direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic42, C42⋊2Q8, C6⋊2Dic14, C14⋊2Dic6, C4.11D42, C28.46D6, C12.46D14, C22.8D42, C42.28C23, C84.53C22, Dic21.7C22, C21⋊5(C2×Q8), C7⋊3(C2×Dic6), (C2×C84).6C2, (C2×C28).4S3, (C2×C12).4D7, (C2×C4).4D21, C3⋊3(C2×Dic14), (C2×C14).26D6, (C2×C6).26D14, C6.28(C22×D7), C2.3(C22×D21), C14.28(C22×S3), (C2×C42).27C22, (C2×Dic21).3C2, SmallGroup(336,194)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic42
G = < a,b,c | a2=b84=1, c2=b42, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 384 in 76 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C7, C2×C4, C2×C4, Q8, Dic3, C12, C2×C6, C14, C14, C2×Q8, C21, Dic6, C2×Dic3, C2×C12, Dic7, C28, C2×C14, C42, C42, C2×Dic6, Dic14, C2×Dic7, C2×C28, Dic21, C84, C2×C42, C2×Dic14, Dic42, C2×Dic21, C2×C84, C2×Dic42
Quotients: C1, C2, C22, S3, Q8, C23, D6, D7, C2×Q8, Dic6, C22×S3, D14, D21, C2×Dic6, Dic14, C22×D7, D42, C2×Dic14, Dic42, C22×D21, C2×Dic42
►Regular action on 336 points
Generators in S336
(1 180)(2 181)(3 182)(4 183)(5 184)(6 185)(7 186)(8 187)(9 188)(10 189)(11 190)(12 191)(13 192)(14 193)(15 194)(16 195)(17 196)(18 197)(19 198)(20 199)(21 200)(22 201)(23 202)(24 203)(25 204)(26 205)(27 206)(28 207)(29 208)(30 209)(31 210)(32 211)(33 212)(34 213)(35 214)(36 215)(37 216)(38 217)(39 218)(40 219)(41 220)(42 221)(43 222)(44 223)(45 224)(46 225)(47 226)(48 227)(49 228)(50 229)(51 230)(52 231)(53 232)(54 233)(55 234)(56 235)(57 236)(58 237)(59 238)(60 239)(61 240)(62 241)(63 242)(64 243)(65 244)(66 245)(67 246)(68 247)(69 248)(70 249)(71 250)(72 251)(73 252)(74 169)(75 170)(76 171)(77 172)(78 173)(79 174)(80 175)(81 176)(82 177)(83 178)(84 179)(85 282)(86 283)(87 284)(88 285)(89 286)(90 287)(91 288)(92 289)(93 290)(94 291)(95 292)(96 293)(97 294)(98 295)(99 296)(100 297)(101 298)(102 299)(103 300)(104 301)(105 302)(106 303)(107 304)(108 305)(109 306)(110 307)(111 308)(112 309)(113 310)(114 311)(115 312)(116 313)(117 314)(118 315)(119 316)(120 317)(121 318)(122 319)(123 320)(124 321)(125 322)(126 323)(127 324)(128 325)(129 326)(130 327)(131 328)(132 329)(133 330)(134 331)(135 332)(136 333)(137 334)(138 335)(139 336)(140 253)(141 254)(142 255)(143 256)(144 257)(145 258)(146 259)(147 260)(148 261)(149 262)(150 263)(151 264)(152 265)(153 266)(154 267)(155 268)(156 269)(157 270)(158 271)(159 272)(160 273)(161 274)(162 275)(163 276)(164 277)(165 278)(166 279)(167 280)(168 281)
(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)
