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G = C3×C7⋊C16order 336 = 24·3·7

Direct product of C3 and C7⋊C16

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C3×C7⋊C16, C73C48, C212C16, C42.2C8, C84.5C4, C56.6C6, C24.4D7, C14.3C24, C168.5C2, C28.8C12, C12.5Dic7, C6.2(C7⋊C8), C8.2(C3×D7), C4.2(C3×Dic7), C2.(C3×C7⋊C8), SmallGroup(336,4)

Series: Derived Chief Lower central Upper central

C1C7 — C3×C7⋊C16
C1C7C14C28C56C168 — C3×C7⋊C16
C7 — C3×C7⋊C16
C1C24

Generators and relations for C3×C7⋊C16
 G = < a,b,c | a3=b7=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

7C16
7C48

Smallest permutation representation of C3×C7⋊C16
Regular action on 336 points
Generators in S336
(1 167 127)(2 168 128)(3 169 113)(4 170 114)(5 171 115)(6 172 116)(7 173 117)(8 174 118)(9 175 119)(10 176 120)(11 161 121)(12 162 122)(13 163 123)(14 164 124)(15 165 125)(16 166 126)(17 186 205)(18 187 206)(19 188 207)(20 189 208)(21 190 193)(22 191 194)(23 192 195)(24 177 196)(25 178 197)(26 179 198)(27 180 199)(28 181 200)(29 182 201)(30 183 202)(31 184 203)(32 185 204)(33 95 157)(34 96 158)(35 81 159)(36 82 160)(37 83 145)(38 84 146)(39 85 147)(40 86 148)(41 87 149)(42 88 150)(43 89 151)(44 90 152)(45 91 153)(46 92 154)(47 93 155)(48 94 156)(49 314 243)(50 315 244)(51 316 245)(52 317 246)(53 318 247)(54 319 248)(55 320 249)(56 305 250)(57 306 251)(58 307 252)(59 308 253)(60 309 254)(61 310 255)(62 311 256)(63 312 241)(64 313 242)(65 97 143)(66 98 144)(67 99 129)(68 100 130)(69 101 131)(70 102 132)(71 103 133)(72 104 134)(73 105 135)(74 106 136)(75 107 137)(76 108 138)(77 109 139)(78 110 140)(79 111 141)(80 112 142)(209 289 284)(210 290 285)(211 291 286)(212 292 287)(213 293 288)(214 294 273)(215 295 274)(216 296 275)(217 297 276)(218 298 277)(219 299 278)(220 300 279)(221 301 280)(222 302 281)(223 303 282)(224 304 283)(225 332 257)(226 333 258)(227 334 259)(228 335 260)(229 336 261)(230 321 262)(231 322 263)(232 323 264)(233 324 265)(234 325 266)(235 326 267)(236 327 268)(237 328 269)(238 329 270)(239 330 271)(240 331 272)
(1 243 200 97 239 299 48)(2 33 300 240 98 201 244)(3 245 202 99 225 301 34)(4 35 302 226 100 203 246)(5 247 204 101 227 303 36)(6 37 304 228 102 205 248)(7 249 206 103 229 289 38)(8 39 290 230 104 207 250)(9 251 208 105 231 291 40)(10 41 292 232 106 193 252)(11 253 194 107 233 293 42)(12 43 294 234 108 195 254)(13 255 196 109 235 295 44)(14 45 296 236 110 197 256)(15 241 198 111 237 297 46)(16 47 298 238 112 199 242)(17 54 172 83 283 335 132)(18 133 336 284 84 173 55)(19 56 174 85 285 321 134)(20 135 322 286 86 175 57)(21 58 176 87 287 323 136)(22 137 324 288 88 161 59)(23 60 162 89 273 325 138)(24 139 326 274 90 163 61)(25 62 164 91 275 327 140)(26 141 328 276 92 165 63)(27 64 166 93 277 329 142)(28 143 330 278 94 167 49)(29 50 168 95 279 331 144)(30 129 332 280 96 169 51)(31 52 170 81 281 333 130)(32 131 334 282 82 171 53)(65 271 219 156 127 314 181)(66 182 315 128 157 220 272)(67 257 221 158 113 316 183)(68 184 317 114 159 222 258)(69 259 223 160 115 318 185)(70 186 319 116 145 224 260)(71 261 209 146 117 320 187)(72 188 305 118 147 210 262)(73 263 211 148 119 306 189)(74 190 307 120 149 212 264)(75 265 213 150 121 308 191)(76 192 309 122 151 214 266)(77 267 215 152 123 310 177)(78 178 311 124 153 216 268)(79 269 217 154 125 312 179)(80 180 313 126 155 218 270)
(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)

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

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

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

120 conjugacy classes

class 1  2 3A3B4A4B6A6B7A7B7C8A8B8C8D12A12B12C12D14A14B14C16A···16H21A···21F24A···24H28A···28F42A···42F48A···48P56A···56L84A···84L168A···168X
order1233446677788881212121214141416···1621···2124···2428···2842···4248···4856···5684···84168···168
size11111111222111111112227···72···21···12···22···27···72···22···22···2

120 irreducible representations

dim111111111122222222
type+++-
imageC1C2C3C4C6C8C12C16C24C48D7Dic7C3×D7C7⋊C8C3×Dic7C7⋊C16C3×C7⋊C8C3×C7⋊C16
kernelC3×C7⋊C16C168C7⋊C16C84C56C42C28C21C14C7C24C12C8C6C4C3C2C1
# reps1122244881633666121224

Matrix representation of C3×C7⋊C16 in GL4(𝔽337) generated by

208000
0100
0010
0001
,
1000
0100
00144336
00145336
,
111000
019100
0015327
00151184
G:=sub<GL(4,GF(337))| [208,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,144,145,0,0,336,336],[111,0,0,0,0,191,0,0,0,0,153,151,0,0,27,184] >;

C3×C7⋊C16 in GAP, Magma, Sage, TeX

C_3\times C_7\rtimes C_{16}
% in TeX

G:=Group("C3xC7:C16");
// GroupNames label

G:=SmallGroup(336,4);
// by ID

G=gap.SmallGroup(336,4);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,-2,-7,36,50,69,10373]);
// Polycyclic

G:=Group<a,b,c|a^3=b^7=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×C7⋊C16 in TeX

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