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G = C19⋊C16order 304 = 24·19

The semidirect product of C19 and C16 acting via C16/C8=C2

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

C1C19 — C19⋊C16
C1C19C38C76C152 — C19⋊C16
C19 — C19⋊C16
C1C8

Generators and relations for C19⋊C16
 G = < a,b | a19=b16=1, bab-1=a-1 >

19C16

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)
(1 286 143 210 76 248 99 172 31 267 125 191 53 229 95 153)(2 304 144 228 58 266 100 190 32 285 126 209 54 247 77 171)(3 303 145 227 59 265 101 189 33 284 127 208 55 246 78 170)(4 302 146 226 60 264 102 188 34 283 128 207 56 245 79 169)(5 301 147 225 61 263 103 187 35 282 129 206 57 244 80 168)(6 300 148 224 62 262 104 186 36 281 130 205 39 243 81 167)(7 299 149 223 63 261 105 185 37 280 131 204 40 242 82 166)(8 298 150 222 64 260 106 184 38 279 132 203 41 241 83 165)(9 297 151 221 65 259 107 183 20 278 133 202 42 240 84 164)(10 296 152 220 66 258 108 182 21 277 115 201 43 239 85 163)(11 295 134 219 67 257 109 181 22 276 116 200 44 238 86 162)(12 294 135 218 68 256 110 180 23 275 117 199 45 237 87 161)(13 293 136 217 69 255 111 179 24 274 118 198 46 236 88 160)(14 292 137 216 70 254 112 178 25 273 119 197 47 235 89 159)(15 291 138 215 71 253 113 177 26 272 120 196 48 234 90 158)(16 290 139 214 72 252 114 176 27 271 121 195 49 233 91 157)(17 289 140 213 73 251 96 175 28 270 122 194 50 232 92 156)(18 288 141 212 74 250 97 174 29 269 123 193 51 231 93 155)(19 287 142 211 75 249 98 173 30 268 124 192 52 230 94 154)

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), (1,286,143,210,76,248,99,172,31,267,125,191,53,229,95,153)(2,304,144,228,58,266,100,190,32,285,126,209,54,247,77,171)(3,303,145,227,59,265,101,189,33,284,127,208,55,246,78,170)(4,302,146,226,60,264,102,188,34,283,128,207,56,245,79,169)(5,301,147,225,61,263,103,187,35,282,129,206,57,244,80,168)(6,300,148,224,62,262,104,186,36,281,130,205,39,243,81,167)(7,299,149,223,63,261,105,185,37,280,131,204,40,242,82,166)(8,298,150,222,64,260,106,184,38,279,132,203,41,241,83,165)(9,297,151,221,65,259,107,183,20,278,133,202,42,240,84,164)(10,296,152,220,66,258,108,182,21,277,115,201,43,239,85,163)(11,295,134,219,67,257,109,181,22,276,116,200,44,238,86,162)(12,294,135,218,68,256,110,180,23,275,117,199,45,237,87,161)(13,293,136,217,69,255,111,179,24,274,118,198,46,236,88,160)(14,292,137,216,70,254,112,178,25,273,119,197,47,235,89,159)(15,291,138,215,71,253,113,177,26,272,120,196,48,234,90,158)(16,290,139,214,72,252,114,176,27,271,121,195,49,233,91,157)(17,289,140,213,73,251,96,175,28,270,122,194,50,232,92,156)(18,288,141,212,74,250,97,174,29,269,123,193,51,231,93,155)(19,287,142,211,75,249,98,173,30,268,124,192,52,230,94,154)>;

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), (1,286,143,210,76,248,99,172,31,267,125,191,53,229,95,153)(2,304,144,228,58,266,100,190,32,285,126,209,54,247,77,171)(3,303,145,227,59,265,101,189,33,284,127,208,55,246,78,170)(4,302,146,226,60,264,102,188,34,283,128,207,56,245,79,169)(5,301,147,225,61,263,103,187,35,282,129,206,57,244,80,168)(6,300,148,224,62,262,104,186,36,281,130,205,39,243,81,167)(7,299,149,223,63,261,105,185,37,280,131,204,40,242,82,166)(8,298,150,222,64,260,106,184,38,279,132,203,41,241,83,165)(9,297,151,221,65,259,107,183,20,278,133,202,42,240,84,164)(10,296,152,220,66,258,108,182,21,277,115,201,43,239,85,163)(11,295,134,219,67,257,109,181,22,276,116,200,44,238,86,162)(12,294,135,218,68,256,110,180,23,275,117,199,45,237,87,161)(13,293,136,217,69,255,111,179,24,274,118,198,46,236,88,160)(14,292,137,216,70,254,112,178,25,273,119,197,47,235,89,159)(15,291,138,215,71,253,113,177,26,272,120,196,48,234,90,158)(16,290,139,214,72,252,114,176,27,271,121,195,49,233,91,157)(17,289,140,213,73,251,96,175,28,270,122,194,50,232,92,156)(18,288,141,212,74,250,97,174,29,269,123,193,51,231,93,155)(19,287,142,211,75,249,98,173,30,268,124,192,52,230,94,154) );

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)], [(1,286,143,210,76,248,99,172,31,267,125,191,53,229,95,153),(2,304,144,228,58,266,100,190,32,285,126,209,54,247,77,171),(3,303,145,227,59,265,101,189,33,284,127,208,55,246,78,170),(4,302,146,226,60,264,102,188,34,283,128,207,56,245,79,169),(5,301,147,225,61,263,103,187,35,282,129,206,57,244,80,168),(6,300,148,224,62,262,104,186,36,281,130,205,39,243,81,167),(7,299,149,223,63,261,105,185,37,280,131,204,40,242,82,166),(8,298,150,222,64,260,106,184,38,279,132,203,41,241,83,165),(9,297,151,221,65,259,107,183,20,278,133,202,42,240,84,164),(10,296,152,220,66,258,108,182,21,277,115,201,43,239,85,163),(11,295,134,219,67,257,109,181,22,276,116,200,44,238,86,162),(12,294,135,218,68,256,110,180,23,275,117,199,45,237,87,161),(13,293,136,217,69,255,111,179,24,274,118,198,46,236,88,160),(14,292,137,216,70,254,112,178,25,273,119,197,47,235,89,159),(15,291,138,215,71,253,113,177,26,272,120,196,48,234,90,158),(16,290,139,214,72,252,114,176,27,271,121,195,49,233,91,157),(17,289,140,213,73,251,96,175,28,270,122,194,50,232,92,156),(18,288,141,212,74,250,97,174,29,269,123,193,51,231,93,155),(19,287,142,211,75,249,98,173,30,268,124,192,52,230,94,154)])

88 conjugacy classes

class 1  2 4A4B8A8B8C8D16A···16H19A···19I38A···38I76A···76R152A···152AJ
order1244888816···1619···1938···3876···76152···152
size1111111119···192···22···22···22···2

88 irreducible representations

dim111112222
type+++-
imageC1C2C4C8C16D19Dic19C19⋊C8C19⋊C16
kernelC19⋊C16C152C76C38C19C8C4C2C1
# reps11248991836

Matrix representation of C19⋊C16 in GL2(𝔽1217) generated by

01
12161118
,
367892
1069850
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

Export

Subgroup lattice of C19⋊C16 in TeX

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