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## G = C2×C6×Dic7order 336 = 24·3·7

### Direct product of C2×C6 and Dic7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C2×C6×Dic7
 Chief series C1 — C7 — C14 — C42 — C3×Dic7 — C6×Dic7 — C2×C6×Dic7
 Lower central C7 — C2×C6×Dic7
 Upper central C1 — C22×C6

Generators and relations for C2×C6×Dic7
G = < a,b,c,d | a2=b6=c14=1, d2=c7, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 240 in 108 conjugacy classes, 86 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C7, C2×C4, C23, C12, C2×C6, C14, C14, C22×C4, C21, C2×C12, C22×C6, Dic7, C2×C14, C42, C42, C22×C12, C2×Dic7, C22×C14, C3×Dic7, C2×C42, C22×Dic7, C6×Dic7, C22×C42, C2×C6×Dic7
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, D7, C22×C4, C2×C12, C22×C6, Dic7, D14, C3×D7, C22×C12, C2×Dic7, C22×D7, C3×Dic7, C6×D7, C22×Dic7, C6×Dic7, C2×C6×D7, C2×C6×Dic7

Smallest permutation representation of C2×C6×Dic7
Regular action on 336 points
Generators in S336
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 85)(9 86)(10 87)(11 88)(12 89)(13 90)(14 91)(15 112)(16 99)(17 100)(18 101)(19 102)(20 103)(21 104)(22 105)(23 106)(24 107)(25 108)(26 109)(27 110)(28 111)(29 124)(30 125)(31 126)(32 113)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 121)(41 122)(42 123)(43 139)(44 140)(45 127)(46 128)(47 129)(48 130)(49 131)(50 132)(51 133)(52 134)(53 135)(54 136)(55 137)(56 138)(57 143)(58 144)(59 145)(60 146)(61 147)(62 148)(63 149)(64 150)(65 151)(66 152)(67 153)(68 154)(69 141)(70 142)(71 157)(72 158)(73 159)(74 160)(75 161)(76 162)(77 163)(78 164)(79 165)(80 166)(81 167)(82 168)(83 155)(84 156)(169 260)(170 261)(171 262)(172 263)(173 264)(174 265)(175 266)(176 253)(177 254)(178 255)(179 256)(180 257)(181 258)(182 259)(183 267)(184 268)(185 269)(186 270)(187 271)(188 272)(189 273)(190 274)(191 275)(192 276)(193 277)(194 278)(195 279)(196 280)(197 288)(198 289)(199 290)(200 291)(201 292)(202 293)(203 294)(204 281)(205 282)(206 283)(207 284)(208 285)(209 286)(210 287)(211 302)(212 303)(213 304)(214 305)(215 306)(216 307)(217 308)(218 295)(219 296)(220 297)(221 298)(222 299)(223 300)(224 301)(225 309)(226 310)(227 311)(228 312)(229 313)(230 314)(231 315)(232 316)(233 317)(234 318)(235 319)(236 320)(237 321)(238 322)(239 323)(240 324)(241 325)(242 326)(243 327)(244 328)(245 329)(246 330)(247 331)(248 332)(249 333)(250 334)(251 335)(252 336)
