,97,42,7204]);
// Polycyclic
G:=Group<a,b,c|a^19=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export