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G = C7×Dic15order 420 = 22·3·5·7

Direct product of C7 and Dic15

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C7×Dic15, C153C28, C1058C4, C70.3S3, C42.3D5, C355Dic3, C213Dic5, C210.4C2, C30.1C14, C14.2D15, C6.(C7×D5), C3⋊(C7×Dic5), C10.(S3×C7), C2.(C7×D15), C52(C7×Dic3), SmallGroup(420,10)

Series: Derived Chief Lower central Upper central

C1C15 — C7×Dic15
C1C5C15C30C210 — C7×Dic15
C15 — C7×Dic15
C1C14

Generators and relations for C7×Dic15
 G = < a,b,c | a7=b30=1, c2=b15, ab=ba, ac=ca, cbc-1=b-1 >

15C4
5Dic3
3Dic5
15C28
5C7×Dic3
3C7×Dic5

Smallest permutation representation of C7×Dic15
Regular action on 420 points
Generators in S420
(1 195 172 143 112 82 31)(2 196 173 144 113 83 32)(3 197 174 145 114 84 33)(4 198 175 146 115 85 34)(5 199 176 147 116 86 35)(6 200 177 148 117 87 36)(7 201 178 149 118 88 37)(8 202 179 150 119 89 38)(9 203 180 121 120 90 39)(10 204 151 122 91 61 40)(11 205 152 123 92 62 41)(12 206 153 124 93 63 42)(13 207 154 125 94 64 43)(14 208 155 126 95 65 44)(15 209 156 127 96 66 45)(16 210 157 128 97 67 46)(17 181 158 129 98 68 47)(18 182 159 130 99 69 48)(19 183 160 131 100 70 49)(20 184 161 132 101 71 50)(21 185 162 133 102 72 51)(22 186 163 134 103 73 52)(23 187 164 135 104 74 53)(24 188 165 136 105 75 54)(25 189 166 137 106 76 55)(26 190 167 138 107 77 56)(27 191 168 139 108 78 57)(28 192 169 140 109 79 58)(29 193 170 141 110 80 59)(30 194 171 142 111 81 60)(211 401 371 336 321 286 241)(212 402 372 337 322 287 242)(213 403 373 338 323 288 243)(214 404 374 339 324 289 244)(215 405 375 340 325 290 245)(216 406 376 341 326 291 246)(217 407 377 342 327 292 247)(218 408 378 343 328 293 248)(219 409 379 344 329 294 249)(220 410 380 345 330 295 250)(221 411 381 346 301 296 251)(222 412 382 347 302 297 252)(223 413 383 348 303 298 253)(224 414 384 349 304 299 254)(225 415 385 350 305 300 255)(226 416 386 351 306 271 256)(227 417 387 352 307 272 257)(228 418 388 353 308 273 258)(229 419 389 354 309 274 259)(230 420 390 355 310 275 260)(231 391 361 356 311 276 261)(232 392 362 357 312 277 262)(233 393 363 358 313 278 263)(234 394 364 359 314 279 264)(235 395 365 360 315 280 265)(236 396 366 331 316 281 266)(237 397 367 332 317 282 267)(238 398 368 333 318 283 268)(239 399 369 334 319 284 269)(240 400 370 335 320 285 270)
(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)(331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360)(361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390)(391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420)
(1 231 16 216)(2 230 17 215)(3 229 18 214)(4 228 19 213)(5 227 20 212)(6 226 21 211)(7 225 22 240)(8 224 23 239)(9 223 24 238)(10 222 25 237)(11 221 26 236)(12 220 27 235)(13 219 28 234)(14 218 29 233)(15 217 30 232)(31 261 46 246)(32 260 47 245)(33 259 48 244)(34 258 49 243)(35 257 50 242)(36 256 51 241)(37 255 52 270)(38 254 53 269)(39 253 54 268)(40 252 55 267)(41 251 56 266)(42 250 57 265)(43 249 58 264)(44 248 59 263)(45 247 60 262)(61 297 76 282)(62 296 77 281)(63 295 78 280)(64 294 79 279)(65 293 80 278)(66 292 81 277)(67 291 82 276)(68 290 83 275)(69 289 84 274)(70 288 85 273)(71 287 86 272)(72 286 87 271)(73 285 88 300)(74 284 89 299)(75 283 90 298)(91 302 106 317)(92 301 107 316)(93 330 108 315)(94 329 109 314)(95 328 110 313)(96 327 111 312)(97 326 112 311)(98 325 113 310)(99 324 114 309)(100 323 115 308)(101 322 116 307)(102 321 117 306)(103 320 118 305)(104 319 119 304)(105 318 120 303)(121 348 136 333)(122 347 137 332)(123 346 138 331)(124 345 139 360)(125 344 140 359)(126 343 141 358)(127 342 142 357)(128 341 143 356)(129 340 144 355)(130 339 145 354)(131 338 146 353)(132 337 147 352)(133 336 148 351)(134 335 149 350)(135 334 150 349)(151 382 166 367)(152 381 167 366)(153 380 168 365)(154 379 169 364)(155 378 170 363)(156 377 171 362)(157 376 172 361)(158 375 173 390)(159 374 174 389)(160 373 175 388)(161 372 176 387)(162 371 177 386)(163 370 178 385)(164 369 179 384)(165 368 180 383)(181 405 196 420)(182 404 197 419)(183 403 198 418)(184 402 199 417)(185 401 200 416)(186 400 201 415)(187 399 202 414)(188 398 203 413)(189 397 204 412)(190 396 205 411)(191 395 206 410)(192 394 207 409)(193 393 208 408)(194 392 209 407)(195 391 210 406)

