direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C4⋊Dic7, C84⋊3C4, C28⋊5C12, C42.7Q8, C12⋊3Dic7, C6.16D28, C42.26D4, C6.7Dic14, C4⋊(C3×Dic7), C21⋊7(C4⋊C4), (C2×C12).9D7, C2.1(C3×D28), C14.5(C3×Q8), C42.35(C2×C4), (C2×C84).10C2, (C2×C28).14C6, C14.20(C3×D4), (C2×C6).33D14, C2.4(C6×Dic7), C22.5(C6×D7), C14.22(C2×C12), (C2×Dic7).6C6, (C6×Dic7).7C2, C6.14(C2×Dic7), C2.2(C3×Dic14), (C2×C42).34C22, C7⋊5(C3×C4⋊C4), (C2×C4).3(C3×D7), (C2×C14).22(C2×C6), SmallGroup(336,67)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4⋊Dic7
G = < a,b,c,d | a3=b4=c14=1, d2=c7, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Smallest permutation representation of C3×C4⋊Dic7
►Regular action on 336 points
Generators in S336
(1 211 83)(2 212 84)(3 213 71)(4 214 72)(5 215 73)(6 216 74)(7 217 75)(8 218 76)(9 219 77)(10 220 78)(11 221 79)(12 222 80)(13 223 81)(14 224 82)(15 127 333)(16 128 334)(17 129 335)(18 130 336)(19 131 323)(20 132 324)(21 133 325)(22 134 326)(23 135 327)(24 136 328)(25 137 329)(26 138 330)(27 139 331)(28 140 332)(29 92 43)(30 93 44)(31 94 45)(32 95 46)(33 96 47)(34 97 48)(35 98 49)(36 85 50)(37 86 51)(38 87 52)(39 88 53)(40 89 54)(41 90 55)(42 91 56)(57 210 195)(58 197 196)(59 198 183)(60 199 184)(61 200 185)(62 201 186)(63 202 187)(64 203 188)(65 204 189)(66 205 190)(67 206 191)(68 207 192)(69 208 193)(70 209 194)(99 160 229)(100 161 230)(101 162 231)(102 163 232)(103 164 233)(104 165 234)(105 166 235)(106 167 236)(107 168 237)(108 155 238)(109 156 225)(110 157 226)(111 158 227)(112 159 228)(113 302 291)(114 303 292)(115 304 293)(116 305 294)(117 306 281)(118 307 282)(119 308 283)(120 295 284)(121 296 285)(122 297 286)(123 298 287)(124 299 288)(125 300 289)(126 301 290)(141 280 182)(142 267 169)(143 268 170)(144 269 171)(145 270 172)(146 271 173)(147 272 174)(148 273 175)(149 274 176)(150 275 177)(151 276 178)(152 277 179)(153 278 180)(154 279 181)(239 262 311)(240 263 312)(241 264 313)(242 265 314)(243 266 315)(244 253 316)(245 254 317)(246 255 318)(247 256 319)(248 257 320)(249 258 321)(250 259 322)(251 260 309)(252 261 310)
(1 178 264 41)(2 179 265 42)(3 180 266 29)(4 181 253 30)(5 182 254 31)(6 169 255 32)(7 170 256 33)(8 171 257 34)(9 172 258 35)(10 173 259 36)(11 174 260 37)(12 175 261 38)(13 176 262 39)(14 177 263 40)(15 155 209 118)(16 156 210 119)(17 157 197 120)(18 158 198 121)(19 159 199 122)(20 160 200 123)(21 161 201 124)(22 162 202 125)(23 163 203 126)(24 164 204 113)(25 165 205 114)(26 166 206 115)(27 167 207 116)(28 168 208 117)(43 71 278 243)(44 72 279 244)(45 73 280 245)(46 74 267 246)(47 75 268 247)(48 76 269 248)(49 77 270 249)(50 78 271 250)(51 79 272 251)(52 80 273 252)(53 81 274 239)(54 82 275 240)(55 83 276 241)(56 84 277 242)(57 283 334 109)(58 284 335 110)(59 285 336 111)(60 286 323 112)(61 287 324 99)(62 288 325 100)(63 289 326 101)(64 290 327 102)(65 291 328 103)(66 292 329 104)(67 293 330 105)(68 294 331 106)(69 281 332 107)(70 282 333 108)(85 220 146 322)(86 221 147 309)(87 222 148 310)(88 223 149 311)(89 224 150 312)(90 211 151 313)(91 212 152 314)(92 213 153 315)(93 214 154 316)(94 215 141 317)(95 216 142 318)(96 217 143 319)(97 218 144 320)(98 219 145 321)(127 238 194 307)(128 225 195 308)(129 226 196 295)(130 227 183 296)(131 228 184 297)(132 229 185 298)(133 230 186 299)(134 231 187 300)(135 232 188 301)(136 233 189 302)(137 234 190 303)(138 235 191 304)(139 236 192 305)(140 237 193 306)
