direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7×Dic3⋊C4, Dic3⋊C28, C42.8Q8, C42.45D4, C14.6Dic6, C21⋊8(C4⋊C4), C6.5(C7×D4), C6.1(C7×Q8), C2.4(S3×C28), C6.3(C2×C28), (C2×C84).2C2, (C2×C28).1S3, C14.18(C4×S3), C42.27(C2×C4), (C2×C12).1C14, (C7×Dic3)⋊3C4, (C2×C14).32D6, C2.1(C7×Dic6), C22.4(S3×C14), C14.21(C3⋊D4), (C2×C42).43C22, (C2×Dic3).1C14, (Dic3×C14).6C2, C3⋊1(C7×C4⋊C4), (C2×C4).1(S3×C7), C2.1(C7×C3⋊D4), (C2×C6).4(C2×C14), SmallGroup(336,82)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×Dic3⋊C4
G = < a,b,c,d | a7=b6=d4=1, c2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >
Smallest permutation representation of C7×Dic3⋊C4
►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)
(1 148 135 55 30 80)(2 149 136 56 31 81)(3 150 137 50 32 82)(4 151 138 51 33 83)(5 152 139 52 34 84)(6 153 140 53 35 78)(7 154 134 54 29 79)(8 37 335 280 262 26)(9 38 336 274 263 27)(10 39 330 275 264 28)(11 40 331 276 265 22)(12 41 332 277 266 23)(13 42 333 278 260 24)(14 36 334 279 261 25)(15 113 64 88 99 120)(16 114 65 89 100 121)(17 115 66 90 101 122)(18 116 67 91 102 123)(19 117 68 85 103 124)(20 118 69 86 104 125)(21 119 70 87 105 126)(43 63 310 323 297 76)(44 57 311 324 298 77)(45 58 312 325 299 71)(46 59 313 326 300 72)(47 60 314 327 301 73)(48 61 315 328 295 74)(49 62 309 329 296 75)(92 110 165 202 233 220)(93 111 166 203 234 221)(94 112 167 197 235 222)(95 106 168 198 236 223)(96 107 162 199 237 224)(97 108 163 200 238 218)(98 109 164 201 232 219)(127 160 147 173 184 205)(128 161 141 174 185 206)(129 155 142 175 186 207)(130 156 143 169 187 208)(131 157 144 170 188 209)(132 158 145 171 189 210)(133 159 146 172 183 204)(176 304 317 286 249 194)(177 305 318 287 250 195)(178 306 319 281 251 196)(179 307 320 282 252 190)(180 308 321 283 246 191)(181 302 322 284 247 192)(182 303 316 285 248 193)(211 289 268 257 231 244)(212 290 269 258 225 245)(213 291 270 259 226 239)(214 292 271 253 227 240)(215 293 272 254 228 241)(216 294 273 255 229 242)(217 288 267 256 230 243)
(1 255 55 216)(2 256 56 217)(3 257 50 211)(4 258 51 212)(5 259 52 213)(6 253 53 214)(7 254 54 215)(8 161 280 185)(9 155 274 186)(10 156 275 187)(11 157 276 188)(12 158 277 189)(13 159 278 183)(14 160 279 184)(15 176 88 286)(16 177 89 287)(17 178 90 281)(18 179 91 282)(19 180 85 283)(20 181 86 284)(21 182 87 285)(22 144 331 209)(23 145 332 210)(24 146 333 204)(25 147 334 205)(26 141 335 206)(27 142 336 207)(28 143 330 208)(29 241 154 272)(30 242 148 273)(31 243 149 267)(32 244 150 268)(33 245 151 269)(34 239 152 270)(35 240 153 271)(36 127 261 173)(37 128 262 174)(38 129 263 175)(39 130 264 169)(40 131 265 170)(41 132 266 171)(42 133 260 172)(43 106 323 236)(44 107 324 237)(45 108 325 238)(46 109 326 232)(47 110 327 233)(48 111 328 234)(49 112 329 235)(57 96 298 199)(58 97 299 200)(59 98 300 201)(60 92 301 202)(61 93 295 203)(62 94 296 197)(63 95 297 198)(64 249 120 304)(65 250 121 305)(66 251 122 306)(67 252 123 307)(68 246 124 308)(69 247 125 302)(70 248 126 303)(71 163 312 218)(72 164 313 219)(73 165 314 220)(74 166 315 221)(75 167 309 222)(76 168 310 223)(77 162 311 224)(78 227 140 292)(79 228 134 293)(80 229 135 294)(81 230 136 288)(82 231 137 289)(83 225 138 290)(84 226 139 291)(99 317 113 194)(100 318 114 195)(101 319 115 196)(102 320 116 190)(103 321 117 191)(104 322 118 192)(105 316 119 193)