(1 68 29 53 23 77)(2 69 30 54 24 78)(3 70 31 55 25 79)(4 57 32 56 26 80)(5 58 33 43 27 81)(6 59 34 44 28 82)(7 60 35 45 15 83)(8 61 36 46 16 84)(9 62 37 47 17 71)(10 63 38 48 18 72)(11 64 39 49 19 73)(12 65 40 50 20 74)(13 66 41 51 21 75)(14 67 42 52 22 76)(85 147 117 128 99 156)(86 148 118 129 100 157)(87 149 119 130 101 158)(88 150 120 131 102 159)(89 151 121 132 103 160)(90 152 122 133 104 161)(91 153 123 134 105 162)(92 154 124 135 106 163)(93 141 125 136 107 164)(94 142 126 137 108 165)(95 143 113 138 109 166)(96 144 114 139 110 167)(97 145 115 140 111 168)(98 146 116 127 112 155)(169 225 204 218 190 246)(170 226 205 219 191 247)(171 227 206 220 192 248)(172 228 207 221 193 249)(173 229 208 222 194 250)(174 230 209 223 195 251)(175 231 210 224 196 252)(176 232 197 211 183 239)(177 233 198 212 184 240)(178 234 199 213 185 241)(179 235 200 214 186 242)(180 236 201 215 187 243)(181 237 202 216 188 244)(182 238 203 217 189 245)(253 316 288 302 267 323)(254 317 289 303 268 324)(255 318 290 304 269 325)(256 319 291 305 270 326)(257 320 292 306 271 327)(258 321 293 307 272 328)(259 322 294 308 273 329)(260 309 281 295 274 330)(261 310 282 296 275 331)(262 311 283 297 276 332)(263 312 284 298 277 333)(264 313 285 299 278 334)(265 314 286 300 279 335)(266 315 287 301 280 336)
(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)
(1 218 8 211)(2 217 9 224)(3 216 10 223)(4 215 11 222)(5 214 12 221)(6 213 13 220)(7 212 14 219)(15 233 22 226)(16 232 23 225)(17 231 24 238)(18 230 25 237)(19 229 26 236)(20 228 27 235)(21 227 28 234)(29 246 36 239)(30 245 37 252)(31 244 38 251)(32 243 39 250)(33 242 40 249)(34 241 41 248)(35 240 42 247)(43 179 50 172)(44 178 51 171)(45 177 52 170)(46 176 53 169)(47 175 54 182)(48 174 55 181)(49 173 56 180)(57 187 64 194)(58 186 65 193)(59 185 66 192)(60 184 67 191)(61 183 68 190)(62 196 69 189)(63 195 70 188)(71 210 78 203)(72 209 79 202)(73 208 80 201)(74 207 81 200)(75 206 82 199)(76 205 83 198)(77 204 84 197)(85 302 92 295)(86 301 93 308)(87 300 94 307)(88 299 95 306)(89 298 96 305)(90 297 97 304)(91 296 98 303)(99 316 106 309)(100 315 107 322)(101 314 108 321)(102 313 109 320)(103 312 110 319)(104 311 111 318)(105 310 112 317)(113 327 120 334)(114 326 121 333)(115 325 122 332)(116 324 123 331)(117 323 124 330)(118 336 125 329)(119 335 126 328)(127 254 134 261)(128 253 135 260)(129 266 136 259)(130 265 137 258)(131 264 138 257)(132 263 139 256)(133 262 140 255)(141 273 148 280)(142 272 149 279)(143 271 150 278)(144 270 151 277)(145 269 152 276)(146 268 153 275)(147 267 154 274)(155 289 162 282)(156 288 163 281)(157 287 164 294)(158 286 165 293)(159 285 166 292)(160 284 167 291)(161 283 168 290)

