direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: Dic3×C2×C14, C42.56C23, (C2×C42)⋊7C4, C6⋊2(C2×C28), C42⋊9(C2×C4), (C2×C6)⋊3C28, C3⋊2(C22×C28), C21⋊10(C22×C4), (C2×C14).39D6, C23.3(S3×C7), C6.9(C22×C14), (C22×C14).6S3, (C22×C6).3C14, (C22×C42).7C2, C22.11(S3×C14), C14.46(C22×S3), (C2×C42).51C22, C2.2(S3×C2×C14), (C2×C6).12(C2×C14), SmallGroup(336,192)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C2×C14 |
Generators and relations for Dic3×C2×C14
G = < a,b,c,d | a2=b14=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 152 in 108 conjugacy classes, 86 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C7, C2×C4, C23, Dic3, C2×C6, C14, C14, C22×C4, C21, C2×Dic3, C22×C6, C28, C2×C14, C42, C42, C22×Dic3, C2×C28, C22×C14, C7×Dic3, C2×C42, C22×C28, Dic3×C14, C22×C42, Dic3×C2×C14
Quotients: C1, C2, C4, C22, S3, C7, C2×C4, C23, Dic3, D6, C14, C22×C4, C2×Dic3, C22×S3, C28, C2×C14, S3×C7, C22×Dic3, C2×C28, C22×C14, C7×Dic3, S3×C14, C22×C28, Dic3×C14, S3×C2×C14, Dic3×C2×C14
►Regular action on 336 points
Generators in S336
(1 316)(2 317)(3 318)(4 319)(5 320)(6 321)(7 322)(8 309)(9 310)(10 311)(11 312)(12 313)(13 314)(14 315)(15 224)(16 211)(17 212)(18 213)(19 214)(20 215)(21 216)(22 217)(23 218)(24 219)(25 220)(26 221)(27 222)(28 223)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 82)(40 83)(41 84)(42 71)(43 188)(44 189)(45 190)(46 191)(47 192)(48 193)(49 194)(50 195)(51 196)(52 183)(53 184)(54 185)(55 186)(56 187)(57 324)(58 325)(59 326)(60 327)(61 328)(62 329)(63 330)(64 331)(65 332)(66 333)(67 334)(68 335)(69 336)(70 323)(85 102)(86 103)(87 104)(88 105)(89 106)(90 107)(91 108)(92 109)(93 110)(94 111)(95 112)(96 99)(97 100)(98 101)(113 199)(114 200)(115 201)(116 202)(117 203)(118 204)(119 205)(120 206)(121 207)(122 208)(123 209)(124 210)(125 197)(126 198)(127 272)(128 273)(129 274)(130 275)(131 276)(132 277)(133 278)(134 279)(135 280)(136 267)(137 268)(138 269)(139 270)(140 271)(141 262)(142 263)(143 264)(144 265)(145 266)(146 253)(147 254)(148 255)(149 256)(150 257)(151 258)(152 259)(153 260)(154 261)(155 281)(156 282)(157 283)(158 284)(159 285)(160 286)(161 287)(162 288)(163 289)(164 290)(165 291)(166 292)(167 293)(168 294)(169 245)(170 246)(171 247)(172 248)(173 249)(174 250)(175 251)(176 252)(177 239)(178 240)(179 241)(180 242)(181 243)(182 244)(225 301)(226 302)(227 303)(228 304)(229 305)(230 306)(231 307)(232 308)(233 295)(234 296)(235 297)(236 298)(237 299)(238 300)
(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)
(1 326 77 123 91 150)(2 327 78 124 92 151)(3 328 79 125 93 152)(4 329 80 126 94 153)(5 330 81 113 95 154)(6 331 82 114 96 141)(7 332 83 115 97 142)(8 333 84 116 98 143)(9 334 71 117 85 144)(10 335 72 118 86 145)(11 336 73 119 87 146)(12 323 74 120 88 147)(13 324 75 121 89 148)(14 325 76 122 90 149)(15 164 270 241 305 187)(16 165 271 242 306 188)(17 166 272 243 307 189)(18 167 273 244 308 190)(19 168 274 245 295 191)(20 155 275 246 296 192)(21 156 276 247 297 193)(22 157 277 248 298 194)(23 158 278 249 299 195)(24 159 279 250 300 196)(25 160 280 251 301 183)(26 161 267 252 302 184)(27 162 268 239 303 185)(28 163 269 240 304 186)(29 204 103 266 311 68)(30 205 104 253 312 69)(31 206 105 254 313 70)(32 207 106 255 314 