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G = Q16×C21order 336 = 24·3·7

Direct product of C21 and Q16

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: Q16×C21, C8.C42, C56.7C6, C24.2C14, C168.6C2, C42.56D4, Q8.2C42, C84.79C22, C4.3(C2×C42), C6.16(C7×D4), C2.5(D4×C21), (C7×Q8).8C6, C28.42(C2×C6), C14.32(C3×D4), (Q8×C21).4C2, (C3×Q8).2C14, C12.19(C2×C14), SmallGroup(336,113)

Series: Derived Chief Lower central Upper central

C1C4 — Q16×C21
C1C2C4C28C84Q8×C21 — Q16×C21
C1C2C4 — Q16×C21
C1C42C84 — Q16×C21

Generators and relations for Q16×C21
 G = < a,b,c | a21=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

2C4
2C4
2C12
2C12
2C28
2C28
2C84
2C84

Smallest permutation representation of Q16×C21
Regular action on 336 points
Generators in S336
(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)(295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315)(316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336)
(1 133 170 91 316 271 311 126)(2 134 171 92 317 272 312 106)(3 135 172 93 318 273 313 107)(4 136 173 94 319 253 314 108)(5 137 174 95 320 254 315 109)(6 138 175 96 321 255 295 110)(7 139 176 97 322 256 296 111)(8 140 177 98 323 257 297 112)(9 141 178 99 324 258 298 113)(10 142 179 100 325 259 299 114)(11 143 180 101 326 260 300 115)(12 144 181 102 327 261 301 116)(13 145 182 103 328 262 302 117)(14 146 183 104 329 263 303 118)(15 147 184 105 330 264 304 119)(16 127 185 85 331 265 305 120)(17 128 186 86 332 266 306 121)(18 129 187 87 333 267 307 122)(19 130 188 88 334 268 308 123)(20 131 189 89 335 269 309 124)(21 132 169 90 336 270 310 125)(22 286 225 200 66 51 161 251)(23 287 226 201 67 52 162 252)(24 288 227 202 68 53 163 232)(25 289 228 203 69 54 164 233)(26 290 229 204 70 55 165 234)(27 291 230 205 71 56 166 235)(28 292 231 206 72 57 167 236)(29 293 211 207 73 58 168 237)(30 294 212 208 74 59 148 238)(31 274 213 209 75 60 149 239)(32 275 214 210 76 61 150 240)(33 276 215 190 77 62 151 241)(34 277 216 191 78 63 152 242)(35 278 217 192 79 43 153 243)(36 279 218 193 80 44 154 244)(37 280 219 194 81 45 155 245)(38 281 220 195 82 46 156 246)(39 282 221 196 83 47 157 247)(40 283 222 197 84 48 158 248)(41 284 223 198 64 49 159 249)(42 285 224 199 65 50 160 250)
(1 278 316 43)(2 279 317 44)(3 280 318 45)(4 281 319 46)(5 282 320 47)(6 283 321 48)(7 284 322 49)(8 285 323 50)(9 286 324 51)(10 287 325 52)(11 288 326 53)(12 289 327 54)(13 290 328 55)(14 291 329 56)(15 292 330 57)(16 293 331 58)(17 294 332 59)(18 274 333 60)(19 275 334 61)(20 276 335 62)(21 277 336 63)(22 258 66 141)(23 259 67 142)(24 260 68 143)(25 261 69 144)(26 262 70 145)(27 263 71 146)(28 264 72 147)(29 265 73 127)(30 266 74 128)(31 267 75 129)(32 268 76 130)(33 269 77 131)(34 270 78 132)(35 271 79 133)(36 272 80 134)(37 273 81 135)(38 253 82 136)(39 254 83 137)(40 255 84 138)(41 256 64 139)(42 257 65 140)(85 168 120 211)(86 148 121 212)(87 149 122 213)(88 150 123 214)(89 151 124 215)(90 152 125 216)(91 153 126 217)(92 154 106 218)(93 155 107 219)(94 156 108 220)(95 157 109 221)(96 158 110 222)(97 159 111 223)(98 160 112 224)(99 161 113 225)(100 162 114 226)(101 163 115 227)(102 164 116 228)(103 165 117 229)(104 166 118 230)(105 167 119 231)(169 242 310 191)(170 243 311 192)(171 244 312 193)(172 245 313 194)(173 246 314 195)(174 247 315 196)(175 248 295 197)(176 249 296 198)(177 250 297 199)(178 251 298 200)(179 252 299 201)(180 232 300 202)(181 233 301 203)(182 234 302 204)(183 235 303 205)(184 236 304 206)(185 237 305 207)(186 238 306 208)(187 239 307 209)(188 240 308 210)(189 241 309 190)