G:=sub<Sym(336)| (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,112)(16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,109)(27,110)(28,111)(29,124)(30,125)(31,126)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,121)(41,122)(42,123)(43,139)(44,140)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,137)(56,138)(57,143)(58,144)(59,145)(60,146)(61,147)(62,148)(63,149)(64,150)(65,151)(66,152)(67,153)(68,154)(69,141)(70,142)(71,157)(72,158)(73,159)(74,160)(75,161)(76,162)(77,163)(78,164)(79,165)(80,166)(81,167)(82,168)(83,155)(84,156)(169,260)(170,261)(171,262)(172,263)(173,264)(174,265)(175,266)(176,253)(177,254)(178,255)(179,256)(180,257)(181,258)(182,259)(183,267)(184,268)(185,269)(186,270)(187,271)(188,272)(189,273)(190,274)(191,275)(192,276)(193,277)(194,278)(195,279)(196,280)(197,288)(198,289)(199,290)(200,291)(201,292)(202,293)(203,294)(204,281)(205,282)(206,283)(207,284)(208,285)(209,286)(210,287)(211,302)(212,303)(213,304)(214,305)(215,306)(216,307)(217,308)(218,295)(219,296)(220,297)(221,298)(222,299)(223,300)(224,301)(225,309)(226,310)(227,311)(228,312)(229,313)(230,314)(231,315)(232,316)(233,317)(234,318)(235,319)(236,320)(237,321)(238,322)(239,323)(240,324)(241,325)(242,326)(243,327)(244,328)(245,329)(246,330)(247,331)(248,332)(249,333)(250,334)(251,335)(252,336), (1,68,29,53,23,77)(2,69,30,54,24,78)(3,70,31,55,25,79)(4,57,32,56,26,80)(5,58,33,43,27,81)(6,59,34,44,28,82)(7,60,35,45,15,83)(8,61,36,46,16,84)(9,62,37,47,17,71)(10,63,38,48,18,72)(11,64,39,49,19,73)(12,65,40,50,20,74)(13,66,41,51,21,75)(14,67,42,52,22,76)(85,147,117,128,99,156)(86,148,118,129,100,157)(87,149,119,130,101,158)(88,150,120,131,102,159)(89,151,121,132,103,160)(90,152,122,133,104,161)(91,153,123,134,105,162)(92,154,124,135,106,163)(93,141,125,136,107,164)(94,142,126,137,108,165)(95,143,113,138,109,166)(96,144,114,139,110,167)(97,145,115,140,111,168)(98,146,116,127,112,155)(169,225,204,218,190,246)(170,226,205,219,191,247)(171,227,206,220,192,248)(172,228,207,221,193,249)(173,229,208,222,194,250)(174,230,209,223,195,251)(175,231,210,224,196,252)(176,232,197,211,183,239)(177,233,198,212,184,240)(178,234,199,213,185,241)(179,235,200,214,186,242)(180,236,201,215,187,243)(181,237,202,216,188,244)(182,238,203,217,189,245)(253,316,288,302,267,323)(254,317,289,303,268,324)(255,318,290,304,269,325)(256,319,291,305,270,326)(257,320,292,306,271,327)(258,321,293,307,272,328)(259,322,294,308,273,329)(260,309,281,295,274,330)(261,310,282,296,275,331)(262,311,283,297,276,332)(263,312,284,298,277,333)(264,313,285,299,278,334)(265,314,286,300,279,335)(266,315,287,301,280,336), (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), (1,218,8,211)(2,217,9,224)(3,216,10,223)(4,215,11,222)(5,214,12,221)(6,213,13,220)(7,212,14,219)(15,233,22,226)(16,232,23,225)(17,231,24,238)(18,230,25,237)(19,229,26,236)(20,228,27,235)(21,227,28,234)(29,246,36,239)(30,245,37,252)(31,244,38,251)(32,243,39,250)(33,242,40,249)(34,241,41,248)(35,240,42,247)(43,179,50,172)(44,178,51,171)(45,177,52,170)(46,176,53,169)(47,175,54,182)(48,174,55,181)(49,173,56,180)(57,187,64,194)(58,186,65,193)(59,185,66,192)(60,184,67,191)(61,183,68,190)(62,196,69,189)(63,195,70,188)(71,210,78,203)(72,209,79,202)(73,208,80,201)(74,207,81,200)(75,206,82,199)(76,205,83,198)(77,204,84,197)(85,302,92,295)(86,301,93,308)(87,300,94,307)(88,299,95,306)(89,298,96,305)(90,297,97,304)(91,296,98,303)(99,316,106,309)(100,315,107,322)(101,314,108,321)(102,313,109,320)(103,312,110,319)(104,311,111,318)(105,310,112,317)(113,327,120,334)(114,326,121,333)(115,325,122,332)(116,324,123,331)(117,323,124,330)(118,336,125,329)(119,335,126,328)(127,254,134,261)(128,253,135,260)(129,266,136,259)(130,265,137,258)(131,264,138,257)(132,263,139,256)(133,262,140,255)(141,273,148,280)(142,272,149,279)(143,271,150,278)(144,270,151,277)(145,269,152,276)(146,268,153,275)(147,267,154,274)(155,289,162,282)(156,288,163,281)(157,287,164,294)(158,286,165,293)(159,285,166,292)(160,284,167,291)(161,283,168,290)>;