57)(33 208 107 256 315 58)(34 209 108 257 316 59)(35 210 109 258 317 60)(36 197 110 259 318 61)(37 198 111 260 319 62)(38 199 112 261 320 63)(39 200 99 262 321 64)(40 201 100 263 322 65)(41 202 101 264 309 66)(42 203 102 265 310 67)(43 211 291 140 180 230)(44 212 292 127 181 231)(45 213 293 128 182 232)(46 214 294 129 169 233)(47 215 281 130 170 234)(48 216 282 131 171 235)(49 217 283 132 172 236)(50 218 284 133 173 237)(51 219 285 134 174 238)(52 220 286 135 175 225)(53 221 287 136 176 226)(54 222 288 137 177 227)(55 223 289 138 178 228)(56 224 290 139 179 229)
(1 24 123 250)(2 25 124 251)(3 26 125 252)(4 27 126 239)(5 28 113 240)(6 15 114 241)(7 16 115 242)(8 17 116 243)(9 18 117 244)(10 19 118 245)(11 20 119 246)(12 21 120 247)(13 22 121 248)(14 23 122 249)(29 233 266 294)(30 234 253 281)(31 235 254 282)(32 236 255 283)(33 237 256 284)(34 238 257 285)(35 225 258 286)(36 226 259 287)(37 227 260 288)(38 228 261 289)(39 229 262 290)(40 230 263 291)(41 231 264 292)(42 232 265 293)(43 100 140 65)(44 101 127 66)(45 102 128 67)(46 103 129 68)(47 104 130 69)(48 105 131 70)(49 106 132 57)(50 107 133 58)(51 108 134 59)(52 109 135 60)(53 110 136 61)(54 111 137 62)(55 112 138 63)(56 99 139 64)(71 308 144 167)(72 295 145 168)(73 296 146 155)(74 297 147 156)(75 298 148 157)(76 299 149 158)(77 300 150 159)(78 301 151 160)(79 302 152 161)(80 303 153 162)(81 304 154 163)(82 305 141 164)(83 306 142 165)(84 307 143 166)(85 273 334 190)(86 274 335 191)(87 275 336 192)(88 276 323 193)(89 277 324 194)(90 278 325 195)(91 279 326 196)(92 280 327 183)(93 267 328 184)(94 268 329 185)(95 269 330 186)(96 270 331 187)(97 271 332 188)(98 272 333 189)(169 311 214 204)(170 312 215 205)(171 313 216 206)(172 314 217 207)(173 315 218 208)(174 316 219 209)(175 317 220 210)(176 318 221 197)(177 319 222 198)(178 320 223 199)(179 321 224 200)(180 322 211 201)(181 309 212 202)(182 310 213 203)
G:=sub<Sym(336)| (1,316)(2,317)(3,318)(4,319)(5,320)(6,321)(7,322)(8,309)(9,310)(10,311)(11,312)(12,313)(13,314)(14,315)(15,224)(16,211)(17,212)(18,213)(19,214)(20,215)(21,216)(22,217)(23,218)(24,219)(25,220)(26,221)(27,222)(28,223)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,71)(43,188)(44,189)(45,190)(46,191)(47,192)(48,193)(49,194)(50,195)(51,196)(52,183)(53,184)(54,185)(55,186)(56,187)(57,324)(58,325)(59,326)(60,327)(61,328)(62,329)(63,330)(64,331)(65,332)(66,333)(67,334)(68,335)(69,336)(70,323)(85,102)(86,103)(87,104)(88,105)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,99)(97,100)(98,101)(113,199)(114,200)(115,201)(116,202)(117,203)(118,204)(119,205)(120,206)(121,207)(122,208)(123,209)(124,210)(125,197)(126,198)(127,272)(128,273)(129,274)(130,275)(131,276)(132,277)(133,278)(134,279)(135,280)(136,267)(137,268)(138,269)(139,270)(140,271)(141,262)(142,263)(143,264)(144,265)(145,266)(146,253)(147,254)(148,255)(149,256)(150,257)(151,258)(152,259)(153,260)(154,261)(155,281)(156,282)(157,283)(158,284)(159,285)(160,286)(161,287)(162,288)(163,289)(164,290)(165,291)(166,292)(167,293)(168,294)(169,245)(170,246)(171,247)(172,248)(173,249)(174,250)(175,251)(176,252)(177,239)(178,240)(179,241)(180,242)(181,243)(182,244)(225,301)(226,302)(227,303)(228,304)(229,305)(230,306)(231,307)(232,308)(233,295