(1 132 21 201)(2 133 15 202)(3 127 16 203)(4 128 17 197)(5 129 18 198)(6 130 19 199)(7 131 20 200)(8 196 329 269)(9 190 323 270)(10 191 324 271)(11 192 325 272)(12 193 326 273)(13 194 327 267)(14 195 328 268)(22 247 312 293)(23 248 313 294)(24 249 314 288)(25 250 315 289)(26 251 309 290)(27 252 310 291)(28 246 311 292)(29 188 104 108)(30 189 105 109)(31 183 99 110)(32 184 100 111)(33 185 101 112)(34 186 102 106)(35 187 103 107)(36 177 295 257)(37 178 296 258)(38 179 297 259)(39 180 298 253)(40 181 299 254)(41 182 300 255)(42 176 301 256)(43 239 274 320)(44 240 275 321)(45 241 276 322)(46 242 277 316)(47 243 278 317)(48 244 279 318)(49 245 280 319)(50 173 89 93)(51 174 90 94)(52 175 91 95)(53 169 85 96)(54 170 86 97)(55 171 87 98)(56 172 88 92)(57 214 264 283)(58 215 265 284)(59 216 266 285)(60 217 260 286)(61 211 261 287)(62 212 262 281)(63 213 263 282)(64 220 136 146)(65 221 137 147)(66 222 138 141)(67 223 139 142)(68 224 140 143)(69 218 134 144)(70 219 135 145)(71 228 331 302)(72 229 332 303)(73 230 333 304)(74 231 334 305)(75 225 335 306)(76 226 336 307)(77 227 330 308)(78 208 124 162)(79 209 125 163)(80 210 126 164)(81 204 120 165)(82 205 121 166)(83 206 122 167)(84 207 123 168)(113 233 149 159)(114 234 150 160)(115 235 151 161)(116 236 152 155)(117 237 153 156)(118 238 154 157)(119 232 148 158)
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), (1,148,135,55,30,80)(2,149,136,56,31,81)(3,150,137,50,32,82)(4,151,138,51,33,83)(5,152,139,52,34,84)(6,153,140,53,35,78)(7,154,134,54,29,79)(8,37,335,280,262,26)(9,38,336,274,263,27)(10,39,330,275,264,28)(11,40,331,276,265,22)(12,41,332,277,266,23)(13,42,333,278,260,24)(14,36,334,279,261,25)(15,113,64,88,99,120)(16,114,65,89,100,121)(17,115,66,90,101,122)(18,116,67,91,102,123)(19,117,68,85,103,124)(20,118,69,86,104,125)(21,119,70,87,105,126)(43,63,310,323,297,76)(44,57,311,324,298,77)(45,58,312,325,299,71)(46,59,313,326,300,72)(47,60,314,327,301,73)(48,61,315,328,295,74)(49,62,309,329,296,75)(92,110,165,202,233,220)(93,111,166,203,234,221)(94,112,167,197,235,222)(95,106,168,198,236,223)(96,107,162,199,237,224)(97,108,163,200,238,218)(98,109,164,201,232,219)(127,160,147,173,184,205)(128,161,141,174,185,206)(129,155,142,175,186,207)(130,156,143,169,187,208)(131,157,144,170,188,209)(132,158,145,171,189,210)(133,159,146,172,183,204)(176,304,317,286,249,194)(177,305,318,287,250,195)(178,306,319,281,251,196)(179,307,320,282,252,190)(180,308,321,283,246,191)(181,302,322,284,247,192)(182,303,316,285,248,193)(211,289,268,257,231,244)(212,290,269,258,225,245)(213,291,270,259,226,239)(214,292,271,253,227,240)(215,293,272,254,228,241)(216,294,273,255,229,242)(217,288,267,256,230,243), (1,255,55,