G:=Group( (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,112)(16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,109)(27,110)(28,111)(29,124)(30,125)(31,126)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,121)(41,122)(42,123)(43,139)(44,140)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,137)(56,138)(57,143)(58,144)(59,145)(60,146)(61,147)(62,148)(63,149)(64,150)(65,151)(66,152)(67,153)(68,154)(69,141)(70,142)(71,157)(72,158)(73,159)(74,160)(75,161)(76,162)(77,163)(78,164)(79,165)(80,166)(81,167)(82,168)(83,155)(84,156)(169,260)(170,261)(171,262)(172,263)(173,264)(174,265)(175,266)(176,253)(177,254)(178,255)(179,256)(180,257)(181,258)(182,259)(183,267)(184,268)(185,269)(186,270)(187,271)(188,272)(189,273)(190,274)(191,275)(192,276)(193,277)(194,278)(195,279)(196,280)(197,288)(198,289)(199,290)(200,291)(201,292)(202,293)(203,294)(204,281)(205,282)(206,283)(207,284)(208,285)(209,286)(210,287)(211,302)(212,303)(213,304)(214,305)(215,306)(216,307)(217,308)(218,295)(219,296)(220,297)(221,298)(222,299)(223,300)(224,301)(225,309)(226,310)(227,311)(228,312)(229,313)(230,314)(231,315)(232,316)(233,317)(234,318)(235,319)(236,320)(237,321)(238,322)(239,323)(240,324)(241,325)(242,326)(243,327)(244,328)(245,329)(246,330)(247,331)(248,332)(249,333)(250,334)(251,335)(252,336), (1,68,29,53,23,77)(2,69,30,54,24,78)(3,70,31,55,25,79)(4,57,32,56,26,80)(5,58,33,43,27,81)(6,59,34,44,28,82)(7,60,35,45,15,83)(8,61,36,46,16,84)(9,62,37,47,17,71)(10,63,38,48,18,72)(11,64,39,49,19,73)(12,65,40,50,20,74)(13,66,41,51,21,75)(14,67,42,52,22,76)(85,147,117,128,99,156)(86,148,118,129,100,157)(87,149,119,130,101,158)(88,150,120,131,102,159)(89,151,121,132,103,160)(90,152,122,133,104,161)(91,153,123,134,105,162)(92,154,124,135,106,163)(93,141,125,136,107,164)(94,142,126,137,108,165)(95,143,113,138,109,166)(96,144,114,139,110,167)(97,145,115,140,111,168)(98,146,116,127,112,155)(169,225,204,218,190,246)(170,226,205,219,191,247)(171,227,206,220,192,248)(172,228,207,221,193,249)(173,229,208,222,194,250)(174,230,209,223,195,251)(175,231,210,224,196,252)(176,232,197,211,183,239)(177,233,198,212,184,240)(178,234,199,213,185,241)(179,235,200,214,186,242)(180,236,201,215,187,243)(181,237,202,216,188,244)(182,238,203,217,189,245)(253,316,288,302,267,323)(254,317,289,303,268,324)(255,318,290,304,269,325)(256,319,291,305,270,326)(257,320,292,306,271,327)(258,321,293,307,272,328)(259,322,294,308,273,329)(260,309,281,295,274,330)(261,310,282,296,275,331)(262,311,283,297,276,332)(263,312,284,298,277,333)(264,313,285,299,278,334)(265,314,286,300,279,335)(266,315,287,301,280,336), (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), (1,218,8,211)(2,217,9,224)(3,216,10,223)(4,215,11,222)(5,214,12,221)(6,213,13,220)(7,212,14,219)(15,233,22,226)(16,232,23,225)(17,231,24,238)(18,230,25,237)(19,229,26,236)(20,228,27,235)(21,227,28,234)(29,246,36,239)(30,245,37,252)(31,244,38,251)(32,243,39,250)(33,242,40,249)(34,241,41,248)(35,240,42,247)(43,179,50,172)(44,178,51,171)(45,177,52,170)(46,176,53,169)(47,175,54,182)(48,174,55,181)(49,173,56,180)(57,187,64,194)(58,186,65,193)(59,185,66,192)(60,184,67,191)(61,183,68,190)(62,196,69,189)(63,195,70,188)(71,210,78,203)(72,209,79,202)(73,208,80,201)(74,207,81,200)(75,206,82,199)(76,205,83,198)(77,204,84,197)(85,302,92,295)(86,301,93,308)(87,300,94,307)(88,299,95,306)(89,298,96,305)(90,297,97,304)(91,296,98,303)(99,316,106,309)(100,315,107,322)(101,314,108,321)(102,313,109,320)(103,312,110,319)(104,311,111,318)(105,310,112,317)(113,327,120,334)(114,326,121,333)(115,325,122,332)(116,324,123,331)(117,323,124,330)(118,336,125,329)(119,335,126,328)(127,254,134,261)(128,253,135,260)(129,266,136,259)(130,265,137,258)(131,264,138,257)(132,263,139,256)(133,262,140,255)(141,273,148,280)(142,272,149,279)(143,271,150,278)(144,270,151,277)(145,269,152,276)(146,268,153,275)(147,267,154,274)(155,289,162,282)(156,288,163,281)(157,287,164,294)(158,286,165,293)(159,285,166,292)(160,284,167,291)(161,283,168,290) );