G:=sub<Sym(420)| (1,195,172,143,112,82,31)(2,196,173,144,113,83,32)(3,197,174,145,114,84,33)(4,198,175,146,115,85,34)(5,199,176,147,116,86,35)(6,200,177,148,117,87,36)(7,201,178,149,118,88,37)(8,202,179,150,119,89,38)(9,203,180,121,120,90,39)(10,204,151,122,91,61,40)(11,205,152,123,92,62,41)(12,206,153,124,93,63,42)(13,207,154,125,94,64,43)(14,208,155,126,95,65,44)(15,209,156,127,96,66,45)(16,210,157,128,97,67,46)(17,181,158,129,98,68,47)(18,182,159,130,99,69,48)(19,183,160,131,100,70,49)(20,184,161,132,101,71,50)(21,185,162,133,102,72,51)(22,186,163,134,103,73,52)(23,187,164,135,104,74,53)(24,188,165,136,105,75,54)(25,189,166,137,106,76,55)(26,190,167,138,107,77,56)(27,191,168,139,108,78,57)(28,192,169,140,109,79,58)(29,193,170,141,110,80,59)(30,194,171,142,111,81,60)(211,401,371,336,321,286,241)(212,402,372,337,322,287,242)(213,403,373,338,323,288,243)(214,404,374,339,324,289,244)(215,405,375,340,325,290,245)(216,406,376,341,326,291,246)(217,407,377,342,327,292,247)(218,408,378,343,328,293,248)(219,409,379,344,329,294,249)(220,410,380,345,330,295,250)(221,411,381,346,301,296,251)(222,412,382,347,302,297,252)(223,413,383,348,303,298,253)(224,414,384,349,304,299,254)(225,415,385,350,305,300,255)(226,416,386,351,306,271,256)(227,417,387,352,307,272,257)(228,418,388,353,308,273,258)(229,419,389,354,309,274,259)(230,420,390,355,310,275,260)(231,391,361,356,311,276,261)(232,392,362,357,312,277,262)(233,393,363,358,313,278,263)(234,394,364,359,314,279,264)(235,395,365,360,315,280,265)(236,396,366,331,316,281,266)(237,397,367,332,317,282,267)(238,398,368,333,318,283,268)(239,399,369,334,319,284,269)(240,400,370,335,320,285,270), (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)(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390)(391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420), (1,231,16,216)(2,230,17,215)(3,229,18,214)(4,228,19,213)(5,227,20,212)(6,226,21,211)(7,225,22,240)(8,224,23,239)(9,223,24,238)(10,222,25,237)(11,221,26,236)(12,220,27,235)(13,219,28,234)(14,218,29,233)(15,217,30,232)(31,261,46,246)(32,260,47,245)(33,259,48,244)(34,258,49,243)(35,257,50,242)(36,256,51,241)(37,255,52,270)(38,254,53,269)(39,253,54,268)(40,252,55,267)(41,251,56,266)(42,250,57,265)(43,249,58,264)(44,248,59,263)(45,247,60,262)(61,297,76,282)(62,296,77,281)(63,295,78,280)(64,294,79,279)(65,293,80,278)(66,292,81,277)(67,291,82,276)(68,290,83,275)(69,289,84,274)(70,288,85,273)(71,287,86,272)(72,286,87,271)(73,285,88,300)(74,284,89,299)(75,283,90,298)(91,302,106,317)(92,301,107,316)(93,330,108,315)(94,329,109,314)(95,328,110,313)(96,327,111,312)(97,326,112,311)(98,325,113,310)(99,324,114,309)(100,323,115,308)(101,322,116,307)(102,321,117,306)(103,320,118,305)(104,319,119,304)(105,318,120,303)(121,348,136,333)(122,347,137,332)(123,346,138,331)(124,345,139,360)(125,344,140,359)(126,343,141,358)(127,342,142,357)(128,341,143,356)(129,340,144,355)(130,339,145,354)(131,338,146,353)(132,337,147,352)(133,336,148,351)(134,335,149,350)(135,334,150,349)(151,382,166,367)(152,381,167,366)(153,380,168,365)(154,379,169,364)(155,378,170,363)(156,377,171,362)(157,376,172,361)(158,375,173,390)(159,374,174,389)(160,373,175,388)(161,372,176,387)(162,371,177,386)(163,370,178,385)(164,369,179,384)(165,368,180,383)(181,405,196,420)(182,404,197,419)(183,403,198,418)(184,402,199,417)(185,401,200,416)(186,400,201,415)(187,399,202,414)(188,398,203,413)(189,397,204,412)(190,396,205,411)(191,395,206,410)(192,394,207,409)(193,393,208,408)(194,392,209,407)(195,391,210,406)>;