216)(2,256,56,217)(3,257,50,211)(4,258,51,212)(5,259,52,213)(6,253,53,214)(7,254,54,215)(8,161,280,185)(9,155,274,186)(10,156,275,187)(11,157,276,188)(12,158,277,189)(13,159,278,183)(14,160,279,184)(15,176,88,286)(16,177,89,287)(17,178,90,281)(18,179,91,282)(19,180,85,283)(20,181,86,284)(21,182,87,285)(22,144,331,209)(23,145,332,210)(24,146,333,204)(25,147,334,205)(26,141,335,206)(27,142,336,207)(28,143,330,208)(29,241,154,272)(30,242,148,273)(31,243,149,267)(32,244,150,268)(33,245,151,269)(34,239,152,270)(35,240,153,271)(36,127,261,173)(37,128,262,174)(38,129,263,175)(39,130,264,169)(40,131,265,170)(41,132,266,171)(42,133,260,172)(43,106,323,236)(44,107,324,237)(45,108,325,238)(46,109,326,232)(47,110,327,233)(48,111,328,234)(49,112,329,235)(57,96,298,199)(58,97,299,200)(59,98,300,201)(60,92,301,202)(61,93,295,203)(62,94,296,197)(63,95,297,198)(64,249,120,304)(65,250,121,305)(66,251,122,306)(67,252,123,307)(68,246,124,308)(69,247,125,302)(70,248,126,303)(71,163,312,218)(72,164,313,219)(73,165,314,220)(74,166,315,221)(75,167,309,222)(76,168,310,223)(77,162,311,224)(78,227,140,292)(79,228,134,293)(80,229,135,294)(81,230,136,288)(82,231,137,289)(83,225,138,290)(84,226,139,291)(99,317,113,194)(100,318,114,195)(101,319,115,196)(102,320,116,190)(103,321,117,191)(104,322,118,192)(105,316,119,193), (1,132,21,201)(2,133,15,202)(3,127,16,203)(4,128,17,197)(5,129,18,198)(6,130,19,199)(7,131,20,200)(8,196,329,269)(9,190,323,270)(10,191,324,271)(11,192,325,272)(12,193,326,273)(13,194,327,267)(14,195,328,268)(22,247,312,293)(23,248,313,294)(24,249,314,288)(25,250,315,289)(26,251,309,290)(27,252,310,291)(28,246,311,292)(29,188,104,108)(30,189,105,109)(31,183,99,110)(32,184,100,111)(33,185,101,112)(34,186,102,106)(35,187,103,107)(36,177,295,257)(37,178,296,258)(38,179,297,259)(39,180,298,253)(40,181,299,254)(41,182,300,255)(42,176,301,256)(43,239,274,320)(44,240,275,321)(45,241,276,322)(46,242,277,316)(47,243,278,317)(48,244,279,318)(49,245,280,319)(50,173,89,93)(51,174,90,94)(52,175,91,95)(53,169,85,96)(54,170,86,97)(55,171,87,98)(56,172,88,92)(57,214,264,283)(58,215,265,284)(59,216,266,285)(60,217,260,286)(61,211,261,287)(62,212,262,281)(63,213,263,282)(64,220,136,146)(65,221,137,147)(66,222,138,141)(67,223,139,142)(68,224,140,143)(69,218,134,144)(70,219,135,145)(71,228,331,302)(72,229,332,303)(73,230,333,304)(74,231,334,305)(75,225,335,306)(76,226,336,307)(77,227,330,308)(78,208,124,162)(79,209,125,163)(80,210,126,164)(81,204,120,165)(82,205,121,166)(83,206,122,167)(84,207,123,168)(113,233,149,159)(114,234,150,160)(115,235,151,161)(116,236,152,155)(117,237,153,156)(118,238,154,157)(119,232,148,158)>;