)(234,296)(235,297)(236,298)(237,299)(238,300), (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), (1,326,77,123,91,150)(2,327,78,124,92,151)(3,328,79,125,93,152)(4,329,80,126,94,153)(5,330,81,113,95,154)(6,331,82,114,96,141)(7,332,83,115,97,142)(8,333,84,116,98,143)(9,334,71,117,85,144)(10,335,72,118,86,145)(11,336,73,119,87,146)(12,323,74,120,88,147)(13,324,75,121,89,148)(14,325,76,122,90,149)(15,164,270,241,305,187)(16,165,271,242,306,188)(17,166,272,243,307,189)(18,167,273,244,308,190)(19,168,274,245,295,191)(20,155,275,246,296,192)(21,156,276,247,297,193)(22,157,277,248,298,194)(23,158,278,249,299,195)(24,159,279,250,300,196)(25,160,280,251,301,183)(26,161,267,252,302,184)(27,162,268,239,303,185)(28,163,269,240,304,186)(29,204,103,266,311,68)(30,205,104,253,312,69)(31,206,105,254,313,70)(32,207,106,255,314,57)(33,208,107,256,315,58)(34,209,108,257,316,59)(35,210,109,258,317,60)(36,197,110,259,318,61)(37,198,111,260,319,62)(38,199,112,261,320,63)(39,200,99,262,321,64)(40,201,100,263,322,65)(41,202,101,264,309,66)(42,203,102,265,310,67)(43,211,291,140,180,230)(44,212,292,127,181,231)(45,213,293,128,182,232)(46,214,294,129,169,233)(47,215,281,130,170,234)(48,216,282,131,171,235)(49,217,283,132,172,236)(50,218,284,133,173,237)(51,219,285,134,174,238)(52,220,286,135,175,225)(53,221,287,136,176,226)(54,222,288,137,177,227)(55,223,289,138,178,228)(56,224,290,139,179,229), (1,24,123,250)(2,25,124,251)(3,26,125,252)(4,27,126,239)(5,28,113,240)(6,15,114,241)(7,16,115,242)(8,17,116,243)(9,18,117,244)(10,19,118,245)(11,20,119,246)(12,21,120,247)(13,22,121,248)(14,23,122,249)(29,233,266,294)(30,234,253,281)(31,235,254,282)(32,236,255,283)(33,237,256,284)(34,238,257,285)(35,225,258,286)(36,226,259,287)(37,227,260,288)(38,228,261,289)(39,229,262,290)(40,230,263,291)(41,231,264,292)(42,232,265,293)(43,100,140,65)(44,101,127,66)(45,102,128,67)(46,103,129,68)(47,104,130,69)(48,105,131,70)(49,106,132,57)(50,107,133,58)(51,108,134,59)(52,109,135,60)(53,110,136,61)(54,111,137,62)(55,112,138,63)(56,99,139,64)(71,308,144,167)(72,295,145,168)(73,296,146,155)(74,297,147,156)(75,298,148,157)(76,299,149,158)(77,300,150,159)(78,301,151,160)(79,302,152,161)(80,303,153,162)(81,304,154,163)(82,305,141,164)(83,306,142,165)(84,307,143,166)(85,273,334,190)(86,274,335,191)(87,275,336,192)(88,276,323,193)(89,277,324,194)(90,278,325,195)(91,279,326,196)(92,280,327,183)(93,267,328,184)(94,268,329,185)(95,269,330,186)(96,270,331,187)(97,271,332,188)(98,272,333,189)(169,311,214,204)(170,312,215,205)(171,313,216,206)(172,314,217,207)(173,315,218,208)(174,316,219,209)(175,317,220,210)(176,318,221,197)(177,319,222,198)(178,320,223,199)(179,321,224,200)(180,322,211,201)(181,309,212,202)(182,310,213,203)>;
G:=Group( (1,316)(2,317)(3,318)(4,319)(5,320)(6,321)(7,322)(8,309)(9,310)(10,311)(11,312)(12,313)(13,314)(14,315)(15,224)(16,211)(17,212)(18,213)(19,214)(20,215)(21,216)(22,217)(23,218)(24,219)(25,220)(26,221)(27,222)(28,223)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,71)(43,188)(44,189)(45,190)(46,191)(47,192)(48,193)(49,194)(50,195)(51,196)(52,183)(53,184)(54,185)(55,186)(56,187)(57,324)(58,325)(59,326)(60,327)(61,328)(62,329)(63,330)(64,331)(65,332)(66,333)(67,334)(68,335)(69