G:=Group( (1,195,172,143,112,82,31)(2,196,173,144,113,83,32)(3,197,174,145,114,84,33)(4,198,175,146,115,85,34)(5,199,176,147,116,86,35)(6,200,177,148,117,87,36)(7,201,178,149,118,88,37)(8,202,179,150,119,89,38)(9,203,180,121,120,90,39)(10,204,151,122,91,61,40)(11,205,152,123,92,62,41)(12,206,153,124,93,63,42)(13,207,154,125,94,64,43)(14,208,155,126,95,65,44)(15,209,156,127,96,66,45)(16,210,157,128,97,67,46)(17,181,158,129,98,68,47)(18,182,159,130,99,69,48)(19,183,160,131,100,70,49)(20,184,161,132,101,71,50)(21,185,162,133,102,72,51)(22,186,163,134,103,73,52)(23,187,164,135,104,74,53)(24,188,165,136,105,75,54)(25,189,166,137,106,76,55)(26,190,167,138,107,77,56)(27,191,168,139,108,78,57)(28,192,169,140,109,79,58)(29,193,170,141,110,80,59)(30,194,171,142,111,81,60)(211,401,371,336,321,286,241)(212,402,372,337,322,287,242)(213,403,373,338,323,288,243)(214,404,374,339,324,289,244)(215,405,375,340,325,290,245)(216,406,376,341,326,291,246)(217,407,377,342,327,292,247)(218,408,378,343,328,293,248)(219,409,379,344,329,294,249)(220,410,380,345,330,295,250)(221,411,381,346,301,296,251)(222,412,382,347,302,297,252)(223,413,383,348,303,298,253)(224,414,384,349,304,299,254)(225,415,385,350,305,300,255)(226,416,386,351,306,271,256)(227,417,387,352,307,272,257)(228,418,388,353,308,273,258)(229,419,389,354,309,274,259)(230,420,390,355,310,275,260)(231,391,361,356,311,276,261)(232,392,362,357,312,277,262)(233,393,363,358,313,278,263)(234,394,364,359,314,279,264)(235,395,365,360,315,280,265)(236,396,366,331,316,281,266)(237,397,367,332,317,282,267)(238,398,368,333,318,283,268)(239,399,369,334,319,284,269)(240,400,370,335,320,285,270), (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)(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390)(391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420), (1,231,16,216)(2,230,17,215)(3,229,18,214)(4,228,19,213)(5,227,20,212)(6,226,21,211)(7,225,22,240)(8,224,23,239)(9,223,24,238)(10,222,25,237)(11,221,26,236)(12,220,27,235)(13,219,28,234)(14,218,29,233)(15,217,30,232)(31,261,46,246)(32,260,47,245)(33,259,48,244)(34,258,49,243)(35,257,50,242)(36,256,51,241)(37,255,52,270)(38,254,53,269)(39,253,54,268)(40,252,55,267)(41,251,56,266)(42,250,57,265)(43,249,58,264)(44,248,59,263)(45,247,60,262)(61,297,76,282)(62,296,77,281)(63,295,78,280)(64,294,79,279)(65,293,80,278)(66,292,81,277)(67,291,82,276)(68,290,83,275)(69,289,84,274)(70,288,85,273)(71,287,86,272)(72,286,87,271)(73,285,88,300)(74,284,89,299)(75,283,90,298)(91,302,106,317)(92,301,107,316)(93,330,108,315)(94,329,109,314)(95,328,110,313)(96,327,111,312)(97,326,112,311)(98,325,113,310)(99,324,114,309)(100,323,115,308)(101,322,116,307)(102,321,117,306)(103,320,118,305)(104,319,119,304)(105,318,120,303)(121,348,136,333)(122,347,137,332)(123,346,138,331)(124,345,139,360)(125,344,140,359)(126,343,141,358)(127,342,142,357)(128,341,143,356)(129,340,144,355)(130,339,145,354)(131,338,146,353)(132,337,147,352)(133,336,148,351)(134,335,149,350)(135,334,150,349)(151,382,166,367)(152,381,167,366)(153,380,168,365)(154,379,169,364)(155,378,170,363)(156,377,171,362)(157,376,172,361)(158,375,173,390)(159,374,174,389)(160,373,175,388)(161,372,176,387)(162,371,177,386)(163,370,178,385)(164,369,179,384)(165,368,180,383)(181,405,196,420)(182,404,197,419)(183,403,198,418)(184,402,199,417)(185,401,200,416)(186,400,201,415)(187,399,202,414)(188,398,203,413)(189,397,204,412)(190,396,205,411)(191,395,206,410)(192,394,207,409)(193,393,208,408)(194,392,209,407)(195,391,210,406) );