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), (1,148,135,55,30,80)(2,149,136,56,31,81)(3,150,137,50,32,82)(4,151,138,51,33,83)(5,152,139,52,34,84)(6,153,140,53,35,78)(7,154,134,54,29,79)(8,37,335,280,262,26)(9,38,336,274,263,27)(10,39,330,275,264,28)(11,40,331,276,265,22)(12,41,332,277,266,23)(13,42,333,278,260,24)(14,36,334,279,261,25)(15,113,64,88,99,120)(16,114,65,89,100,121)(17,115,66,90,101,122)(18,116,67,91,102,123)(19,117,68,85,103,124)(20,118,69,86,104,125)(21,119,70,87,105,126)(43,63,310,323,297,76)(44,57,311,324,298,77)(45,58,312,325,299,71)(46,59,313,326,300,72)(47,60,314,327,301,73)(48,61,315,328,295,74)(49,62,309,329,296,75)(92,110,165,202,233,220)(93,111,166,203,234,221)(94,112,167,197,235,222)(95,106,168,198,236,223)(96,107,162,199,237,224)(97,108,163,200,238,218)(98,109,164,201,232,219)(127,160,147,173,184,205)(128,161,141,174,185,206)(129,155,142,175,186,207)(130,156,143,169,187,208)(131,157,144,170,188,209)(132,158,145,171,189,210)(133,159,146,172,183,204)(176,304,317,286,249,194)(177,305,318,287,250,195)(178,306,319,281,251,196)(179,307,320,282,252,190)(180,308,321,283,246,191)(181,302,322,284,247,192)(182,303,316,285,248,193)(211,289,268,257,231,244)(212,290,269,258,225,245)(213,291,270,259,226,239)(214,292,271,253,227,240)(215,293,272,254,228,241)(216,294,273,255,229,242)(217,288,267,256,230,243), (1,255,55,216)(2,256,56,217)(3,257,50,211)(4,258,51,212)(5,259,52,213)(6,253,53,214)(7,254,54,215)(8,161,280,185)(9,155,274,186)(10,156,275,187)(11,157,276,188)(12,158,277,189)(13,159,278,183)(14,160,279,184)(15,176,88,286)(16,177,89,287)(17,178,90,281)(18,179,91,282)(19,180,85,283)(20,181,86,284)(21,182,87,285)(22,144,331,209)(23,145,332,210)(24,146,333,204)(25,147,334,205)(26,141,335,206)(27,142,336,207)(28,143,330,208)(29,241,154,272)(30,242,148,273)(31,243,149,267)(32,244,150,268)(33,245,151,269)(34,239,152,270)(35,240,153,271)(36,127,261,173)(37,128,262,174)(38,129,263,175)(39,130,264,169)(40,131,265,170)(41,132,266,171)(42,133,260,172)(43,106,323,236)(44,107,324,237)(45,108,325,238)(46,109,326,232)(47,110,327,233)(48,111,328,234)(49,112,329,235)(57,96,298,199)(58,97,299,200)(59,98,300,201)(60,92,301,202)(61,93,295,203)(62,94,296,197)(63,95,297,198)(64,249,120,304)(65,250,121,305)(66,251,122,306)(67,252,123,307)(68,246,124,308)(69,247,125,302)(70,248,126,303)(71,163,312,218)(72,164,313,219)(73,165,314,220)(74,166,315,221)(75,167,309,222)(76,168,310,223)(77,162,311,224)(78,227,140,292)(79,228,134,293)(80,229,135,294)(81,230,136,288)(82,231,137,289)(83,225,138,290)(84,226,139,291)(99,317,113,194)(100,318,114,195)(101,319,115,196)(102,320,116,190)(103,321,117,191)(104,322,118,192)(105,316,119,193), (1,132,21,201)(2,133,15,202)(3,127,16,203)(4,128,17,197)(5,129,18,198)(6,130,19,199)(7,131,20,200)(8,196,329,269)(9,190,323,270)(10,191,324,271)(11,192,325,272)(12,193,326,273)(13,194,327,267)(14,195,328,268)(22,247,312,293)(23,248,313,294)(24,249,314,288)(25,250,315,289)(26,251,309,290)(27,252,310,291)(28,246,311,292)(29,188,104,108)(30,189,105,109)(31,183,99,110)(32,184,100,111)(33,185,101,112)(34,186,102,106)(35,187,103,107)(36,177,295,257)(37,178,296,258)(38,179,297,259)(39,180,298,253)(40,181,299,254)(41,182,300,255)(42,176,301,256)(43,239,274,320)(44,240,275,321)(45,241,276,322)(46,242,277,316)(47,243,278,317)(48,244,279,318)(49,245,280,319)(50,173,89,93)(51,174,90,94)(52,175,91,95)(53