G=PermutationGroup([[(1,92),(2,93),(3,94),(4,95),(5,96),(6,97),(7,98),(8,85),(9,86),(10,87),(11,88),(12,89),(13,90),(14,91),(15,112),(16,99),(17,100),(18,101),(19,102),(20,103),(21,104),(22,105),(23,106),(24,107),(25,108),(26,109),(27,110),(28,111),(29,124),(30,125),(31,126),(32,113),(33,114),(34,115),(35,116),(36,117),(37,118),(38,119),(39,120),(40,121),(41,122),(42,123),(43,139),(44,140),(45,127),(46,128),(47,129),(48,130),(49,131),(50,132),(51,133),(52,134),(53,135),(54,136),(55,137),(56,138),(57,143),(58,144),(59,145),(60,146),(61,147),(62,148),(63,149),(64,150),(65,151),(66,152),(67,153),(68,154),(69,141),(70,142),(71,157),(72,158),(73,159),(74,160),(75,161),(76,162),(77,163),(78,164),(79,165),(80,166),(81,167),(82,168),(83,155),(84,156),(169,260),(170,261),(171,262),(172,263),(173,264),(174,265),(175,266),(176,253),(177,254),(178,255),(179,256),(180,257),(181,258),(182,259),(183,267),(184,268),(185,269),(186,270),(187,271),(188,272),(189,273),(190,274),(191,275),(192,276),(193,277),(194,278),(195,279),(196,280),(197,288),(198,289),(199,290),(200,291),(201,292),(202,293),(203,294),(204,281),(205,282),(206,283),(207,284),(208,285),(209,286),(210,287),(211,302),(212,303),(213,304),(214,305),(215,306),(216,307),(217,308),(218,295),(219,296),(220,297),(221,298),(222,299),(223,300),(224,301),(225,309),(226,310),(227,311),(228,312),(229,313),(230,314),(231,315),(232,316),(233,317),(234,318),(235,319),(236,320),(237,321),(238,322),(239,323),(240,324),(241,325),(242,326),(243,327),(244,328),(245,329),(246,330),(247,331),(248,332),(249,333),(250,334),(251,335),(252,336)], [(1,68,29,53,23,77),(2,69,30,54,24,78),(3,70,31,55,25,79),(4,57,32,56,26,80),(5,58,33,43,27,81),(6,59,34,44,28,82),(7,60,35,45,15,83),(8,61,36,46,16,84),(9,62,37,47,17,71),(10,63,38,48,18,72),(11,64,39,49,19,73),(12,65,40,50,20,74),(13,66,41,51,21,75),(14,67,42,52,22,76),(85,147,117,128,99,156),(86,148,118,129,100,157),(87,149,119,130,101,158),(88,150,120,131,102,159),(89,151,121,132,103,160),(90,152,122,133,104,161),(91,153,123,134,105,162),(92,154,124,135,106,163),(93,141,125,136,107,164),(94,142,126,137,108,165),(95,143,113,138,109,166),(96,144,114,139,110,167),(97,145,115,140,111,168),(98,146,116,127,112,155),(169,225,204,218,190,246),(170,226,205,219,191,247),(171,227,206,220,192,248),(172,228,207,221,193,249),(173,229,208,222,194,250),(174,230,209,223,195,251),(175,231,210,224,196,252),(176,232,197,211,183,239),(177,233,198,212,184,240),(178,234,199,213,185,241),(179,235,200,214,186,242),(180,236,201,215,187,243),(181,237,202,216,188,244),(182,238,203,217,189,245),(253,316,288,302,267,323),(254,317,289,303,268,324),(255,318,290,304,269,325),(256,319,291,305,270,326),(257,320,292,306,271,327),(258,321,293,307,272,328),(259,322,294,308,273,329),(260,309,281,295,274,330),(261,310,282,296,275,331),(262,311,283,297,276,332),(263,312,284,298,277,333),(264,313,285,299,278,334),(265,314,286,300,279,335),(266,315,287,301,280,336)], [(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)], [(1,218,8,211),(2,217,9,224),(3,216,10,223),(4,215,11,222),(5,214,12,221),(6,213,13,220),(7,212,14,219),(15,233,22,226),(16,232,23,225),(17,231,24,238),(18,230,25,237),(19,229,26,236),(20,228,27,235),(21,227,28,234),(29,246,36,239),(30,245,37,252),(31,244,38,251),(32,243,39,250),(33,242,40,249),(34,241,41,248),(35,240,42,247),(43,179,50,172),(44,178,51,171),(45,177,52,170),(46,176,53,169),(47,175,54,182),(48,174,55,181),(49,173,56,180),(57,187,64,194),(58,186,65,193),(59,185,66,192),(60,184,67,191),(61,183,68,190),(62,196,69,189),(63,195,70,188),(71,210,78,203),(72,209,79,202),(73,208,80,201),(74,207,81,200),(75,206,82,199),(76,205,83,198),(77,204,84,197),(85,302,92,295),(86,301,93,308),(87,300,94,307),(88,299,95,306),(89,298,96,305),(90,297,97,304),(91,296,98,303),(99,316,106,309),(100,315,107,322),(101,314,108,321),(102,313,109,320),(103,312,110,319),(104,311,111,318),(105,310,112,317),(113,327,120,334),(114,326,121,333),(115,325,122,332),(116,324,123,331),(117,323,124,330),(118,336,125,329),(119,335,126,328),(127,254,134,261),(128,253,135,260),(129,266,136,259),(130,265,137,258),(131,264,138,257),(132,263,139,256),(133,262,140,255),(141,273,148,280),(142,272,149,279),(143,271,150,278),(144,270,151,277),(145,269,152,276),(146,268,153,275),(147,267,154,274),(155,289,162,282),(156,288,163,281),(157,287,164,294),(158,286,165,293),(159,285,166,292),(160,284,167,291),(161,283,168,290)]])