G:=sub<Sym(336)| (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)(295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315)(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336), (1,133,170,91,316,271,311,126)(2,134,171,92,317,272,312,106)(3,135,172,93,318,273,313,107)(4,136,173,94,319,253,314,108)(5,137,174,95,320,254,315,109)(6,138,175,96,321,255,295,110)(7,139,176,97,322,256,296,111)(8,140,177,98,323,257,297,112)(9,141,178,99,324,258,298,113)(10,142,179,100,325,259,299,114)(11,143,180,101,326,260,300,115)(12,144,181,102,327,261,301,116)(13,145,182,103,328,262,302,117)(14,146,183,104,329,263,303,118)(15,147,184,105,330,264,304,119)(16,127,185,85,331,265,305,120)(17,128,186,86,332,266,306,121)(18,129,187,87,333,267,307,122)(19,130,188,88,334,268,308,123)(20,131,189,89,335,269,309,124)(21,132,169,90,336,270,310,125)(22,286,225,200,66,51,161,251)(23,287,226,201,67,52,162,252)(24,288,227,202,68,53,163,232)(25,289,228,203,69,54,164,233)(26,290,229,204,70,55,165,234)(27,291,230,205,71,56,166,235)(28,292,231,206,72,57,167,236)(29,293,211,207,73,58,168,237)(30,294,212,208,74,59,148,238)(31,274,213,209,75,60,149,239)(32,275,214,210,76,61,150,240)(33,276,215,190,77,62,151,241)(34,277,216,191,78,63,152,242)(35,278,217,192,79,43,153,243)(36,279,218,193,80,44,154,244)(37,280,219,194,81,45,155,245)(38,281,220,195,82,46,156,246)(39,282,221,196,83,47,157,247)(40,283,222,197,84,48,158,248)(41,284,223,198,64,49,159,249)(42,285,224,199,65,50,160,250), (1,278,316,43)(2,279,317,44)(3,280,318,45)(4,281,319,46)(5,282,320,47)(6,283,321,48)(7,284,322,49)(8,285,323,50)(9,286,324,51)(10,287,325,52)(11,288,326,53)(12,289,327,54)(13,290,328,55)(14,291,329,56)(15,292,330,57)(16,293,331,58)(17,294,332,59)(18,274,333,60)(19,275,334,61)(20,276,335,62)(21,277,336,63)(22,258,66,141)(23,259,67,142)(24,260,68,143)(25,261,69,144)(26,262,70,145)(27,263,71,146)(28,264,72,147)(29,265,73,127)(30,266,74,128)(31,267,75,129)(32,268,76,130)(33,269,77,131)(34,270,78,132)(35,271,79,133)(36,272,80,134)(37,273,81,135)(38,253,82,136)(39,254,83,137)(40,255,84,138)(41,256,64,139)(42,257,65,140)(85,168,120,211)(86,148,121,212)(87,149,122,213)(88,150,123,214)(89,151,124,215)(90,152,125,216)(91,153,126,217)(92,154,106,218)(93,155,107,219)(94,156,108,220)(95,157,109,221)(96,158,110,222)(97,159,111,223)(98,160,112,224)(99,161,113,225)(100,162,114,226)(101,163,115,227)(102,164,116,228)(103,165,117,229)(104,166,118,230)(105,167,119,231)(169,242,310,191)(170,243,311,192)(171,244,312,193)(172,245,313,194)(173,246,314,195)(174,247,315,196)(175,248,295,197)(176,249,296,198)(177,250,297,199)(178,251,298,200)(179,252,299,201)(180,232,300,202)(181,233,301,203)(182,234,302,204)(183,235,303,205)(184,236,304,206)(185,237,305,207)(186,238,306,208)(187,239,307,209)(188,240,308,210)(189,241,309,190)>;