,169,85,96)(54,170,86,97)(55,171,87,98)(56,172,88,92)(57,214,264,283)(58,215,265,284)(59,216,266,285)(60,217,260,286)(61,211,261,287)(62,212,262,281)(63,213,263,282)(64,220,136,146)(65,221,137,147)(66,222,138,141)(67,223,139,142)(68,224,140,143)(69,218,134,144)(70,219,135,145)(71,228,331,302)(72,229,332,303)(73,230,333,304)(74,231,334,305)(75,225,335,306)(76,226,336,307)(77,227,330,308)(78,208,124,162)(79,209,125,163)(80,210,126,164)(81,204,120,165)(82,205,121,166)(83,206,122,167)(84,207,123,168)(113,233,149,159)(114,234,150,160)(115,235,151,161)(116,236,152,155)(117,237,153,156)(118,238,154,157)(119,232,148,158) );
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)], [(1,148,135,55,30,80),(2,149,136,56,31,81),(3,150,137,50,32,82),(4,151,138,51,33,83),(5,152,139,52,34,84),(6,153,140,53,35,78),(7,154,134,54,29,79),(8,37,335,280,262,26),(9,38,336,274,263,27),(10,39,330,275,264,28),(11,40,331,276,265,22),(12,41,332,277,266,23),(13,42,333,278,260,24),(14,36,334,279,261,25),(15,113,64,88,99,120),(16,114,65,89,100,121),(17,115,66,90,101,122),(18,116,67,91,102,123),(19,117,68,85,103,124),(20,118,69,86,104,125),(21,119,70,87,105,126),(43,63,310,323,297,76),(44,57,311,324,298,77),(45,58,312,325,299,71),(46,59,313,326,300,72),(47,60,314,327,301,73),(48,61,315,328,295,74),(49,62,309,329,296,75),(92,110,165,202,233,220),(93,111,166,203,234,221),(94,112,167,197,235,222),(95,106,168,198,236,223),(96,107,162,199,237,224),(97,108,163,200,238,218),(98,109,164,201,232,219),(127,160,147,173,184,205),(128,161,141,174,185,206),(129,155,142,175,186,207),(130,156,143,169,187,208),(131,157,144,170,188,209),(132,158,145,171,189,210),(133,159,146,172,183,204),(176,304,317,286,249,194),(177,305,318,287,250,195),(178,306,319,281,251,196),(179,307,320,282,252,190),(180,308,321,283,246,191),(181,302,322,284,247,192),(182,303,316,285,248,193),(211,289,268,257,231,244),(212,290,269,258,225,245),(213,291,270,259,226,239),(214,292,271,253,227,240),(215,293,272,254,228,241),(216,294,273,255,229,242),(217,288,267,256,230,243)], [(1,255,55,216),(2,256,56,217),(3,257,50,211),(4,258,51,212),(5,259,52,213),(6,253,53,214),(7,254,54,215),(8,161,280,185),(9,155,274,186),(10,156,275,187),(11,157,276,188),(12,158,277,189),(13,159,278,183),(14,160,279,184),(15,176,88,286),(16,177,89,287),(17,178,90,281),(18,179,91,282),(19,180,85,283),(20,181,86,284),(21,182,87,285),(22,144,331,209),(23,145,332,210),(24,146,333,204),(25,147,334,205),(26,141,335,206),(27,142,336,207),(28,143,330,208),(29,241,154,272),(30,242,148,273),(31,243,149,267),(32,244,150,268),(33,245,151,269),(34,239,152,270),(35,240,153,271),(36,127,261,173),(37,128,262,174),(38,129,263,175),(39,130,264,169),(40,131,265,170),(41,132,266,171),(42,133,260,172),(43,106,323,236),(44,107,324,237),(45,108,325,238),(46,109,326,232),(47,110,327,233),(48,111,328,234),(49,112,329,235),(57,96,298,199),(58,97,299,200),(59,98,300,201),(60,92,301,202),(61,93,295,203),(62,94,296,197),(63,95,297,198),(64,249,120,304),(65,250,121,305),(66,251,122,306