120 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4H 6A ··· 6N 7A 7B 7C 12A ··· 12P 14A ··· 14U 21A ··· 21F 42A ··· 42AP order 1 2 ··· 2 3 3 4 ··· 4 6 ··· 6 7 7 7 12 ··· 12 14 ··· 14 21 ··· 21 42 ··· 42 size 1 1 ··· 1 1 1 7 ··· 7 1 ··· 1 2 2 2 7 ··· 7 2 ··· 2 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 D7 Dic7 D14 C3×D7 C3×Dic7 C6×D7 kernel C2×C6×Dic7 C6×Dic7 C22×C42 C22×Dic7 C2×C42 C2×Dic7 C22×C14 C2×C14 C22×C6 C2×C6 C2×C6 C23 C22 C22 # reps 1 6 1 2 8 12 2 16 3 12 9 6 24 18

Matrix representation of C2×C6×Dic7 in GL4(𝔽337) generated by

 336 0 0 0 0 1 0 0 0 0 336 0 0 0 0 336
,
 209 0 0 0 0 336 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 336 0 0 1 34
,
 1 0 0 0 0 336 0 0 0 0 142 131 0 0 21 195
G:=sub<GL(4,GF(337))| [336,0,0,0,0,1,0,0,0,0,336,0,0,0,0,336],[209,0,0,0,0,336,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,336,34],[1,0,0,0,0,336,0,0,0,0,142,21,0,0,131,195] >;

C2×C6×Dic7 in GAP, Magma, Sage, TeX

C_2\times C_6\times {\rm Dic}_7
% in TeX

G:=Group("C2xC6xDic7");
// GroupNames label

G:=SmallGroup(336,182);
// by ID

G=gap.SmallGroup(336,182);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-7,144,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^14=1,d^2=c^7,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

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