,336)(70,323)(85,102)(86,103)(87,104)(88,105)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,99)(97,100)(98,101)(113,199)(114,200)(115,201)(116,202)(117,203)(118,204)(119,205)(120,206)(121,207)(122,208)(123,209)(124,210)(125,197)(126,198)(127,272)(128,273)(129,274)(130,275)(131,276)(132,277)(133,278)(134,279)(135,280)(136,267)(137,268)(138,269)(139,270)(140,271)(141,262)(142,263)(143,264)(144,265)(145,266)(146,253)(147,254)(148,255)(149,256)(150,257)(151,258)(152,259)(153,260)(154,261)(155,281)(156,282)(157,283)(158,284)(159,285)(160,286)(161,287)(162,288)(163,289)(164,290)(165,291)(166,292)(167,293)(168,294)(169,245)(170,246)(171,247)(172,248)(173,249)(174,250)(175,251)(176,252)(177,239)(178,240)(179,241)(180,242)(181,243)(182,244)(225,301)(226,302)(227,303)(228,304)(229,305)(230,306)(231,307)(232,308)(233,295)(234,296)(235,297)(236,298)(237,299)(238,300), (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), (1,326,77,123,91,150)(2,327,78,124,92,151)(3,328,79,125,93,152)(4,329,80,126,94,153)(5,330,81,113,95,154)(6,331,82,114,96,141)(7,332,83,115,97,142)(8,333,84,116,98,143)(9,334,71,117,85,144)(10,335,72,118,86,145)(11,336,73,119,87,146)(12,323,74,120,88,147)(13,324,75,121,89,148)(14,325,76,122,90,149)(15,164,270,241,305,187)(16,165,271,242,306,188)(17,166,272,243,307,189)(18,167,273,244,308,190)(19,168,274,245,295,191)(20,155,275,246,296,192)(21,156,276,247,297,193)(22,157,277,248,298,194)(23,158,278,249,299,195)(24,159,279,250,300,196)(25,160,280,251,301,183)(26,161,267,252,302,184)(27,162,268,239,303,185)(28,163,269,240,304,186)(29,204,103,266,311,68)(30,205,104,253,312,69)(31,206,105,254,313,70)(32,207,106,255,314,57)(33,208,107,256,315,58)(34,209,108,257,316,59)(35,210,109,258,317,60)(36,197,110,259,318,61)(37,198,111,260,319,62)(38,199,112,261,320,63)(39,200,99,262,321,64)(40,201,100,263,322,65)(41,202,101,264,309,66)(42,203,102,265,310,67)(43,211,291,140,180,230)(44,212,292,127,181,231)(45,213,293,128,182,232)(46,214,294,129,169,233)(47,215,281,130,170,234)(48,216,282,131,171,235)(49,217,283,132,172,236)(50,218,284,133,173,237)(51,219,285,134,174,238)(52,220,286,135,175,225)(53,221,287,136,176,226)(54,222,288,137,177,227)(55,223,289,138,178,228)(56,224,290,139,179,229), (1,24,123,250)(2,25,124,251)(3,26,125,252)(4,27,126,239)(5,28,113,240)(6,15,114,241)(7,16,115,242)(8,17,116,243)(9,18,117,244)(10,19,118,245)(11,20,119,246)(12,21,120,247)(13,22,121,248)(14,23,122,249)(29,233,266,294)(30,234,253,281)(31,235,254,282)(32,236,255,283)(33,237,256,284)(34,238,257,285)(35,225,258,286)(36,226,259,287)(37,227,260,288)(38,228,261,289)(39,229,262,290)(40,230,263,291)(41,231,264,292)(42,232,265,293)(43,100,140,65)(44,101,127,66)(45,102,128,67)(46,103,129,68)(47,104,130,69)(48,105,131,70)(49,106,132,57)(50,107,133,58)(51,108,134,59)(52,109,135,60)(53,110,136,61)(54,111,137,62)(55,112,138,63)(56,99,139,64)(71,308,144,167)(72,295,145,168)(73,296,146,155)(74,297,147,156)(75,298,148,157)(76,299,149,158)(77,300,150,159)(78,301,151,160)(79,302,152,161)(80,303,153,162)(81,304,154,163)(82,305,141,164)(83,306,142,165)(84,307,143,166)(85,273,334,190)(86,274,335,191)(87,275,336,192)(88,276,323,193)(89,277,324,194)(90,278,325,195)(91,279,326,196)(92,280,327,183)(93,267,328,184)(94,268,329,185)(95,269,330,186)(96,270,331,187)(97,271,332,188)(98,272,333,189)(169,311,214,204)(170,312,215,205)(171,313,216,206)(172,314,217,207)(173,315,218,208)(174,316,219,209)(175,317,220,210)(176,318,221,197)(177,319,222,198)(178,320,223,199)(179,321,224,200)(180,322,211,201)(181,309,212,202)(182,310,213,203) );