),(67,252,123,307),(68,246,124,308),(69,247,125,302),(70,248,126,303),(71,163,312,218),(72,164,313,219),(73,165,314,220),(74,166,315,221),(75,167,309,222),(76,168,310,223),(77,162,311,224),(78,227,140,292),(79,228,134,293),(80,229,135,294),(81,230,136,288),(82,231,137,289),(83,225,138,290),(84,226,139,291),(99,317,113,194),(100,318,114,195),(101,319,115,196),(102,320,116,190),(103,321,117,191),(104,322,118,192),(105,316,119,193)], [(1,132,21,201),(2,133,15,202),(3,127,16,203),(4,128,17,197),(5,129,18,198),(6,130,19,199),(7,131,20,200),(8,196,329,269),(9,190,323,270),(10,191,324,271),(11,192,325,272),(12,193,326,273),(13,194,327,267),(14,195,328,268),(22,247,312,293),(23,248,313,294),(24,249,314,288),(25,250,315,289),(26,251,309,290),(27,252,310,291),(28,246,311,292),(29,188,104,108),(30,189,105,109),(31,183,99,110),(32,184,100,111),(33,185,101,112),(34,186,102,106),(35,187,103,107),(36,177,295,257),(37,178,296,258),(38,179,297,259),(39,180,298,253),(40,181,299,254),(41,182,300,255),(42,176,301,256),(43,239,274,320),(44,240,275,321),(45,241,276,322),(46,242,277,316),(47,243,278,317),(48,244,279,318),(49,245,280,319),(50,173,89,93),(51,174,90,94),(52,175,91,95),(53,169,85,96),(54,170,86,97),(55,171,87,98),(56,172,88,92),(57,214,264,283),(58,215,265,284),(59,216,266,285),(60,217,260,286),(61,211,261,287),(62,212,262,281),(63,213,263,282),(64,220,136,146),(65,221,137,147),(66,222,138,141),(67,223,139,142),(68,224,140,143),(69,218,134,144),(70,219,135,145),(71,228,331,302),(72,229,332,303),(73,230,333,304),(74,231,334,305),(75,225,335,306),(76,226,336,307),(77,227,330,308),(78,208,124,162),(79,209,125,163),(80,210,126,164),(81,204,120,165),(82,205,121,166),(83,206,122,167),(84,207,123,168),(113,233,149,159),(114,234,150,160),(115,235,151,161),(116,236,152,155),(117,237,153,156),(118,238,154,157),(119,232,148,158)]])
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 | D6 | Dic6 | C4×S3 | C3⋊D4 | S3×C7 | C7×D4 | C7×Q8 | S3×C14 | C7×Dic6 | S3×C28 | C7×C3⋊D4 |
| kernel | C7×Dic3⋊C4 | Dic3×C14 | C2×C84 | C7×Dic3 | Dic3⋊C4 | C2×Dic3 | C2×C12 | Dic3 | C2×C28 | C42 | C42 | C2×C14 | C14 | C14 | C14 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 6 | 12 | 6 | 24 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 12 |
Matrix representation of C7×Dic3⋊C4 ►in GL3(𝔽337) generated by
| 1 | 0 | 0 |
| 0 | 79 | 0 |
| 0 | 0 | 79 |
| 1 | 0 | 0 |
| 0 | 1 | 1 |
| 0 | 336 | 0 |
| 1 | 0 | 0 |
| 0 | 308 | 316 |
| 0 | 8 | 29 |
| 189 | 0 | 0 |
| 0 | 15 | 30 |
| 0 | 307 | 322 |
G:=sub<GL(3,GF(337))| [1,0,0,0,79,0,0,0,79],[1,0,0,0,1,336,0,1,0],[1,0,0,0,308,8,0,316,29],[189,0,0,0,15,307,0,30,322] >;
C7×Dic3⋊C4 in GAP, Magma, Sage, TeX
C_7\times {\rm Dic}_3\rtimes C_4 % in TeX
G:=Group("C7xDic3:C4"); // GroupNames label
G:=SmallGroup(336,82);
// by ID
G=gap.SmallGroup(336,82);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-3,336,697,175,8069]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^6=d^4=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations
Export