(1 336 43 294)(2 335 44 293)(3 334 45 292)(4 333 46 291)(5 332 47 290)(6 331 48 289)(7 330 49 288)(8 329 50 287)(9 328 51 286)(10 327 52 285)(11 326 53 284)(12 325 54 283)(13 324 55 282)(14 323 56 281)(15 322 57 280)(16 321 58 279)(17 320 59 278)(18 319 60 277)(19 318 61 276)(20 317 62 275)(21 316 63 274)(22 315 64 273)(23 314 65 272)(24 313 66 271)(25 312 67 270)(26 311 68 269)(27 310 69 268)(28 309 70 267)(29 308 71 266)(30 307 72 265)(31 306 73 264)(32 305 74 263)(33 304 75 262)(34 303 76 261)(35 302 77 260)(36 301 78 259)(37 300 79 258)(38 299 80 257)(39 298 81 256)(40 297 82 255)(41 296 83 254)(42 295 84 253)(85 192 127 234)(86 191 128 233)(87 190 129 232)(88 189 130 231)(89 188 131 230)(90 187 132 229)(91 186 133 228)(92 185 134 227)(93 184 135 226)(94 183 136 225)(95 182 137 224)(96 181 138 223)(97 180 139 222)(98 179 140 221)(99 178 141 220)(100 177 142 219)(101 176 143 218)(102 175 144 217)(103 174 145 216)(104 173 146 215)(105 172 147 214)(106 171 148 213)(107 170 149 212)(108 169 150 211)(109 252 151 210)(110 251 152 209)(111 250 153 208)(112 249 154 207)(113 248 155 206)(114 247 156 205)(115 246 157 204)(116 245 158 203)(117 244 159 202)(118 243 160 201)(119 242 161 200)(120 241 162 199)(121 240 163 198)(122 239 164 197)(123 238 165 196)(124 237 166 195)(125 236 167 194)(126 235 168 193)
G:=sub<Sym(336)| (1,180)(2,181)(3,182)(4,183)(5,184)(6,185)(7,186)(8,187)(9,188)(10,189)(11,190)(12,191)(13,192)(14,193)(15,194)(16,195)(17,196)(18,197)(19,198)(20,199)(21,200)(22,201)(23,202)(24,203)(25,204)(26,205)(27,206)(28,207)(29,208)(30,209)(31,210)(32,211)(33,212)(34,213)(35,214)(36,215)(37,216)(38,217)(39,218)(40,219)(41,220)(42,221)(43,222)(44,223)(45,224)(46,225)(47,226)(48,227)(49,228)(50,229)(51,230)(52,231)(53,232)(54,233)(55,234)(56,235)(57,236)(58,237)(59,238)(60,239)(61,240)(62,241)(63,242)(64,243)(65,244)(66,245)(67,246)(68,247)(69,248)(70,249)(71,250)(72,251)(73,252)(74,169)(75,170)(76,171)(77,172)(78,173)(79,174)(80,175)(81,176)(82,177)(83,178)(84,179)(85,282)(86,283)(87,284)(88,285)(89,286)(90,287)(91,288)(92,289)(93,290)(94,291)(95,292)(96,293)(97,294)(98,295)(99,296)(100,297)(101,298)(102,299)(103,300)(104,301)(105,302)(106,303)(107,304)(108,305)(109,306)(110,307)(111,308)(112,309)(113,310)(114,311)(115,312)(116,313)(117,314)(118,315)(119,316)(120,317)(121,318)(122,319)(123,320)(124,321)(125,322)(126,323)(127,324)(128,325)(129,326)(130,327)(131,328)(132,329)(133,330)(134,331)(135,332)(136,333)(137,334)(138,335)(139,336)(140,253)(141,254)(142,255)(143,256)(144,257)(145,258)(146,259)(147,260)(148,261)(149,262)(150,263)(151,264)(152,265)(153,266)(154,267)(155,268)(156,269)(157,270)(158,271)(159,272)(160,273)(161,274)(162,275)(163,276)(164,277)(165,278)(166,279)(167,280)(168,281), (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), (1,336,43,294)(2,335,44,293)(3,334,45,292)(4,333,46,291)(5,332,47,290)(6,331,48,289)(7,330,49,288)(8,329,50,287)(9,328,51,286)(10,327,52,285)(11,326,53,284)(12,325,54,283)(13,324,55,282)(14,323,56,281)(15,322,57,280)(16,321,58,279)(17,320,59,278)(18,319,60,277)(19,318,61,276)(20,317,