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)(295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315)(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336), (1,133,170,91,316,271,311,126)(2,134,171,92,317,272,312,106)(3,135,172,93,318,273,313,107)(4,136,173,94,319,253,314,108)(5,137,174,95,320,254,315,109)(6,138,175,96,321,255,295,110)(7,139,176,97,322,256,296,111)(8,140,177,98,323,257,297,112)(9,141,178,99,324,258,298,113)(10,142,179,100,325,259,299,114)(11,143,180,101,326,260,300,115)(12,144,181,102,327,261,301,116)(13,145,182,103,328,262,302,117)(14,146,183,104,329,263,303,118)(15,147,184,105,330,264,304,119)(16,127,185,85,331,265,305,120)(17,128,186,86,332,266,306,121)(18,129,187,87,333,267,307,122)(19,130,188,88,334,268,308,123)(20,131,189,89,335,269,309,124)(21,132,169,90,336,270,310,125)(22,286,225,200,66,51,161,251)(23,287,226,201,67,52,162,252)(24,288,227,202,68,53,163,232)(25,289,228,203,69,54,164,233)(26,290,229,204,70,55,165,234)(27,291,230,205,71,56,166,235)(28,292,231,206,72,57,167,236)(29,293,211,207,73,58,168,237)(30,294,212,208,74,59,148,238)(31,274,213,209,75,60,149,239)(32,275,214,210,76,61,150,240)(33,276,215,190,77,62,151,241)(34,277,216,191,78,63,152,242)(35,278,217,192,79,43,153,243)(36,279,218,193,80,44,154,244)(37,280,219,194,81,45,155,245)(38,281,220,195,82,46,156,246)(39,282,221,196,83,47,157,247)(40,283,222,197,84,48,158,248)(41,284,223,198,64,49,159,249)(42,285,224,199,65,50,160,250), (1,278,316,43)(2,279,317,44)(3,280,318,45)(4,281,319,46)(5,282,320,47)(6,283,321,48)(7,284,322,49)(8,285,323,50)(9,286,324,51)(10,287,325,52)(11,288,326,53)(12,289,327,54)(13,290,328,55)(14,291,329,56)(15,292,330,57)(16,293,331,58)(17,294,332,59)(18,274,333,60)(19,275,334,61)(20,276,335,62)(21,277,336,63)(22,258,66,141)(23,259,67,142)(24,260,68,143)(25,261,69,144)(26,262,70,145)(27,263,71,146)(28,264,72,147)(29,265,73,127)(30,266,74,128)(31,267,75,129)(32,268,76,130)(33,269,77,131)(34,270,78,132)(35,271,79,133)(36,272,80,134)(37,273,81,135)(38,253,82,136)(39,254,83,137)(40,255,84,138)(41,256,64,139)(42,257,65,140)(85,168,120,211)(86,148,121,212)(87,149,122,213)(88,150,123,214)(89,151,124,215)(90,152,125,216)(91,153,126,217)(92,154,106,218)(93,155,107,219)(94,156,108,220)(95,157,109,221)(96,158,110,222)(97,159,111,223)(98,160,112,224)(99,161,113,225)(100,162,114,226)(101,163,115,227)(102,164,116,228)(103,165,117,229)(104,166,118,230)(105,167,119,231)(169,242,310,191)(170,243,311,192)(171,244,312,193)(172,245,313,194)(173,246,314,195)(174,247,315,196)(175,248,295,197)(176,249,296,198)(177,250,297,199)(178,251,298,200)(179,252,299,201)(180,232,300,202)(181,233,301,203)(182,234,302,204)(183,235,303,205)(184,236,304,206)(185,237,305,207)(186,238,306,208)(187,239,307,209)(188,240,308,210)(189,241,309,190) );

