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## G = C3×Dic7⋊C4order 336 = 24·3·7

### Direct product of C3 and Dic7⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C3×Dic7⋊C4
 Chief series C1 — C7 — C14 — C2×C14 — C2×C42 — C6×Dic7 — C3×Dic7⋊C4
 Lower central C7 — C14 — C3×Dic7⋊C4
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C3×Dic7⋊C4
G = < a,b,c,d | a3=b14=d4=1, c2=b7, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b7c >

Smallest permutation representation of C3×Dic7⋊C4
Regular action on 336 points
Generators in S336
(1 57 29)(2 58 30)(3 59 31)(4 60 32)(5 61 33)(6 62 34)(7 63 35)(8 64 36)(9 65 37)(10 66 38)(11 67 39)(12 68 40)(13 69 41)(14 70 42)(15 71 50)(16 72 51)(17 73 52)(18 74 53)(19 75 54)(20 76 55)(21 77 56)(22 78 43)(23 79 44)(24 80 45)(25 81 46)(26 82 47)(27 83 48)(28 84 49)(85 148 113)(86 149 114)(87 150 115)(88 151 116)(89 152 117)(90 153 118)(91 154 119)(92 141 120)(93 142 121)(94 143 122)(95 144 123)(96 145 124)(97 146 125)(98 147 126)(99 155 134)(100 156 135)(101 157 136)(102 158 137)(103 159 138)(104 160 139)(105 161 140)(106 162 127)(107 163 128)(108 164 129)(109 165 130)(110 166 131)(111 167 132)(112 168 133)(169 235 202)(170 236 203)(171 237 204)(172 238 205)(173 225 206)(174 226 207)(175 227 208)(176 228 209)(177 229 210)(178 230 197)(179 231 198)(180 232 199)(181 233 200)(182 234 201)(183 242 211)(184 243 212)(185 244 213)(186 245 214)(187 246 215)(188 247 216)(189 248 217)(190 249 218)(191 250 219)(192 251 220)(193 252 221)(194 239 222)(195 240 223)(196 241 224)(253 314 286)(254 315 287)(255 316 288)(256 317 289)(257 318 290)(258 319 291)(259 320 292)(260 321 293)(261 322 294)(262 309 281)(263 310 282)(264 311 283)(265 312 284)(266 313 285)(267 328 303)(268 329 304)(269 330 305)(270 331 306)(271 332 307)(272 333 308)(273 334 295)(274 335 296)(275 336 297)(276 323 298)(277 324 299)(278 325 300)(279 326 301)(280 327 302)
(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)
(1 185 8 192)(2 184 9 191)(3 183 10 190)(4 196 11 189)(5 195 12 188)(6 194 13 187)(7 193 14 186)(15 170 22 177)(16 169 23 176)(17 182 24 175)(18 181 25 174)(19 180 26 173)(20 179 27 172)(21 178 28 171)(29 213 36 220)(30 212 37 219)(31 211 38 218)(32 224 39 217)(33 223 40 216)(34 222 41 215)(35 221 42 214)(43 210 50 203)(44 209 51 202)(45 208 52 201)(46 207 53 200)(47 206 54 199)(48 205 55 198)(49 204 56 197)(57 244 64 251)(58 243 65 250)(59 242 66 249)(60 241 67 248)(61 240 68 247)(62 239 69 246)(63 252 70 245)(71 236 78 229)(72 235 79 228)(73 234 80 227)(74 233 81 226)(75 232 82 225)(76 231 83 238)(77 230 84 237)(85 278 92 271)(86 277 93 270)(87 276 94 269)(88 275 95 268)(89 274 96 267)(90 273 97 280)(91 272 98 279)(99 266 106 259)(100 265 107 258)(101 264 108 257)(102 263 109 256)(103 262 110 255)(104 261 111 254)(105 260 112 253)(113 300 120 307)(114 299 121 306)(115 298 122 305)(116 297 123 304)(117 296 124 303)(118 295 125 302)(119 308 126 301)(127 292 134 285)(128 291 135 284)(129 290 136 283)(130 289 137 282)(131 288 138 281)(132 287 139 294)(133 286 140 293)(141 332 148 325)(142 331 149 324)(143 330 150 323)(144 329 151 336)(145 328 152 335)(146 327 153 334)(147 326 154 333)(155 313 162 320)(156 312 163 319)(157 311 164 318)(158 310 165 317)(159 309 166 316)(160 322 167 315)(161 321 168 314)
(1 106 15 85)(2 107 16 86)(3 108 17 87)(4 109 18 88)(5 110 19 89)(6 111 20 90)(7 112 21 91)(8 99 22 92)(9 100 23 93)(10 101 24 94)(11 102 25 95)(12 103 26 96)(13 104 27 97)(14 105 28 98)(29 127 50 113)(30 128 51 114)(31 129 52 115)(32 130 53 116)(33 131 54 117)(34 132 55 118)(35 133 56 119)(36 134 43 120)(37 135 44 121)(38 136 45 122)(39 137 46 123)(40 138 47 124)(41 139 48 125)(42 140 49 126)(57 162 71 148)(58 163 72 149)(59 164 73 150)(60 165 74 151)(61 166 75 152)(62 167 76 153)(63 168 77 154)(64 155 78 141)(65 156 79 142)(66 157 80 143)(67 158 81 144)(68 159 82 145)(69 160 83 146)(70 161 84 147)(169 270 184 265)(170 271 185 266)(171 272 186 253)(172 273 187 254)(173 274 188 255)(174 275 189 256)(175 276 190 257)(176 277 191 258)(177 278 192 259)(178 279 193 260)(179 280 194 261)(180 267 195 262)(181 268 196 263)(182 269 183 264)(197 301 221 293)(198 302 222 294)(199 303 223 281)(200 304 224 282)(201 305 211 283)(202 306 212 284)(203 307 213 285)(204 308 214 286)(205 295 215 287)(206 296 216 288)(207 297 217 289)(208 298 218 290)(209 299 219 291)(210 300 220 292)(225 335 247 316)(226 336 248 317)(227 323 249 318)(228 324 250 319)(229 325 251 320)(230 326 252 321)(231 327 239 322)(232 328 240 309)(233 329 241 310)(234 330 242 311)(235 331 243 312)(236 332 244 313)(237 333 245 314)(238 334 246 315)

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

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

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

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 D14 C3×D7 Dic14 C4×D7 C7⋊D4 C6×D7 C3×Dic14 C12×D7 C3×C7⋊D4 kernel C3×Dic7⋊C4 C6×Dic7 C2×C84 Dic7⋊C4 C3×Dic7 C2×Dic7 C2×C28 Dic7 C42 C42 C2×C12 C14 C14 C2×C6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 2 1 2 4 4 2 8 1 1 3 2 2 3 6 6 6 6 6 12 12 12

Matrix representation of C3×Dic7⋊C4 in GL6(𝔽337)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 128 0 0 0 0 0 0 128 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 144 336 0 0 0 0 145 336 0 0 0 0 0 0 143 336 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 336 110
,
 34 194 0 0 0 0 34 303 0 0 0 0 0 0 20 89 0 0 0 0 253 317 0 0 0 0 0 0 185 327 0 0 0 0 120 152
,
 189 0 0 0 0 0 0 189 0 0 0 0 0 0 336 0 0 0 0 0 0 336 0 0 0 0 0 0 243 112 0 0 0 0 225 94

G:=sub<GL(6,GF(337))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,128,0,0,0,0,0,0,128,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[144,145,0,0,0,0,336,336,0,0,0,0,0,0,143,1,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,1,110],[34,34,0,0,0,0,194,303,0,0,0,0,0,0,20,253,0,0,0,0,89,317,0,0,0,0,0,0,185,120,0,0,0,0,327,152],[189,0,0,0,0,0,0,189,0,0,0,0,0,0,336,0,0,0,0,0,0,336,0,0,0,0,0,0,243,225,0,0,0,0,112,94] >;

C3×Dic7⋊C4 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_7\rtimes C_4
% in TeX

G:=Group("C3xDic7:C4");
// GroupNames label

G:=SmallGroup(336,66);
// by ID

G=gap.SmallGroup(336,66);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,144,313,79,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^14=d^4=1,c^2=b^7,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^7*c>;
// generators/relations

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