G=PermutationGroup([[(1,316),(2,317),(3,318),(4,319),(5,320),(6,321),(7,322),(8,309),(9,310),(10,311),(11,312),(12,313),(13,314),(14,315),(15,224),(16,211),(17,212),(18,213),(19,214),(20,215),(21,216),(22,217),(23,218),(24,219),(25,220),(26,221),(27,222),(28,223),(29,72),(30,73),(31,74),(32,75),(33,76),(34,77),(35,78),(36,79),(37,80),(38,81),(39,82),(40,83),(41,84),(42,71),(43,188),(44,189),(45,190),(46,191),(47,192),(48,193),(49,194),(50,195),(51,196),(52,183),(53,184),(54,185),(55,186),(56,187),(57,324),(58,325),(59,326),(60,327),(61,328),(62,329),(63,330),(64,331),(65,332),(66,333),(67,334),(68,335),(69,336),(70,323),(85,102),(86,103),(87,104),(88,105),(89,106),(90,107),(91,108),(92,109),(93,110),(94,111),(95,112),(96,99),(97,100),(98,101),(113,199),(114,200),(115,201),(116,202),(117,203),(118,204),(119,205),(120,206),(121,207),(122,208),(123,209),(124,210),(125,197),(126,198),(127,272),(128,273),(129,274),(130,275),(131,276),(132,277),(133,278),(134,279),(135,280),(136,267),(137,268),(138,269),(139,270),(140,271),(141,262),(142,263),(143,264),(144,265),(145,266),(146,253),(147,254),(148,255),(149,256),(150,257),(151,258),(152,259),(153,260),(154,261),(155,281),(156,282),(157,283),(158,284),(159,285),(160,286),(161,287),(162,288),(163,289),(164,290),(165,291),(166,292),(167,293),(168,294),(169,245),(170,246),(171,247),(172,248),(173,249),(174,250),(175,251),(176,252),(177,239),(178,240),(179,241),(180,242),(181,243),(182,244),(225,301),(226,302),(227,303),(228,304),(229,305),(230,306),(231,307),(232,308),(233,295),(234,296),(235,297),(236,298),(237,299),(238,300)], [(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)], [(1,326,77,123,91,150),(2,327,78,124,92,151),(3,328,79,125,93,152),(4,329,80,126,94,153),(5,330,81,113,95,154),(6,331,82,114,96,141),(7,332,83,115,97,142),(8,333,84,116,98,143),(9,334,71,117,85,144),(10,335,72,118,86,145),(11,336,73,119,87,146),(12,323,74,120,88,147),(13,324,75,121,89,148),(14,325,76,122,90,149),(15,164,270,241,305,187),(16,165,271,242,306,188),(17,166,272,243,307,189),(18,167,273,244,308,190),(19,168,274,245,295,191),(20,155,275,246,296,192),(21,156,276,247,297,193),(22,157,277,248,298,194),(23,158,278,249,299,195),(24,159,279,250,300,196),(25,160,280,251,301,183),(26,161,267,252,302,184),(27,162,268,239,303,185),(28,163,269,240,304,186),(29,204,103,266,311,68),(30,205,104,253,312,69),(31,206,105,254,313,70),(32,207,106,255,314,57),(33,208,107,256,315,58),(34,209,108,257,316,59),(35,210,109,258,317,60),(36,197,110,259,318,61),(37,198,111,260,319,62),(38,199,112,261,320,63),(39,200,99,262,321,64),(40,201,100,263,322,65),(41,202,101,264,309,66),(42,203,102,265,310,67),(43,211,291,140,180,230),(44,212,292,127,181,231),(45,213,293,128,182,232),(46,214,294,129,169,233),(47,215,281,130,170,234),(48,216,282,131,171,235),(49,217,283,132,172,236),(50,218,284,133,173,237),(51,219,285,134,174,238),