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),(295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315),(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336)], [(1,133,170,91,316,271,311,126),(2,134,171,92,317,272,312,106),(3,135,172,93,318,273,313,107),(4,136,173,94,319,253,314,108),(5,137,174,95,320,254,315,109),(6,138,175,96,321,255,295,110),(7,139,176,97,322,256,296,111),(8,140,177,98,323,257,297,112),(9,141,178,99,324,258,298,113),(10,142,179,100,325,259,299,114),(11,143,180,101,326,260,300,115),(12,144,181,102,327,261,301,116),(13,145,182,103,328,262,302,117),(14,146,183,104,329,263,303,118),(15,147,184,105,330,264,304,119),(16,127,185,85,331,265,305,120),(17,128,186,86,332,266,306,121),(18,129,187,87,333,267,307,122),(19,130,188,88,334,268,308,123),(20,131,189,89,335,269,309,124),(21,132,169,90,336,270,310,125),(22,286,225,200,66,51,161,251),(23,287,226,201,67,52,162,252),(24,288,227,202,68,53,163,232),(25,289,228,203,69,54,164,233),(26,290,229,204,70,55,165,234),(27,291,230,205,71,56,166,235),(28,292,231,206,72,57,167,236),(29,293,211,207,73,58,168,237),(30,294,212,208,74,59,148,238),(31,274,213,209,75,60,149,239),(32,275,214,210,76,61,150,240),(33,276,215,190,77,62,151,241),(34,277,216,191,78,63,152,242),(35,278,217,192,79,43,153,243),(36,279,218,193,80,44,154,244),(37,280,219,194,81,45,155,245),(38,281,220,195,82,46,156,246),(39,282,221,196,83,47,157,247),(40,283,222,197,84,48,158,248),(41,284,223,198,64,49,159,249),(42,285,224,199,65,50,160,250)], [(1,278,316,43),(2,279,317,44),(3,280,318,45),(4,281,319,46),(5,282,320,47),(6,283,321,48),(7,284,322,49),(8,285,323,50),(9,286,324,51),(10,287,325,52),(11,288,326,53),(12,289,327,54),(13,290,328,55),(14,291,329,56),(15,292,330,57),(16,293,331,58),(17,294,332,59),(18,274,333,60),(19,275,334,61),(20,276,335,62),(21,277,336,63),(22,258,66,141),(23,259,67,142),(24,260,68,143),(25,261,69,144),(26,262,70,145),(27,263,71,146),(28,264,72,147),(29,265,73,127),(30,266,74,128),(31,267,75,129),(32,268,76,130),(33,269,77,131),(34,270,78,132),(35,271,79,133),(36,272,80,134),(37,273,81,135),(38,253,82,136),(39,254,83,137),(40,255,84,138),(41,256,64,139),(42,257,65,140),(85,168,120,211),(86,148,121,212),(87,149,122,213),(88,150,123,214),(89,151,124,215),(90,152,125,216),(91,153,126,217),(92,154,106,218),(93,155,107,219),(94,156,108,220),(95,157,109,221),(96,158,110,222),(97,159,111,223),(98,160,112,224),(99,161,113,225),(100,162,114,226),(101,163,115,227),(102,164,116,228),(103,165,117,229),(104,166,118,230),(105,167,119,231),(169,242,310,191),(170,243,311,192),(171,244,312,193),(172,245,313,194),(173,246,314,195),(174,247,315,196),(175,248,295,197),(176,249,296,198),(177,250,297,199),(178,251,298,200),(179,252,299,201),(180,232,300,202),(181,233,301,203),(182,234,302,204),(183,235,303,205),(184,236,304,206),(185,237,305,207),(186,238,306,208),(187,239,307,209),(188,240,308,210),(189,241,309,190)])

147 conjugacy classes

class 1  2 3A3B4A4B4C6A6B7A···7F8A8B12A12B12C12D12E12F14A···14F21A···21L24A24B24C24D28A···28F28G···28R42A···42L56A···56L84A···84L84M···84AJ168A···168X
order1233444667···78812121212121214···1421···212424242428···2828···2842···4256···5684···8484···84168···168
size1111244111···1222244441···11···122222···24···41···12···22···24···42···2

147 irreducible representations

dim11111111111122222222
type++++-
imageC1C2C2C3C6C6C7C14C14C21C42C42D4Q16C3×D4C3×Q16C7×D4C7×Q16D4×C21Q16×C21
kernelQ16×C21C168Q8×C21C7×Q16C56C7×Q8C3×Q16C24C3×Q8Q16C8Q8C42C21C14C7C6C3C2C1
# reps112224661212122412246121224

Matrix representation of Q16×C21 in GL2(𝔽337) generated by

260
026
,
32413
324324
,
244198
19893
G:=sub<GL(2,GF(337))| [26,0,0,26],[324,324,13,324],[244,198,198,93] >;

Q16×C21 in GAP, Magma, Sage, TeX

Q_{16}\times C_{21}
% in TeX

G:=Group("Q16xC21");
// GroupNames label

G:=SmallGroup(336,113);
// by ID

G=gap.SmallGroup(336,113);
# by ID

G:=PCGroup([6,-2,-2,-3,-7,-2,-2,1008,1033,1015,7564,3790,88]);
// Polycyclic

G:=Group<a,b,c|a^21=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Q16×C21 in TeX

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