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G = C3×Dic28order 336 = 24·3·7

Direct product of C3 and Dic28

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

Aliases: C3×Dic28, C215Q16, C56.5C6, C24.2D7, C168.2C2, C42.25D4, C6.15D28, C12.54D14, C84.61C22, Dic14.3C6, C8.(C3×D7), C74(C3×Q16), C4.10(C6×D7), C2.5(C3×D28), C28.33(C2×C6), C14.19(C3×D4), (C3×Dic14).3C2, SmallGroup(336,62)

Series: Derived Chief Lower central Upper central

C1C28 — C3×Dic28
C1C7C14C28C84C3×Dic14 — C3×Dic28
C7C14C28 — C3×Dic28
C1C6C12C24

Generators and relations for C3×Dic28
 G = < a,b,c | a3=b56=1, c2=b28, ab=ba, ac=ca, cbc-1=b-1 >

14C4
14C4
7Q8
7Q8
14C12
14C12
2Dic7
2Dic7
7Q16
7C3×Q8
7C3×Q8
2C3×Dic7
2C3×Dic7
7C3×Q16

Smallest permutation representation of C3×Dic28
Regular action on 336 points
Generators in S336
(1 277 163)(2 278 164)(3 279 165)(4 280 166)(5 225 167)(6 226 168)(7 227 113)(8 228 114)(9 229 115)(10 230 116)(11 231 117)(12 232 118)(13 233 119)(14 234 120)(15 235 121)(16 236 122)(17 237 123)(18 238 124)(19 239 125)(20 240 126)(21 241 127)(22 242 128)(23 243 129)(24 244 130)(25 245 131)(26 246 132)(27 247 133)(28 248 134)(29 249 135)(30 250 136)(31 251 137)(32 252 138)(33 253 139)(34 254 140)(35 255 141)(36 256 142)(37 257 143)(38 258 144)(39 259 145)(40 260 146)(41 261 147)(42 262 148)(43 263 149)(44 264 150)(45 265 151)(46 266 152)(47 267 153)(48 268 154)(49 269 155)(50 270 156)(51 271 157)(52 272 158)(53 273 159)(54 274 160)(55 275 161)(56 276 162)(57 312 208)(58 313 209)(59 314 210)(60 315 211)(61 316 212)(62 317 213)(63 318 214)(64 319 215)(65 320 216)(66 321 217)(67 322 218)(68 323 219)(69 324 220)(70 325 221)(71 326 222)(72 327 223)(73 328 224)(74 329 169)(75 330 170)(76 331 171)(77 332 172)(78 333 173)(79 334 174)(80 335 175)(81 336 176)(82 281 177)(83 282 178)(84 283 179)(85 284 180)(86 285 181)(87 286 182)(88 287 183)(89 288 184)(90 289 185)(91 290 186)(92 291 187)(93 292 188)(94 293 189)(95 294 190)(96 295 191)(97 296 192)(98 297 193)(99 298 194)(100 299 195)(101 300 196)(102 301 197)(103 302 198)(104 303 199)(105 304 200)(106 305 201)(107 306 202)(108 307 203)(109 308 204)(110 309 205)(111 310 206)(112 311 207)
(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)
(1 177 29 205)(2 176 30 204)(3 175 31 203)(4 174 32 202)(5 173 33 201)(6 172 34 200)(7 171 35 199)(8 170 36 198)(9 169 37 197)(10 224 38 196)(11 223 39 195)(12 222 40 194)(13 221 41 193)(14 220 42 192)(15 219 43 191)(16 218 44 190)(17 217 45 189)(18 216 46 188)(19 215 47 187)(20 214 48 186)(21 213 49 185)(22 212 50 184)(23 211 51 183)(24 210 52 182)(25 209 53 181)(26 208 54 180)(27 207 55 179)(28 206 56 178)(57 274 85 246)(58 273 86 245)(59 272 87 244)(60 271 88 243)(61 270 89 242)(62 269 90 241)(63 268 91 240)(64 267 92 239)(65 266 93 238)(66 265 94 237)(67 264 95 236)(68 263 96 235)(69 262 97 234)(70 261 98 233)(71 260 99 232)(72 259 100 231)(73 258 101 230)(74 257 102 229)(75 256 103 228)(76 255 104 227)(77 254 105 226)(78 253 106 225)(79 252 107 280)(80 251 108 279)(81 250 109 278)(82 249 110 277)(83 248 111 276)(84 247 112 275)(113 331 141 303)(114 330 142 302)(115 329 143 301)(116 328 144 300)(117 327 145 299)(118 326 146 298)(119 325 147 297)(120 324 148 296)(121 323 149 295)(122 322 150 294)(123 321 151 293)(124 320 152 292)(125 319 153 291)(126 318 154 290)(127 317 155 289)(128 316 156 288)(129 315 157 287)(130 314 158 286)(131 313 159 285)(132 312 160 284)(133 311 161 283)(134 310 162 282)(135 309 163 281)(136 308 164 336)(137 307 165 335)(138 306 166 334)(139 305 167 333)(140 304 168 332)

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

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

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

93 conjugacy classes

class 1  2 3A3B4A4B4C6A6B7A7B7C8A8B12A12B12C12D12E12F14A14B14C21A···21F24A24B24C24D28A···28F42A···42F56A···56L84A···84L168A···168X
order1233444667778812121212121214141421···212424242428···2842···4256···5684···84168···168
size111122828112222222282828282222···222222···22···22···22···22···2

93 irreducible representations

dim111111222222222222
type+++++-++-
imageC1C2C2C3C6C6D4D7Q16C3×D4D14C3×D7C3×Q16D28C6×D7Dic28C3×D28C3×Dic28
kernelC3×Dic28C168C3×Dic14Dic28C56Dic14C42C24C21C14C12C8C7C6C4C3C2C1
# reps112224132236466121224

Matrix representation of C3×Dic28 in GL2(𝔽337) generated by

1280
0128
,
108224
11369
,
21446
96123
G:=sub<GL(2,GF(337))| [128,0,0,128],[108,113,224,69],[214,96,46,123] >;

C3×Dic28 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{28}
% in TeX

G:=Group("C3xDic28");
// GroupNames label

G:=SmallGroup(336,62);
// by ID

G=gap.SmallGroup(336,62);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,144,169,223,867,69,10373]);
// Polycyclic

G:=Group<a,b,c|a^3=b^56=1,c^2=b^28,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×Dic28 in TeX

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