(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)
(1 164 8 157)(2 163 9 156)(3 162 10 155)(4 161 11 168)(5 160 12 167)(6 159 13 166)(7 158 14 165)(15 180 22 173)(16 179 23 172)(17 178 24 171)(18 177 25 170)(19 176 26 169)(20 175 27 182)(21 174 28 181)(29 202 36 209)(30 201 37 208)(31 200 38 207)(32 199 39 206)(33 198 40 205)(34 197 41 204)(35 210 42 203)(43 63 50 70)(44 62 51 69)(45 61 52 68)(46 60 53 67)(47 59 54 66)(48 58 55 65)(49 57 56 64)(71 101 78 108)(72 100 79 107)(73 99 80 106)(74 112 81 105)(75 111 82 104)(76 110 83 103)(77 109 84 102)(85 194 92 187)(86 193 93 186)(87 192 94 185)(88 191 95 184)(89 190 96 183)(90 189 97 196)(91 188 98 195)(113 257 120 264)(114 256 121 263)(115 255 122 262)(116 254 123 261)(117 253 124 260)(118 266 125 259)(119 265 126 258)(127 153 134 146)(128 152 135 145)(129 151 136 144)(130 150 137 143)(131 149 138 142)(132 148 139 141)(133 147 140 154)(211 233 218 226)(212 232 219 225)(213 231 220 238)(214 230 221 237)(215 229 222 236)(216 228 223 235)(217 227 224 234)(239 293 246 286)(240 292 247 285)(241 291 248 284)(242 290 249 283)(243 289 250 282)(244 288 251 281)(245 287 252 294)(267 323 274 330)(268 336 275 329)(269 335 276 328)(270 334 277 327)(271 333 278 326)(272 332 279 325)(273 331 280 324)(295 313 302 320)(296 312 303 319)(297 311 304 318)(298 310 305 317)(299 309 306 316)(300 322 307 315)(301 321 308 314)
G:=sub<Sym(336)| (1,211,83)(2,212,84)(3,213,71)(4,214,72)(5,215,73)(6,216,74)(7,217,75)(8,218,76)(9,219,77)(10,220,78)(11,221,79)(12,222,80)(13,223,81)(14,224,82)(15,127,333)(16,128,334)(17,129,335)(18,130,336)(19,131,323)(20,132,324)(21,133,325)(22,134,326)(23,135,327)(24,136,328)(25,137,329)(26,138,330)(27,139,331)(28,140,332)(29,92,43)(30,93,44)(31,94,45)(32,95,46)(33,96,47)(34,97,48)(35,98,49)(36,85,50)(37,86,51)(38,87,52)(39,88,53)(40,89,54)(41,90,55)(42,91,56)(57,210,195)(58,197,196)(59,198,183)(60,199,184)(61,200,185)(62,201,186)(63,202,187)(64,203,188)(65,204,189)(66,205,190)(67,206,191)(68,207,192)(69,208,193)(70,209,194)(99,160,229)(100,161,230)(101,162,231)(102,163,232)(103,164,233)(104,165,234)(105,166,235)(106,167,236)(107,168,237)(108,155,238)(109,156,225)(110,157,226)(111,158,227)(112,159,228)(113,302,291)(114,303,292)(115,304,293)(116,305,294)(117,306,281)(118,307,282)(119,308,283)(120,295,284)(121,296,285)(122,297,286)(123,298,287)(124,299,288)(125,300,289)(126,301,290)(141,280,182)(142,267,169)(143,268,170)(144,269,171)(145,270,172)(146,271,173)(147,272,174)(148,273,175)(149,274,176)(150,275,177)(151,276,178)(152,277,179)(153,278,180)(154,279,181)(239,262,311)(240,263,312)(241,264,313)(242,265,314)(243,266,315)(244,253,316)(245,254,317)(246,255,318)(247,256,319)(248,257,320)(249,258,321)(250,259,322)(251,260,309)(252,261,310), (1,178,264,41)(2,179,265,42)(3,180,266,29)(4,181,253,30)(5,182,254,31)(6,169,255,32)(7,170,256,33)(8,171,257,34)(9,172,258,35)(10,173,259,36)(11,174,260,37)(12,175,261,38)(13,176,262,39)(14,177,263,