62,275)(21,316,63,274)(22,315,64,273)(23,314,65,272)(24,313,66,271)(25,312,67,270)(26,311,68,269)(27,310,69,268)(28,309,70,267)(29,308,71,266)(30,307,72,265)(31,306,73,264)(32,305,74,263)(33,304,75,262)(34,303,76,261)(35,302,77,260)(36,301,78,259)(37,300,79,258)(38,299,80,257)(39,298,81,256)(40,297,82,255)(41,296,83,254)(42,295,84,253)(85,192,127,234)(86,191,128,233)(87,190,129,232)(88,189,130,231)(89,188,131,230)(90,187,132,229)(91,186,133,228)(92,185,134,227)(93,184,135,226)(94,183,136,225)(95,182,137,224)(96,181,138,223)(97,180,139,222)(98,179,140,221)(99,178,141,220)(100,177,142,219)(101,176,143,218)(102,175,144,217)(103,174,145,216)(104,173,146,215)(105,172,147,214)(106,171,148,213)(107,170,149,212)(108,169,150,211)(109,252,151,210)(110,251,152,209)(111,250,153,208)(112,249,154,207)(113,248,155,206)(114,247,156,205)(115,246,157,204)(116,245,158,203)(117,244,159,202)(118,243,160,201)(119,242,161,200)(120,241,162,199)(121,240,163,198)(122,239,164,197)(123,238,165,196)(124,237,166,195)(125,236,167,194)(126,235,168,193)>;
G:=Group( (1,180)(2,181)(3,182)(4,183)(5,184)(6,185)(7,186)(8,187)(9,188)(10,189)(11,190)(12,191)(13,192)(14,193)(15,194)(16,195)(17,196)(18,197)(19,198)(20,199)(21,200)(22,201)(23,202)(24,203)(25,204)(26,205)(27,206)(28,207)(29,208)(30,209)(31,210)(32,211)(33,212)(34,213)(35,214)(36,215)(37,216)(38,217)(39,218)(40,219)(41,220)(42,221)(43,222)(44,223)(45,224)(46,225)(47,226)(48,227)(49,228)(50,229)(51,230)(52,231)(53,232)(54,233)(55,234)(56,235)(57,236)(58,237)(59,238)(60,239)(61,240)(62,241)(63,242)(64,243)(65,244)(66,245)(67,246)(68,247)(69,248)(70,249)(71,250)(72,251)(73,252)(74,169)(75,170)(76,171)(77,172)(78,173)(79,174)(80,175)(81,176)(82,177)(83,178)(84,179)(85,282)(86,283)(87,284)(88,285)(89,286)(90,287)(91,288)(92,289)(93,290)(94,291)(95,292)(96,293)(97,294)(98,295)(99,296)(100,297)(101,298)(102,299)(103,300)(104,301)(105,302)(106,303)(107,304)(108,305)(109,306)(110,307)(111,308)(112,309)(113,310)(114,311)(115,312)(116,313)(117,314)(118,315)(119,316)(120,317)(121,318)(122,319)(123,320)(124,321)(125,322)(126,323)(127,324)(128,325)(129,326)(130,327)(131,328)(132,329)(133,330)(134,331)(135,332)(136,333)(137,334)(138,335)(139,336)(140,253)(141,254)(142,255)(143,256)(144,257)(145,258)(146,259)(147,260)(148,261)(149,262)(150,263)(151,264)(152,265)(153,266)(154,267)(155,268)(156,269)(157,270)(158,271)(159,272)(160,273)(161,274)(162,275)(163,276)(164,277)(165,278)(166,279)(167,280)(168,281), (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), (1,336,43,294)(2,335,44,293)(3,334,45,292)(4,333,46,291)(5,332,47,290)(6,331,48,289)(7,330,49,288)(8,329,50,287)(9,328,51,286)(10,327,52,285)(11,326,53,284)(12,325,54,283)(13,324,55,282)(14,323,56,281)(15,322,57,280)(16,321,58,279)(17,320,59,278)(18,319,60,277)(19