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## G = Q16×C19order 304 = 24·19

### Direct product of C19 and Q16

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

 Derived series C1 — C4 — Q16×C19
 Chief series C1 — C2 — C4 — C76 — Q8×C19 — Q16×C19
 Lower central C1 — C2 — C4 — Q16×C19
 Upper central C1 — C38 — C76 — Q16×C19

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

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

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

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

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

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