40)(15,155,209,118)(16,156,210,119)(17,157,197,120)(18,158,198,121)(19,159,199,122)(20,160,200,123)(21,161,201,124)(22,162,202,125)(23,163,203,126)(24,164,204,113)(25,165,205,114)(26,166,206,115)(27,167,207,116)(28,168,208,117)(43,71,278,243)(44,72,279,244)(45,73,280,245)(46,74,267,246)(47,75,268,247)(48,76,269,248)(49,77,270,249)(50,78,271,250)(51,79,272,251)(52,80,273,252)(53,81,274,239)(54,82,275,240)(55,83,276,241)(56,84,277,242)(57,283,334,109)(58,284,335,110)(59,285,336,111)(60,286,323,112)(61,287,324,99)(62,288,325,100)(63,289,326,101)(64,290,327,102)(65,291,328,103)(66,292,329,104)(67,293,330,105)(68,294,331,106)(69,281,332,107)(70,282,333,108)(85,220,146,322)(86,221,147,309)(87,222,148,310)(88,223,149,311)(89,224,150,312)(90,211,151,313)(91,212,152,314)(92,213,153,315)(93,214,154,316)(94,215,141,317)(95,216,142,318)(96,217,143,319)(97,218,144,320)(98,219,145,321)(127,238,194,307)(128,225,195,308)(129,226,196,295)(130,227,183,296)(131,228,184,297)(132,229,185,298)(133,230,186,299)(134,231,187,300)(135,232,188,301)(136,233,189,302)(137,234,190,303)(138,235,191,304)(139,236,192,305)(140,237,193,306), (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), (1,164,8,157)(2,163,9,156)(3,162,10,155)(4,161,11,168)(5,160,12,167)(6,159,13,166)(7,158,14,165)(15,180,22,173)(16,179,23,172)(17,178,24,171)(18,177,25,170)(19,176,26,169)(20,175,27,182)(21,174,28,181)(29,202,36,209)(30,201,37,208)(31,200,38,207)(32,199,39,206)(33,198,40,205)(34,197,41,204)(35,210,42,203)(43,63,50,70)(44,62,51,69)(45,61,52,68)(46,60,53,67)(47,59,54,66)(48,58,55,65)(49,57,56,64)(71,101,78,108)(72,100,79,107)(73,99,80,106)(74,112,81,105)(75,111,82,104)(76,110,83,103)(77,109,84,102)(85,194,92,187)(86,193,93,186)(87,192,94,185)(88,191,95,184)(89,190,96,183)(90,189,97,196)(91,188,98,195)(113,257,120,264)(114,256,121,263)(115,255,122,262)(116,254,123,261)(117,253,124,260)(118,266,125,259)(119,265,126,258)(127,153,134,146)(128,152,135,145)(129,151,136,144)(130,150,137,143)(131,149,138,142)(132,148,139,141)(133,147,140,154)(211,233,218,226)(212,232,219,225)(213,231,220,238)(214,230,221,237)(215,229,222,236)(216,228,223,235)(217,227,224,234)(239,293,246,286)(240,292,247,285)(241,291,248,284)(242,290,249,283)(243,289,250,282)(244,288,251,281)(245,287,252,294)(267,323,274,330)(268,336,275,329)(269,335,276,328)(270,334,277,327)(271,333,278,326)(272,332,279,325)(273,331,280,324)(295,313,302,320)(296,312,303,319)(297,311,304,318)(298,310,305,317)(299,309,306,316)(300,322,307,315)(301,321,308,314)>;
G:=Group( (1,211,83)(2,212,84)(3,213,71)(4,214,72)(5,215,73)(6,216,74)(7,217,75)(8,218,76)(9,219,77)(10,220,78)(11,221,79)(12,222,80)(13,223,81)(14,224,82)(15,127,333)(16,128,334)(17,129,335)(18,130,336)(19,131,323)(20,132,324)(21,133,325)(22,134,326)(23,135,327)(24,136,328)(25,137,329)(26,138,330)(27,139,331)(28,140,332)(29,92,43)(30,93,44)(31,94,45)(32,95,46)(33,96,47)(34,97,48)(35,98,49)(36,85,50)(37,86,51)(38,87,52)(39,88,53)(40,89,54)(41,90,55)(42,91,56)(57,210,195)(58,197,196)(59,198,183)(60,199,184)(61,200,185)(62,201,186)(63,202,187)(64,203,188)(65,204,189)(66,205,190)(67,206,191)(68,207,192)(69,208,193)(70,209,194)(99,160,229)(100,161,230)(101,162,231)(102,163,232)(103,164,233)(104,165,234)(105,166,235)(106,167,236)(107,168,237)(108,155,238)(109,156,225)(110,157,226)(111,158,227)(112,159,228)(113,302,291)(114,303,292)(115,304,293)(116,305,294)(117,306,281)(118,307,282)(119,308,283)(120,295,284)(121,296,285)(122,297,286)(123,298,287)(124,299,288)(125,300,289)(126,301,290)(141,280,182)(142,267,169)(143,268,170)(144,269,171)(145,270,172)(146,271,173)(147,272,174)(148,273,175)(149,274,176)(150,275,177)(151,276,178)(152,277,179)(153,278,180)(154,279,181)(239,262,311)(240,263,312)(241,264,313)(242,265,314)(243,266,315)(244,253,316)(245,254,317)(246,255,318)(247,256,319)(248,257,320)(249,258,321)(250,259,322)(251,260,309)(252,261,310), (1,178,264,41)(2,179,265,42)(3,180,266,29)(4,181,253,30)(5,182,254,31)(6,169,255,32)(7,170,256,33)(8,171,257,34)(9,172,258,35)(10,173,259,36)(11,174,260,37)(12,175,261,38)(13,176,262,39)(14,177,263,40)(15,155,209,118)(16,156,210,119)(17,157,197,120)(18,158,198,121)(19,159,199,122)(20,160,200,123)(21,161,201,124)(22,162,202,125)(23,163,203,126)(24,164,204,113)(25,165,205,114)(26,166,206,115)(27,167,207,116)(28,168,208,117)(43,71,278,243)(44,72,279,244)(45,73,280,245)(46,74,267,246)(47,75,268,247)(48,76,269,248)(49,77,270,249)(50,78,271,250)(51,79,272,251)(52,80,273,252)(53,81,274,239)(54,82,275,240)(55,83,276,241)(56,84,277,242)(57,283,334,109)(58,284,335,110)(59,285,336,111)(60,286,323,112)(61,287,324,99)(62,288,325,100)(63,289,326,101)(64,290,327,102)(65,291,328,103)(66,292,329,104)(67,293,330,105)(68,294,331,106)(69,281,332,107)(70,282,333,108)(85,220,146,322)(86,221,147,309)(87,222,148,310)(88,223,149,311)(89,224,150,312)(90,211,151,313)(91,212,152,314)(92,213,153,315)(93,214,154,316)(94,215,141,317)(95,216,142,318)(96,217,143,319)(97,218,144,320)(98,219,145,321)(127,238,194,307)(128,225,195,308)(129,226,196,295)(130,227,183,296)(131,228,184,297)(132,229,185,298)(133,230,186,299)(134,231,187,300)(135,232,188,301)(136,233,189,302)(137,234,190,303)(138,235,191,304)(139,236,192,305)(140,237,193,306), (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), (1,164,8,157)(2,163,9,156)(3,162,10,155)(4,161,11,168)(5,160,12,167)(6,159,13,166)(7,158,14,165)(15,180,22,173)(16,179,23,172)(17,178,24,171)(18,177,25,170)(19,176,26,169)(20,175,27,182)(21,174,28,181)(29,202,36,209)(30,201,37,208)(31,200,38,207)(32,199,39,206)(33,198,40,205)(34,197,41,204)(35,210,42,203)(43,63,50,70)(44,62,51,69)(45,61,52,68)(46,60,53,67)(47,59,54,66)(48,58,55,65)(49,57,56,64)(71,101,78,108)(72,100,79,107)(73,99,80,106)(74,112,81,105)(75,111,82,104)(76,110,83,103)(77,109,84,102)(85,194,92,187)(86,193,93,186)(87,192,94,185)(88,191,95,184)(89,190,96,183)(90,189,97,196)(91,188,98,195)(113,257,120,264)(114,256,121,263)(115,255,122,262)(116,254,123,261)(117,253,124,260)(118,266,125,259)(119,265,126,258)(127,153,134,146)(128,152,135,145)(129,151,136,144)(130,150,137,143)(131,149,138,142)(132,148,139,141)(133,147,140,154)(211,233,218,226)(212,232,219,225)(213,231,220,238)(214,230,221,237)(215,229,222,236)(216,228,223,235)(217,227,224,234)(239,293,246,286)(240,292,247,285)(241,291,248,284)(242,290,249,283)(243,289,250,282)(244,288,251,281)(245,287,252,294)(267,323,274,330)(268,336,275,329)(269,335,276,328)(270,334,277,327)(271,333,278,326)(272,332,279,325)(273,331,280,324)(295,313,302,320)(296,312,303,319)(297,311,304,318)(298,310,305,317)(299,309,306,316)(300,322,307,315)(301,321,308,314) );
G=PermutationGroup([[(1,211,83),(2,212,84),(3,213,71),(4,214,72),(5,215,73),(6,216,74),(7,217,75),(8,218,76),(9,219,77),(10,220,78),(11,221,79),(12,222,80),(13,223,81),(14,224,82),(15,127,333),(16,128,334),(17,129,335),(18,130,336),(19,131,323),(20,132,324),(21,133,325),(22,134,326),(23,135,327),(24,136,328),(25,137,329),(26,138,330),(27,139,331),(28,140,332),(29,92,43),(30,93,44),(31,94,45),(32,95,46),(33,96,47),(34,97,48),(35,98,49),(36,85,50),(37,86,51),(38,87,52),(39,88,53),(40,89,54),(41,90,55),(42,91,56),(57,210,195),(58,197,196),(59,198,183),(60,199,184),(61,200,185),(62,201,186),(63,202,187),(64,203,188),(65,204,189),(66,205,190),(67,206,191),(68,207,192),(69,208,193),(70,209,194),(99,160,229),(100,161,230),(101,162,231),(102,163,232),(103,164,233),(104,165,234),(105,166,235),(106,167,236),(107,168,237),(108,155,238),(109,156,225),(110,157,226),(111,158,227),(112,159,228),(113,302,291),(114,303,292),(115,304,293),(116,305,294),(117,306,281),(118,307,282),(119,308,283),(120,295,284),(121,296,285),(122,297,286),(123,298,287),(124,299,288),(125,300,289),(126,301,290),(141,280,182),(142,267,169),(143,268,170),(144,269,171),(145,270,172),(146,271,173),(147,272,174),(148,273,175),(149,274,176),(150,275,177),(151,276,178),(152,277,179),(153,278,180),(154,279,181),(239,262,311),(240,263,312),(241,264,313),(242,265,314),(243,266,315),(244,253,316),(245,254,317),(246,255,318),(247,256,319),(248,257,320),(249,258,321),(250,259,322),(251,260,309),(252,261,310)], [(1,178,264,41),(2,179,265,42),(3,180,266,29),(4,181,253,30),(5,182,254,31),(6,169,255,32),(7,170,256,33),(8,171,257,34),(9,172,258,35),(10,173,259,36),(11,174,260,37),(12,175,261,38),(13,176,262,39),(14,177,263,40),(15,155,209,118),(16,156,210,119),(17,157,197,120),(18,158,198,121),(19,159,199,122),(20,160,200,123),(21,161,201,124),(22,162,202,125),(23,163,203,126),(24,164,204,113),(25,165,205,114),(26,166,206,115),(27,167,207,116),(28,168,208,117),(43,71,278,243),(44,72,279,244),(45,73,280,245),(46,74,267,246),(47,75,268,247),(48,76,269,248),(49,77,270,249),(50,78,271,250),(51,79,272,251),(52,80,273,252),(53,81,274,239),(54,82,275,240),(55,83,276,241),(56,84,277,242),(57,283,334,109),(58,284,335,110),(59,285,336,111),(60,286,323,112),(61,287,324,99),(62,288,325,100),(63,289,326,101),(64,290,327,102),(65,291,328,103),(66,292,329,104),(67,293,330,105),(68,294,331,106),(69,281,332,107),(70,282,333,108),(85,220,146,322),(86,221,147,309),(87,222,148,310),(88,223,149,311),(89,224,150,312),(90,211,151,313),(91,212,152,314),(92,213,153,315),(93,214,154,316),(94,215,141,317),(95,216,142,318),(96,217,143,319),(97,218,144,320),(98,219,145,321),(127,238,194,307),(128,225,195,308),(129,226,196,295),(130,227,183,296),(131,228,184,297),(132,229,185,298),(133,230,186,299),(134,231,187,300),(135,232,188,301),(136,233,189,302),(137,234,190,303),(138,235,191,304),(139,236,192,305),(140,237,193,306)], [(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)], [(1,164,8,157),(2,163,9,156),(3,162,10,155),(4,161,11,168),(5,160,12,167),(6,159,13,166),(7,158,14,165),(15,180,22,173),(16,179,23,172),(17,178,24,171),(18,177,25,170),(19,176,26,169),(20,175,27,182),(21,174,28,181),(29,202,36,209),(30,201,37,208),(31,200,38,207),(32,199,39,206),(33,198,40,205),(34,197,41,204),(35,210,42,203),(43,63,50,70),(44,62,51,69),(45,61,52,68),(46,60,53,67),(47,59,54,66),(48,58,55,65),(49,57,56,64),(71,101,78,108),(72,100,79,107),(73,99,80,106),(74,112,81,105),(75,111,82,104),(76,110,83,103),(77,109,84,102),(85,194,92,187),(86,193,93,186),(87,192,94,185),(88,191,95,184),(89,190,96,183),(90,189,97,196),(91,188,98,195),(113,257,120,264),(114,256,121,263),(115,255,122,262),(116,254,123,261),(117,253,124,260),(118,266,125,259),(119,265,126,258),(127,153,134,146),(128,152,135,145),(129,151,136,144),(130,150,137,143),(131,149,138,142),(132,148,139,141),(133,147,140,154),(211,233,218,226),(212,232,219,225),(213,231,220,238),(214,230,221,237),(215,229,222,236),(216,228,223,235),(217,227,224,234),(239,293,246,286),(240,292,247,285),(241,291,248,284),(242,290,249,283),(243,289,250,282),(244,288,251,281),(245,287,252,294),(267,323,274,330),(268,336,275,329),(269,335,276,328),(270,334,277,327),(271,333,278,326),(272,332,279,325),(273,331,280,324),(295,313,302,320),(296,312,303,319),(297,311,304,318),(298,310,305,317),(299,309,306,316),(300,322,307,315),(301,321,308,314)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 7A | 7B | 7C | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 14A | ··· | 14I | 21A | ··· | 21F | 28A | ··· | 28L | 42A | ··· | 42R | 84A | ··· | 84X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 7 | 7 | 7 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 14 | 14 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | Q8 | D7 | C3×D4 | C3×Q8 | Dic7 | D14 | C3×D7 | Dic14 | D28 | C3×Dic7 | C6×D7 | C3×Dic14 | C3×D28 |
| kernel | C3×C4⋊Dic7 | C6×Dic7 | C2×C84 | C4⋊Dic7 | C84 | C2×Dic7 | C2×C28 | C28 | C42 | C42 | C2×C12 | C14 | C14 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 1 | 3 | 2 | 2 | 6 | 3 | 6 | 6 | 6 | 12 | 6 | 12 | 12 |
Matrix representation of C3×C4⋊Dic7 ►in GL4(𝔽337) generated by
| 128 | 0 | 0 | 0 |
| 0 | 128 | 0 | 0 |
| 0 | 0 | 128 | 0 |
| 0 | 0 | 0 | 128 |
| 336 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 |
| 0 | 0 | 243 | 225 |
| 0 | 0 | 112 | 94 |
| 110 | 1 | 0 | 0 |
| 336 | 0 | 0 | 0 |
| 0 | 0 | 110 | 1 |
| 0 | 0 | 336 | 0 |
| 118 | 11 | 0 | 0 |
| 174 | 219 | 0 | 0 |
| 0 | 0 | 250 | 287 |
| 0 | 0 | 84 | 87 |
G:=sub<GL(4,GF(337))| [128,0,0,0,0,128,0,0,0,0,128,0,0,0,0,128],[336,0,0,0,0,336,0,0,0,0,243,112,0,0,225,94],[110,336,0,0,1,0,0,0,0,0,110,336,0,0,1,0],[118,174,0,0,11,219,0,0,0,0,250,84,0,0,287,87] >;
C3×C4⋊Dic7 in GAP, Magma, Sage, TeX
C_3\times C_4\rtimes {\rm Dic}_7 % in TeX
G:=Group("C3xC4:Dic7"); // GroupNames label
G:=SmallGroup(336,67);
// by ID
G=gap.SmallGroup(336,67);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,72,313,151,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^14=1,d^2=c^7,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations
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