,318,61,276)(20,317,62,275)(21,316,63,274)(22,315,64,273)(23,314,65,272)(24,313,66,271)(25,312,67,270)(26,311,68,269)(27,310,69,268)(28,309,70,267)(29,308,71,266)(30,307,72,265)(31,306,73,264)(32,305,74,263)(33,304,75,262)(34,303,76,261)(35,302,77,260)(36,301,78,259)(37,300,79,258)(38,299,80,257)(39,298,81,256)(40,297,82,255)(41,296,83,254)(42,295,84,253)(85,192,127,234)(86,191,128,233)(87,190,129,232)(88,189,130,231)(89,188,131,230)(90,187,132,229)(91,186,133,228)(92,185,134,227)(93,184,135,226)(94,183,136,225)(95,182,137,224)(96,181,138,223)(97,180,139,222)(98,179,140,221)(99,178,141,220)(100,177,142,219)(101,176,143,218)(102,175,144,217)(103,174,145,216)(104,173,146,215)(105,172,147,214)(106,171,148,213)(107,170,149,212)(108,169,150,211)(109,252,151,210)(110,251,152,209)(111,250,153,208)(112,249,154,207)(113,248,155,206)(114,247,156,205)(115,246,157,204)(116,245,158,203)(117,244,159,202)(118,243,160,201)(119,242,161,200)(120,241,162,199)(121,240,163,198)(122,239,164,197)(123,238,165,196)(124,237,166,195)(125,236,167,194)(126,235,168,193) );
G=PermutationGroup([[(1,180),(2,181),(3,182),(4,183),(5,184),(6,185),(7,186),(8,187),(9,188),(10,189),(11,190),(12,191),(13,192),(14,193),(15,194),(16,195),(17,196),(18,197),(19,198),(20,199),(21,200),(22,201),(23,202),(24,203),(25,204),(26,205),(27,206),(28,207),(29,208),(30,209),(31,210),(32,211),(33,212),(34,213),(35,214),(36,215),(37,216),(38,217),(39,218),(40,219),(41,220),(42,221),(43,222),(44,223),(45,224),(46,225),(47,226),(48,227),(49,228),(50,229),(51,230),(52,231),(53,232),(54,233),(55,234),(56,235),(57,236),(58,237),(59,238),(60,239),(61,240),(62,241),(63,242),(64,243),(65,244),(66,245),(67,246),(68,247),(69,248),(70,249),(71,250),(72,251),(73,252),(74,169),(75,170),(76,171),(77,172),(78,173),(79,174),(80,175),(81,176),(82,177),(83,178),(84,179),(85,282),(86,283),(87,284),(88,285),(89,286),(90,287),(91,288),(92,289),(93,290),(94,291),(95,292),(96,293),(97,294),(98,295),(99,296),(100,297),(101,298),(102,299),(103,300),(104,301),(105,302),(106,303),(107,304),(108,305),(109,306),(110,307),(111,308),(112,309),(113,310),(114,311),(115,312),(116,313),(117,314),(118,315),(119,316),(120,317),(121,318),(122,319),(123,320),(124,321),(125,322),(126,323),(127,324),(128,325),(129,326),(130,327),(131,328),(132,329),(133,330),(134,331),(135,332),(136,333),(137,334),(138,335),(139,336),(140,253),(141,254),(142,255),(143,256),(144,257),(145,258),(146,259),(147,260),(148,261),(149,262),(150,263),(151,264),(152,265),(153,266),(154,267),(155,268),(156,269),(157,270),(158,271),(159,272),(160,273),(161,274),(162,275),(163,276),(164,277),(165,278),(166,279),(167,280),(168,281)], [(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)], [(1,336,43,294),(2,335,44,293),(3,334,45,292),(4,333,46,291