(52,220,286,135,175,225),(53,221,287,136,176,226),(54,222,288,137,177,227),(55,223,289,138,178,228),(56,224,290,139,179,229)], [(1,24,123,250),(2,25,124,251),(3,26,125,252),(4,27,126,239),(5,28,113,240),(6,15,114,241),(7,16,115,242),(8,17,116,243),(9,18,117,244),(10,19,118,245),(11,20,119,246),(12,21,120,247),(13,22,121,248),(14,23,122,249),(29,233,266,294),(30,234,253,281),(31,235,254,282),(32,236,255,283),(33,237,256,284),(34,238,257,285),(35,225,258,286),(36,226,259,287),(37,227,260,288),(38,228,261,289),(39,229,262,290),(40,230,263,291),(41,231,264,292),(42,232,265,293),(43,100,140,65),(44,101,127,66),(45,102,128,67),(46,103,129,68),(47,104,130,69),(48,105,131,70),(49,106,132,57),(50,107,133,58),(51,108,134,59),(52,109,135,60),(53,110,136,61),(54,111,137,62),(55,112,138,63),(56,99,139,64),(71,308,144,167),(72,295,145,168),(73,296,146,155),(74,297,147,156),(75,298,148,157),(76,299,149,158),(77,300,150,159),(78,301,151,160),(79,302,152,161),(80,303,153,162),(81,304,154,163),(82,305,141,164),(83,306,142,165),(84,307,143,166),(85,273,334,190),(86,274,335,191),(87,275,336,192),(88,276,323,193),(89,277,324,194),(90,278,325,195),(91,279,326,196),(92,280,327,183),(93,267,328,184),(94,268,329,185),(95,269,330,186),(96,270,331,187),(97,271,332,188),(98,272,333,189),(169,311,214,204),(170,312,215,205),(171,313,216,206),(172,314,217,207),(173,315,218,208),(174,316,219,209),(175,317,220,210),(176,318,221,197),(177,319,222,198),(178,320,223,199),(179,321,224,200),(180,322,211,201),(181,309,212,202),(182,310,213,203)]])
168 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | ··· | 4H | 6A | ··· | 6G | 7A | ··· | 7F | 14A | ··· | 14AP | 21A | ··· | 21F | 28A | ··· | 28AV | 42A | ··· | 42AP |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 |
| size | 1 | 1 | ··· | 1 | 2 | 3 | ··· | 3 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
168 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | S3 | Dic3 | D6 | S3×C7 | C7×Dic3 | S3×C14 |
| kernel | Dic3×C2×C14 | Dic3×C14 | C22×C42 | C2×C42 | C22×Dic3 | C2×Dic3 | C22×C6 | C2×C6 | C22×C14 | C2×C14 | C2×C14 | C23 | C22 | C22 |
| # reps | 1 | 6 | 1 | 8 | 6 | 36 | 6 | 48 | 1 | 4 | 3 | 6 | 24 | 18 |
Matrix representation of Dic3×C2×C14 ►in GL4(𝔽337) generated by
| 1 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 |
| 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 336 |
| 258 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 329 | 0 |
| 0 | 0 | 0 | 329 |
| 336 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 336 | 1 |
| 0 | 0 | 336 | 0 |
| 148 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 |
| 0 | 0 | 223 | 259 |
| 0 | 0 | 145 | 114 |
G:=sub<GL(4,GF(337))| [1,0,0,0,0,336,0,0,0,0,336,0,0,0,0,336],[258,0,0,0,0,1,0,0,0,0,329,0,0,0,0,329],[336,0,0,0,0,1,0,0,0,0,336,336,0,0,1,0],[148,0,0,0,0,336,0,0,0,0,223,145,0,0,259,114] >;
Dic3×C2×C14 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_2\times C_{14} % in TeX
G:=Group("Dic3xC2xC14"); // GroupNames label
G:=SmallGroup(336,192);
// by ID
G=gap.SmallGroup(336,192);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-3,336,8069]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^14=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations