metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C19⋊C16, C38.C8, C76.2C4, C8.2D19, C152.2C2, C4.2Dic19, C2.(C19⋊C8), SmallGroup(304,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C19⋊C16 |
Generators and relations for C19⋊C16
G = < a,b | a19=b16=1, bab-1=a-1 >
Smallest permutation representation of C19⋊C16
►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)
(1 286 150 210 62 248 100 172 34 267 116 191 53 229 88 153)(2 304 151 228 63 266 101 190 35 285 117 209 54 247 89 171)(3 303 152 227 64 265 102 189 36 284 118 208 55 246 90 170)(4 302 134 226 65 264 103 188 37 283 119 207 56 245 91 169)(5 301 135 225 66 263 104 187 38 282 120 206 57 244 92 168)(6 300 136 224 67 262 105 186 20 281 121 205 39 243 93 167)(7 299 137 223 68 261 106 185 21 280 122 204 40 242 94 166)(8 298 138 222 69 260 107 184 22 279 123 203 41 241 95 165)(9 297 139 221 70 259 108 183 23 278 124 202 42 240 77 164)(10 296 140 220 71 258 109 182 24 277 125 201 43 239 78 163)(11 295 141 219 72 257 110 181 25 276 126 200 44 238 79 162)(12 294 142 218 73 256 111 180 26 275 127 199 45 237 80 161)(13 293 143 217 74 255 112 179 27 274 128 198 46 236 81 160)(14 292 144 216 75 254 113 178 28 273 129 197 47 235 82 159)(15 291 145 215 76 253 114 177 29 272 130 196 48 234 83 158)(16 290 146 214 58 252 96 176 30 271 131 195 49 233 84 157)(17 289 147 213 59 251 97 175 31 270 132 194 50 232 85 156)(18 288 148 212 60 250 98 174 32 269 133 193 51 231 86 155)(19 287 149 211 61 249 99 173 33 268 115 192 52 230 87 154)
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), (1,286,150,210,62,248,100,172,34,267,116,191,53,229,88,153)(2,304,151,228,63,266,101,190,35,285,117,209,54,247,89,171)(3,303,152,227,64,265,102,189,36,284,118,208,55,246,90,170)(4,302,134,226,65,264,103,188,37,283,119,207,56,245,91,169)(5,301,135,225,66,263,104,187,38,282,120,206,57,244,92,168)(6,300,136,224,67,262,105,186,20,281,121,205,39,243,93,167)(7,299,137,223,68,261,106,185,21,280,122,204,40,242,94,166)(8,298,138,222,69,260,107,184,22,279,123,203,41,241,95,165)(9,297,139,221,70,259,108,183,23,278,124,202,42,240,77,164)(10,296,140,220,71,258,109,182,24,277,125,201,43,239,78,163)(11,295,141,219,72,257,110,181,25,276,126,200,44,238,79,162)(12,294,142,218,73,256,111,180,26,275,127,199,45,237,80,161)(13,293,143,217,74,255,112,179,27,274,128,198,46,236,81,160)(14,292,144,216,75,254,113,178,28,273,129,197,47,235,82,159)(15,291,145,215,76,253,114,177,29,272,130,196,48,234,83,158)(16,290,146,214,58,252,96,176,30,271,131,195,49,233,84,157)(17,289,147,213,59,251,97,175,31,270,132,194,50,232,85,156)(18,288,148,212,60,250,98,174,32,269,133,193,51,231,86,155)(19,287,149,211,61,249,99,173,33,268,115,192,52,230,87,154)>;
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), (1,286,150,210,62,248,100,172,34,267,116,191,53,229,88,153)(2,304,151,228,63,266,101,190,35,285,117,209,54,247,89,171)(3,303,152,227,64,265,102,189,36,284,118,208,55,246,90,170)(4,302,134,226,65,264,103,188,37,283,119,207,56,245,91,169)(5,301,135,225,66,263,104,187,38,282,120,206,57,244,92,168)(6,300,136,224,67,262,105,186,20,281,121,205,39,243,93,167)(7,299,137,223,68,261,106,185,21,280,122,204,40,242,94,166)(8,298,138,222,69,260,107,184,22,279,123,203,41,241,95,165)(9,297,139,221,70,259,108,183,23,278,124,202,42,240,77,164)(10,296,140,220,71,258,109,182,24,277,125,201,43,239,78,163)(11,295,141,219,72,257,110,181,25,276,126,200,44,238,79,162)(12,294,142,218,73,256,111,180,26,275,127,199,45,237,80,161)(13,293,143,217,74,255,112,179,27,274,128,198,46,236,81,160)(14,292,144,216,75,254,113,178,28,273,129,197,47,235,82,159)(15,291,145,215,76,253,114,177,29,272,130,196,48,234,83,158)(16,290,146,214,58,252,96,176,30,271,131,195,49,233,84,157)(17,289,147,213,59,251,97,175,31,270,132,194,50,232,85,156)(18,288,148,212,60,250,98,174,32,269,133,193,51,231,86,155)(19,287,149,211,61,249,99,173,33,268,115,192,52,230,87,154) );
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)], [(1,286,150,210,62,248,100,172,34,267,116,191,53,229,88,153),(2,304,151,228,63,266,101,190,35,285,117,209,54,247,89,171),(3,303,152,227,64,265,102,189,36,284,118,208,55,246,90,170),(4,302,134,226,65,264,103,188,37,283,119,207,56,245,91,169),(5,301,135,225,66,263,104,187,38,282,120,206,57,244,92,168),(6,300,136,224,67,262,105,186,20,281,121,205,39,243,93,167),(7,299,137,223,68,261,106,185,21,280,122,204,40,242,94,166),(8,298,138,222,69,260,107,184,22,279,123,203,41,241,95,165),(9,297,139,221,70,259,108,183,23,278,124,202,42,240,77,164),(10,296,140,220,71,258,109,182,24,277,125,201,43,239,78,163),(11,295,141,219,72,257,110,181,25,276,126,200,44,238,79,162),(12,294,142,218,73,256,111,180,26,275,127,199,45,237,80,161),(13,293,143,217,74,255,112,179,27,274,128,198,46,236,81,160),(14,292,144,216,75,254,113,178,28,273,129,197,47,235,82,159),(15,291,145,215,76,253,114,177,29,272,130,196,48,234,83,158),(16,290,146,214,58,252,96,176,30,271,131,195,49,233,84,157),(17,289,147,213,59,251,97,175,31,270,132,194,50,232,85,156),(18,288,148,212,60,250,98,174,32,269,133,193,51,231,86,155),(19,287,149,211,61,249,99,173,33,268,115,192,52,230,87,154)]])
88 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 16A | ··· | 16H | 19A | ··· | 19I | 38A | ··· | 38I | 76A | ··· | 76R | 152A | ··· | 152AJ |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 | 152 | ··· | 152 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 19 | ··· | 19 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||
| image | C1 | C2 | C4 | C8 | C16 | D19 | Dic19 | C19⋊C8 | C19⋊C16 |
| kernel | C19⋊C16 | C152 | C76 | C38 | C19 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 9 | 9 | 18 | 36 |
Matrix representation of C19⋊C16 ►in GL2(𝔽1217) generated by
| 0 | 1 |
| 1216 | 1118 |
| 367 | 892 |
| 1069 | 850 |
G:=sub<GL(2,GF(1217))| [0,1216,1,1118],[367,1069,892,850] >;
C19⋊C16 in GAP, Magma, Sage, TeX
C_{19}\rtimes C_{16} % in TeX
G:=Group("C19:C16"); // GroupNames label
G:=SmallGroup(304,1);
// by ID
G=gap.SmallGroup(304,1);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,10,26,42,7204]);
// Polycyclic
G:=Group<a,b|a^19=b^16=1,b*a*b^-1=a^-1>;
// generators/relations
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