),(5,332,47,290),(6,331,48,289),(7,330,49,288),(8,329,50,287),(9,328,51,286),(10,327,52,285),(11,326,53,284),(12,325,54,283),(13,324,55,282),(14,323,56,281),(15,322,57,280),(16,321,58,279),(17,320,59,278),(18,319,60,277),(19,318,61,276),(20,317,62,275),(21,316,63,274),(22,315,64,273),(23,314,65,272),(24,313,66,271),(25,312,67,270),(26,311,68,269),(27,310,69,268),(28,309,70,267),(29,308,71,266),(30,307,72,265),(31,306,73,264),(32,305,74,263),(33,304,75,262),(34,303,76,261),(35,302,77,260),(36,301,78,259),(37,300,79,258),(38,299,80,257),(39,298,81,256),(40,297,82,255),(41,296,83,254),(42,295,84,253),(85,192,127,234),(86,191,128,233),(87,190,129,232),(88,189,130,231),(89,188,131,230),(90,187,132,229),(91,186,133,228),(92,185,134,227),(93,184,135,226),(94,183,136,225),(95,182,137,224),(96,181,138,223),(97,180,139,222),(98,179,140,221),(99,178,141,220),(100,177,142,219),(101,176,143,218),(102,175,144,217),(103,174,145,216),(104,173,146,215),(105,172,147,214),(106,171,148,213),(107,170,149,212),(108,169,150,211),(109,252,151,210),(110,251,152,209),(111,250,153,208),(112,249,154,207),(113,248,155,206),(114,247,156,205),(115,246,157,204),(116,245,158,203),(117,244,159,202),(118,243,160,201),(119,242,161,200),(120,241,162,199),(121,240,163,198),(122,239,164,197),(123,238,165,196),(124,237,166,195),(125,236,167,194),(126,235,168,193)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 7A | 7B | 7C | 12A | 12B | 12C | 12D | 14A | ··· | 14I | 21A | ··· | 21F | 28A | ··· | 28L | 42A | ··· | 42R | 84A | ··· | 84X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 42 | 42 | 42 | 42 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | + | - | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D7 | Dic6 | D14 | D14 | D21 | Dic14 | D42 | D42 | Dic42 |
| kernel | C2×Dic42 | Dic42 | C2×Dic21 | C2×C84 | C2×C28 | C42 | C28 | C2×C14 | C2×C12 | C14 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 1 | 2 | 2 | 1 | 3 | 4 | 6 | 3 | 6 | 12 | 12 | 6 | 24 |
Matrix representation of C2×Dic42 ►in GL4(𝔽337) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 336 |
| 142 | 18 | 0 | 0 |
| 86 | 56 | 0 | 0 |
| 0 | 0 | 107 | 100 |
| 0 | 0 | 237 | 253 |
| 278 | 29 | 0 | 0 |
| 31 | 59 | 0 | 0 |
| 0 | 0 | 277 | 32 |
| 0 | 0 | 214 | 60 |
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,336,0,0,0,0,336],[142,86,0,0,18,56,0,0,0,0,107,237,0,0,100,253],[278,31,0,0,29,59,0,0,0,0,277,214,0,0,32,60] >;
C2×Dic42 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{42} % in TeX
G:=Group("C2xDic42"); // GroupNames label
G:=SmallGroup(336,194);
// by ID
G=gap.SmallGroup(336,194);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,48,218,50,964,10373]);
// Polycyclic
G:=Group<a,b,c|a^2=b^84=1,c^2=b^42,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations