direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C7×C4⋊Dic3, C84⋊5C4, C12⋊1C28, C42.9Q8, C28⋊3Dic3, C42.31D4, C14.16D12, C14.7Dic6, C4⋊(C7×Dic3), C21⋊9(C4⋊C4), C6.4(C7×D4), C6.2(C7×Q8), C6.8(C2×C28), (C2×C28).9S3, C2.1(C7×D12), (C2×C12).3C14, (C2×C84).15C2, C42.39(C2×C4), (C2×C14).33D6, C2.2(C7×Dic6), C22.5(S3×C14), C2.4(Dic3×C14), (C2×C42).44C22, (C2×Dic3).2C14, (Dic3×C14).7C2, C14.14(C2×Dic3), C3⋊2(C7×C4⋊C4), (C2×C4).3(S3×C7), (C2×C6).5(C2×C14), SmallGroup(336,83)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4⋊Dic3
G = < a,b,c,d | a7=b4=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Smallest permutation representation of C7×C4⋊Dic3
►Regular action on 336 points
Generators in S336
(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)
(1 122 68 170)(2 123 69 171)(3 124 70 172)(4 125 64 173)(5 126 65 174)(6 120 66 175)(7 121 67 169)(8 204 288 259)(9 205 289 253)(10 206 290 254)(11 207 291 255)(12 208 292 256)(13 209 293 257)(14 210 294 258)(15 236 152 106)(16 237 153 107)(17 238 154 108)(18 232 148 109)(19 233 149 110)(20 234 150 111)(21 235 151 112)(22 247 331 302)(23 248 332 303)(24 249 333 304)(25 250 334 305)(26 251 335 306)(27 252 336 307)(28 246 330 308)(29 198 114 128)(30 199 115 129)(31 200 116 130)(32 201 117 131)(33 202 118 132)(34 203 119 133)(35 197 113 127)(36 193 277 316)(37 194 278 317)(38 195 279 318)(39 196 280 319)(40 190 274 320)(41 191 275 321)(42 192 276 322)(43 239 323 270)(44 240 324 271)(45 241 325 272)(46 242 326 273)(47 243 327 267)(48 244 328 268)(49 245 329 269)(50 182 98 141)(51 176 92 142)(52 177 93 143)(53 178 94 144)(54 179 95 145)(55 180 96 146)(56 181 97 147)(57 214 298 284)(58 215 299 285)(59 216 300 286)(60 217 301 287)(61 211 295 281)(62 212 296 282)(63 213 297 283)(71 228 312 262)(72 229 313 263)(73 230 314 264)(74 231 315 265)(75 225 309 266)(76 226 310 260)(77 227 311 261)(78 224 140 162)(79 218 134 163)(80 219 135 164)(81 220 136 165)(82 221 137 166)(83 222 138 167)(84 223 139 168)(85 186 102 155)(86 187 103 156)(87 188 104 157)(88 189 105 158)(89 183 99 159)(90 184 100 160)(91 185 101 161)
(1 105 135 33 18 54)(2 99 136 34 19 55)(3 100 137 35 20 56)(4 101 138 29 21 50)(5 102 139 30 15 51)(6 103 140 31 16 52)(7 104 134 32 17 53)(8 311 275 298 28 44)(9 312 276 299 22 45)(10 313 277 300 23 46)(11 314 278 301 24 47)(12 315 279 295 25 48)(13 309 280 296 26 49)(14 310 274 297 27 43)(36 59 332 326 290 72)(37 60 333 327 291 73)(38 61 334 328 292 74)(39 62 335 329 293 75)(40 63 336 323 294 76)(41 57 330 324 288 77)(42 58 331 325 289 71)(64 91 83 114 151 98)(65 85 84 115 152 92)(66 86 78 116 153 93)(67 87 79 117 154 94)(68 88 80 118 148 95)(69 89 81 119 149 96)(70 90 82 113 150 97)(106 142 174 186 223 129)(107 143 175 187 224 130)(108 144 169 188 218 131)(109 145 170 189 219 132)(110 146 171 183 220 133)(111 147 172 184 221 127)(112 141 173 185 222 128)(120 156 162 200 237 177)(121 157 163 201 238 178)(122 158 164 202 232 179)(123 159 165 203 233 180)(124 160 166 197 234 181)(125 161 167 198 235 182)(126 155 168 199 236 176)(190 213 307 270 258 226)(191 214 308 271 259 227)(192 215 302 272 253 228)(193 216 303 273 254 229)(194 217 304 267 255 230)(195 211 305 268 256 231)(196 212 306 269 257 225)(204 261 321 284 246 240)(205 262 322 285 247 241)(206 263 316 286 248 242)(207 264 317 287 249 243)(208 265 318 281 250 244)(209 266 319 282 251 245)(210 260 320 283 252 239)
(1 216 33 254)(2 217 34 255)(3 211 35 256)(4 212 29 257)(5 213 30 258)(6 214 31 259)(7 215 32 253)(8 175 298 130)(9 169 299 131)(10 170 300 132)(11 171 301 133)(12 172 295 127)(13 173 296 128)(14 174 297 129)(15 270 102 190)(16 271 103 191)(17 272 104 192)(18 273 105 193)(19 267 99 194)(20 268 100 195)(21 269 101 196)(22 218 312 144)(23 219 313 145)(24 220 314 146)(25 221 315 147)(26 222 309 141)(27 223 310 142)(28 224 311 143)(36 232 326 158)(37 233 327 159)(38 234 328 160)(39 235 329 161)(40 236 323 155)(41 237 324 156)(42 238 325 157)(43 186 274 106)(44 187 275 107)(45 188 276 108)(46 189 277 109)(47 183 278 110)(48 184 279 111)(49 185 280 112)(50 306 138 225)(51 307 139 226)(52 308 140 227)(53 302 134 228)(54 303 135 229)(55 304 136 230)(56 305 137 231)(57 200 288 120)(58 201 289 121)(59 202 290 122)(60 203 291 123)(61 197 292 124)(62 198 293 125)(63 199 294 126)(64 282 114 209)(65 283 115 210)(66 284 116 204)(67 285 117 205)(68 286 118 206)(69 287 119 207)(70 281 113 208)(71 178 331 163)(72 179 332 164)(73 180 333 165)(74 181 334 166)(75 182 335 167)(76 176 336 168)(77 177 330 162)(78 261 93 246)(79 262 94 247)(80 263 95 248)(81 264 96 249)(82 265 97 250)(83 266 98 251)(84 260 92 252)(85 320 152 239)(86 321 153 240)(87 322 154 241)(88 316 148 242)(89 317 149 243)(90 318 150 244)(91 319 151 245)
G:=sub<Sym(336)| (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), (1,122,68,170)(2,123,69,171)(3,124,70,172)(4,125,64,173)(5,126,65,174)(6,120,66,175)(7,121,67,169)(8,204,288,259)(9,205,289,253)(10,206,290,254)(11,207,291,255)(12,208,292,256)(13,209,293,257)(14,210,294,258)(15,236,152,106)(16,237,153,107)(17,238,154,108)(18,232,148,109)(19,233,149,110)(20,234,150,111)(21,235,151,112)(22,247,331,302)(23,248,332,303)(24,249,333,304)(25,250,334,305)(26,251,335,306)(27,252,336,307)(28,246,330,308)(29,198,114,128)(30,199,115,129)(31,200,116,130)(32,201,117,131)(33,202,118,132)(34,203,119,133)(35,197,113,127)(36,193,277,316)(37,194,278,317)(38,195,279,318)(39,196,280,319)(40,190,274,320)(41,191,275,321)(42,192,276,322)(43,239,323,270)(44,240,324,271)(45,241,325,272)(46,242,326,273)(47,243,327,267)(48,244,328,268)(49,245,329,269)(50,182,98,141)(51,176,92,142)(52,177,93,143)(53,178,94,144)(54,179,95,145)(55,180,96,146)(56,181,97,147)(57,214,298,284)(58,215,299,285)(59,216,300,286)(60,217,301,287)(61,211,295,281)(62,212,296,282)(63,213,297,283)(71,228,312,262)(72,229,313,263)(73,230,314,264)(74,231,315,265)(75,225,309,266)(76,226,310,260)(77,227,311,261)(78,224,140,162)(79,218,134,163)(80,219,135,164)(81,220,136,165)(82,221,137,166)(83,222,138,167)(84,223,139,168)(85,186,102,155)(86,187,103,156)(87,188,104,157)(88,189,105,158)(89,183,99,159)(90,184,100,160)(91,185,101,161), (1,105,135,33,18,54)(2,99,136,34,19,55)(3,100,137,35,20,56)(4,101,138,29,21,50)(5,102,139,30,15,51)(6,103,140,31,16,52)(7,104,134,32,17,53)(8,311,275,298,28,44)(9,312,276,299,22,45)(10,313,277,300,23,46)(11,314,278,301,24,47)(12,315,279,295,25,48)(13,309,280,296,26,49)(14,310,274,297,27,43)(36,59,332,326,290,72)(37,60,333,327,291,73)(38,61,334,328,292,74)(39,62,335,329,293,75)(40,63,336,323,294,76)(41,57,330,324,288,77)(42,58,331,325,289,71)(64,91,83,114,151,98)(65,85,84,115,152,92)(66,86,78,116,153,93)(67,87,79,117,154,94)(68,88,80,118,148,95)(69,89,81,119,149,96)(70,90,82,113,150,97)(106,142,174,186,223,129)(107,143,175,187,224,130)(108,144,169,188,218,131)(109,145,170,189,219,132)(110,146,171,183,220,133)(111,147,172,184,221,127)(112,141,173,185,222,128)(120,156,162,200,237,177)(121,157,163,201,238,178)(122,158,164,202,232,179)(123,159,165,203,233,180)(124,160,166,197,234,181)(125,161,167,198,235,182)(126,155,168,199,236,176)(190,213,307,270,258,226)(191,214,308,271,259,227)(192,215,302,272,253,228)(193,216,303,273,254,229)(194,217,304,267,255,230)(195,211,305,268,256,231)(196,212,306,269,257,225)(204,261,321,284,246,240)(205,262,322,285,247,241)(206,263,316,286,248,242)(207,264,317,287,249,243)(208,265,318,281,250,244)(209,266,319,282,251,245)(210,260,320,283,252,239), (1,216,33,254)(2,217,34,255)(3,211,35,256)(4,212,29,257)(5,213,30,258)(6,214,31,259)(7,215,32,253)(8,175,298,130)(9,169,299,131)(10,170,300,132)(11,171,301,133)(12,172,295,127)(13,173,296,128)(14,174,297,129)(15,270,102,190)(16,271,103,191)(17,272,104,192)(18,273,105,193)(19,267,99,194)(20,268,100,195)(21,269,101,196)(22,218,312,144)(23,219,313,145)(24,220,314,146)(25,221,315,147)(26,222,309,141)(27,223,310,142)(28,224,311,143)(36,232,326,158)(37,233,327,159)(38,234,328,160)(39,235,329,161)(40,236,323,155)(41,237,324,156)(42,238,325,157)(43,186,274,106)(44,187,275,107)(45,188,276,108)(46,189,277,109)(47,183,278,110)(48,184,279,111)(49,185,280,112)(50,306,138,225)(51,307,139,226)(52,308,140,227)(53,302,134,228)(54,303,135,229)(55,304,136,230)(56,305,137,231)(57,200,288,120)(58,201,289,121)(59,202,290,122)(60,203,291,123)(61,197,292,124)(62,198,293,125)(63,199,294,126)(64,282,114,209)(65,283,115,210)(66,284,116,204)(67,285,117,205)(68,286,118,206)(69,287,119,207)(70,281,113,208)(71,178,331,163)(72,179,332,164)(73,180,333,165)(74,181,334,166)(75,182,335,167)(76,176,336,168)(77,177,330,162)(78,261,93,246)(79,262,94,247)(80,263,95,248)(81,264,96,249)(82,265,97,250)(83,266,98,251)(84,260,92,252)(85,320,152,239)(86,321,153,240)(87,322,154,241)(88,316,148,242)(89,317,149,243)(90,318,150,244)(91,319,151,245)>;
G:=Group( (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), (1,122,68,170)(2,123,69,171)(3,124,70,172)(4,125,64,173)(5,126,65,174)(6,120,66,175)(7,121,67,169)(8,204,288,259)(9,205,289,253)(10,206,290,254)(11,207,291,255)(12,208,292,256)(13,209,293,257)(14,210,294,258)(15,236,152,106)(16,237,153,107)(17,238,154,108)(18,232,148,109)(19,233,149,110)(20,234,150,111)(21,235,151,112)(22,247,331,302)(23,248,332,303)(24,249,333,304)(25,250,334,305)(26,251,335,306)(27,252,336,307)(28,246,330,308)(29,198,114,128)(30,199,115,129)(31,200,116,130)(32,201,117,131)(33,202,118,132)(34,203,119,133)(35,197,113,127)(36,193,277,316)(37,194,278,317)(38,195,279,318)(39,196,280,319)(40,190,274,320)(41,191,275,321)(42,192,276,322)(43,239,323,270)(44,240,324,271)(45,241,325,272)(46,242,326,273)(47,243,327,267)(48,244,328,268)(49,245,329,269)(50,182,98,141)(51,176,92,142)(52,177,93,143)(53,178,94,144)(54,179,95,145)(55,180,96,146)(56,181,97,147)(57,214,298,284)(58,215,299,285)(59,216,300,286)(60,217,301,287)(61,211,295,281)(62,212,296,282)(63,213,297,283)(71,228,312,262)(72,229,313,263)(73,230,314,264)(74,231,315,265)(75,225,309,266)(76,226,310,260)(77,227,311,261)(78,224,140,162)(79,218,134,163)(80,219,135,164)(81,220,136,165)(82,221,137,166)(83,222,138,167)(84,223,139,168)(85,186,102,155)(86,187,103,156)(87,188,104,157)(88,189,105,158)(89,183,99,159)(90,184,100,160)(91,185,101,161), (1,105,135,33,18,54)(2,99,136,34,19,55)(3,100,137,35,20,56)(4,101,138,29,21,50)(5,102,139,30,15,51)(6,103,140,31,16,52)(7,104,134,32,17,53)(8,311,275,298,28,44)(9,312,276,299,22,45)(10,313,277,300,23,46)(11,314,278,301,24,47)(12,315,279,295,25,48)(13,309,280,296,26,49)(14,310,274,297,27,43)(36,59,332,326,290,72)(37,60,333,327,291,73)(38,61,334,328,292,74)(39,62,335,329,293,75)(40,63,336,323,294,76)(41,57,330,324,288,77)(42,58,331,325,289,71)(64,91,83,114,151,98)(65,85,84,115,152,92)(66,86,78,116,153,93)(67,87,79,117,154,94)(68,88,80,118,148,95)(69,89,81,119,149,96)(70,90,82,113,150,97)(106,142,174,186,223,129)(107,143,175,187,224,130)(108,144,169,188,218,131)(109,145,170,189,219,132)(110,146,171,183,220,133)(111,147,172,184,221,127)(112,141,173,185,222,128)(120,156,162,200,237,177)(121,157,163,201,238,178)(122,158,164,202,232,179)(123,159,165,203,233,180)(124,160,166,197,234,181)(125,161,167,198,235,182)(126,155,168,199,236,176)(190,213,307,270,258,226)(191,214,308,271,259,227)(192,215,302,272,253,228)(193,216,303,273,254,229)(194,217,304,267,255,230)(195,211,305,268,256,231)(196,212,306,269,257,225)(204,261,321,284,246,240)(205,262,322,285,247,241)(206,263,316,286,248,242)(207,264,317,287,249,243)(208,265,318,281,250,244)(209,266,319,282,251,245)(210,260,320,283,252,239), (1,216,33,254)(2,217,34,255)(3,211,35,256)(4,212,29,257)(5,213,30,258)(6,214,31,259)(7,215,32,253)(8,175,298,130)(9,169,299,131)(10,170,300,132)(11,171,301,133)(12,172,295,127)(13,173,296,128)(14,174,297,129)(15,270,102,190)(16,271,103,191)(17,272,104,192)(18,273,105,193)(19,267,99,194)(20,268,100,195)(21,269,101,196)(22,218,312,144)(23,219,313,145)(24,220,314,146)(25,221,315,147)(26,222,309,141)(27,223,310,142)(28,224,311,143)(36,232,326,158)(37,233,327,159)(38,234,328,160)(39,235,329,161)(40,236,323,155)(41,237,324,156)(42,238,325,157)(43,186,274,106)(44,187,275,107)(45,188,276,108)(46,189,277,109)(47,183,278,110)(48,184,279,111)(49,185,280,112)(50,306,138,225)(51,307,139,226)(52,308,140,227)(53,302,134,228)(54,303,135,229)(55,304,136,230)(56,305,137,231)(57,200,288,120)(58,201,289,121)(59,202,290,122)(60,203,291,123)(61,197,292,124)(62,198,293,125)(63,199,294,126)(64,282,114,209)(65,283,115,210)(66,284,116,204)(67,285,117,205)(68,286,118,206)(69,287,119,207)(70,281,113,208)(71,178,331,163)(72,179,332,164)(73,180,333,165)(74,181,334,166)(75,182,335,167)(76,176,336,168)(77,177,330,162)(78,261,93,246)(79,262,94,247)(80,263,95,248)(81,264,96,249)(82,265,97,250)(83,266,98,251)(84,260,92,252)(85,320,152,239)(86,321,153,240)(87,322,154,241)(88,316,148,242)(89,317,149,243)(90,318,150,244)(91,319,151,245) );
G=PermutationGroup([[(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)], [(1,122,68,170),(2,123,69,171),(3,124,70,172),(4,125,64,173),(5,126,65,174),(6,120,66,175),(7,121,67,169),(8,204,288,259),(9,205,289,253),(10,206,290,254),(11,207,291,255),(12,208,292,256),(13,209,293,257),(14,210,294,258),(15,236,152,106),(16,237,153,107),(17,238,154,108),(18,232,148,109),(19,233,149,110),(20,234,150,111),(21,235,151,112),(22,247,331,302),(23,248,332,303),(24,249,333,304),(25,250,334,305),(26,251,335,306),(27,252,336,307),(28,246,330,308),(29,198,114,128),(30,199,115,129),(31,200,116,130),(32,201,117,131),(33,202,118,132),(34,203,119,133),(35,197,113,127),(36,193,277,316),(37,194,278,317),(38,195,279,318),(39,196,280,319),(40,190,274,320),(41,191,275,321),(42,192,276,322),(43,239,323,270),(44,240,324,271),(45,241,325,272),(46,242,326,273),(47,243,327,267),(48,244,328,268),(49,245,329,269),(50,182,98,141),(51,176,92,142),(52,177,93,143),(53,178,94,144),(54,179,95,145),(55,180,96,146),(56,181,97,147),(57,214,298,284),(58,215,299,285),(59,216,300,286),(60,217,301,287),(61,211,295,281),(62,212,296,282),(63,213,297,283),(71,228,312,262),(72,229,313,263),(73,230,314,264),(74,231,315,265),(75,225,309,266),(76,226,310,260),(77,227,311,261),(78,224,140,162),(79,218,134,163),(80,219,135,164),(81,220,136,165),(82,221,137,166),(83,222,138,167),(84,223,139,168),(85,186,102,155),(86,187,103,156),(87,188,104,157),(88,189,105,158),(89,183,99,159),(90,184,100,160),(91,185,101,161)], [(1,105,135,33,18,54),(2,99,136,34,19,55),(3,100,137,35,20,56),(4,101,138,29,21,50),(5,102,139,30,15,51),(6,103,140,31,16,52),(7,104,134,32,17,53),(8,311,275,298,28,44),(9,312,276,299,22,45),(10,313,277,300,23,46),(11,314,278,301,24,47),(12,315,279,295,25,48),(13,309,280,296,26,49),(14,310,274,297,27,43),(36,59,332,326,290,72),(37,60,333,327,291,73),(38,61,334,328,292,74),(39,62,335,329,293,75),(40,63,336,323,294,76),(41,57,330,324,288,77),(42,58,331,325,289,71),(64,91,83,114,151,98),(65,85,84,115,152,92),(66,86,78,116,153,93),(67,87,79,117,154,94),(68,88,80,118,148,95),(69,89,81,119,149,96),(70,90,82,113,150,97),(106,142,174,186,223,129),(107,143,175,187,224,130),(108,144,169,188,218,131),(109,145,170,189,219,132),(110,146,171,183,220,133),(111,147,172,184,221,127),(112,141,173,185,222,128),(120,156,162,200,237,177),(121,157,163,201,238,178),(122,158,164,202,232,179),(123,159,165,203,233,180),(124,160,166,197,234,181),(125,161,167,198,235,182),(126,155,168,199,236,176),(190,213,307,270,258,226),(191,214,308,271,259,227),(192,215,302,272,253,228),(193,216,303,273,254,229),(194,217,304,267,255,230),(195,211,305,268,256,231),(196,212,306,269,257,225),(204,261,321,284,246,240),(205,262,322,285,247,241),(206,263,316,286,248,242),(207,264,317,287,249,243),(208,265,318,281,250,244),(209,266,319,282,251,245),(210,260,320,283,252,239)], [(1,216,33,254),(2,217,34,255),(3,211,35,256),(4,212,29,257),(5,213,30,258),(6,214,31,259),(7,215,32,253),(8,175,298,130),(9,169,299,131),(10,170,300,132),(11,171,301,133),(12,172,295,127),(13,173,296,128),(14,174,297,129),(15,270,102,190),(16,271,103,191),(17,272,104,192),(18,273,105,193),(19,267,99,194),(20,268,100,195),(21,269,101,196),(22,218,312,144),(23,219,313,145),(24,220,314,146),(25,221,315,147),(26,222,309,141),(27,223,310,142),(28,224,311,143),(36,232,326,158),(37,233,327,159),(38,234,328,160),(39,235,329,161),(40,236,323,155),(41,237,324,156),(42,238,325,157),(43,186,274,106),(44,187,275,107),(45,188,276,108),(46,189,277,109),(47,183,278,110),(48,184,279,111),(49,185,280,112),(50,306,138,225),(51,307,139,226),(52,308,140,227),(53,302,134,228),(54,303,135,229),(55,304,136,230),(56,305,137,231),(57,200,288,120),(58,201,289,121),(59,202,290,122),(60,203,291,123),(61,197,292,124),(62,198,293,125),(63,199,294,126),(64,282,114,209),(65,283,115,210),(66,284,116,204),(67,285,117,205),(68,286,118,206),(69,287,119,207),(70,281,113,208),(71,178,331,163),(72,179,332,164),(73,180,333,165),(74,181,334,166),(75,182,335,167),(76,176,336,168),(77,177,330,162),(78,261,93,246),(79,262,94,247),(80,263,95,248),(81,264,96,249),(82,265,97,250),(83,266,98,251),(84,260,92,252),(85,320,152,239),(86,321,153,240),(87,322,154,241),(88,316,148,242),(89,317,149,243),(90,318,150,244),(91,319,151,245)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 7A | ··· | 7F | 12A | 12B | 12C | 12D | 14A | ··· | 14R | 21A | ··· | 21F | 28A | ··· | 28L | 28M | ··· | 28AJ | 42A | ··· | 42R | 84A | ··· | 84X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 7 | ··· | 7 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
126 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 | C4 | C7 | C14 | C14 | C28 | S3 | D4 | Q8 | Dic3 | D6 | Dic6 | D12 | S3×C7 | C7×D4 | C7×Q8 | C7×Dic3 | S3×C14 | C7×Dic6 | C7×D12 |
| kernel | C7×C4⋊Dic3 | Dic3×C14 | C2×C84 | C84 | C4⋊Dic3 | C2×Dic3 | C2×C12 | C12 | C2×C28 | C42 | C42 | C28 | C2×C14 | C14 | C14 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 6 | 12 | 6 | 24 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 6 | 6 | 6 | 12 | 6 | 12 | 12 |
Matrix representation of C7×C4⋊Dic3 ►in GL4(𝔽337) generated by
| 175 | 0 | 0 | 0 |
| 0 | 175 | 0 | 0 |
| 0 | 0 | 79 | 0 |
| 0 | 0 | 0 | 79 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 322 | 307 |
| 0 | 0 | 30 | 15 |
| 0 | 1 | 0 | 0 |
| 336 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 336 | 0 |
| 314 | 157 | 0 | 0 |
| 134 | 23 | 0 | 0 |
| 0 | 0 | 204 | 310 |
| 0 | 0 | 106 | 133 |
G:=sub<GL(4,GF(337))| [175,0,0,0,0,175,0,0,0,0,79,0,0,0,0,79],[1,0,0,0,0,1,0,0,0,0,322,30,0,0,307,15],[0,336,0,0,1,1,0,0,0,0,1,336,0,0,1,0],[314,134,0,0,157,23,0,0,0,0,204,106,0,0,310,133] >;
C7×C4⋊Dic3 in GAP, Magma, Sage, TeX
C_7\times C_4\rtimes {\rm Dic}_3 % in TeX
G:=Group("C7xC4:Dic3"); // GroupNames label
G:=SmallGroup(336,83);
// by ID
G=gap.SmallGroup(336,83);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-3,168,697,343,8069]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^6=1,d^2=c^3,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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