G=PermutationGroup([(1,195,172,143,112,82,31),(2,196,173,144,113,83,32),(3,197,174,145,114,84,33),(4,198,175,146,115,85,34),(5,199,176,147,116,86,35),(6,200,177,148,117,87,36),(7,201,178,149,118,88,37),(8,202,179,150,119,89,38),(9,203,180,121,120,90,39),(10,204,151,122,91,61,40),(11,205,152,123,92,62,41),(12,206,153,124,93,63,42),(13,207,154,125,94,64,43),(14,208,155,126,95,65,44),(15,209,156,127,96,66,45),(16,210,157,128,97,67,46),(17,181,158,129,98,68,47),(18,182,159,130,99,69,48),(19,183,160,131,100,70,49),(20,184,161,132,101,71,50),(21,185,162,133,102,72,51),(22,186,163,134,103,73,52),(23,187,164,135,104,74,53),(24,188,165,136,105,75,54),(25,189,166,137,106,76,55),(26,190,167,138,107,77,56),(27,191,168,139,108,78,57),(28,192,169,140,109,79,58),(29,193,170,141,110,80,59),(30,194,171,142,111,81,60),(211,401,371,336,321,286,241),(212,402,372,337,322,287,242),(213,403,373,338,323,288,243),(214,404,374,339,324,289,244),(215,405,375,340,325,290,245),(216,406,376,341,326,291,246),(217,407,377,342,327,292,247),(218,408,378,343,328,293,248),(219,409,379,344,329,294,249),(220,410,380,345,330,295,250),(221,411,381,346,301,296,251),(222,412,382,347,302,297,252),(223,413,383,348,303,298,253),(224,414,384,349,304,299,254),(225,415,385,350,305,300,255),(226,416,386,351,306,271,256),(227,417,387,352,307,272,257),(228,418,388,353,308,273,258),(229,419,389,354,309,274,259),(230,420,390,355,310,275,260),(231,391,361,356,311,276,261),(232,392,362,357,312,277,262),(233,393,363,358,313,278,263),(234,394,364,359,314,279,264),(235,395,365,360,315,280,265),(236,396,366,331,316,281,266),(237,397,367,332,317,282,267),(238,398,368,333,318,283,268),(239,399,369,334,319,284,269),(240,400,370,335,320,285,270)], [(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),(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360),(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390),(391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420)], [(1,231,16,216),(2,230,17,215),(3,229,18,214),(4,228,19,213),(5,227,20,212),(6,226,21,211),(7,225,22,240),(8,224,23,239),(9,223,24,238),(10,222,25,237),(11,221,26,236),(12,220,27,235),(13,219,28,234),(14,218,29,233),(15,217,30,232),(31,261,46,246),(32,260,47,245),(33,259,48,244),(34,258,49,243),(35,257,50,242),(36,256,51,241),(37,255,52,270),(38,254,53,269),(39,253,54,268),(40,252,55,267),(41,251,56,266),(42,250,57,265),(43,249,58,264),(44,248,59,263),(45,247,60,262),(61,297,76,282),(62,296,77,281),(63,295,78,280),(64,294,79,279),(65,293,80,278),(66,292,81,277),(67,291,82,276),(68,290,83,275),(69,289,84,274),(70,288,85,273),(71,287,86,272),(72,286,87,271),(73,285,88,300),(74,284,89,299),(75,283,90,298),(91,302,106,317),(92,301,107,316),(93,330,108,315),(94,329,109,314),(95,328,110,313),(96,327,111,312),(97,326,112,311),(98,325,113,310),(99,324,114,309),(100,323,115,308),(101,322,116,307),(102,321,117,306),(103,320,118,305),(104,319,119,304),(105,318,120,303),(121,348,136,333),(122,347,137,332),(123,346,138,331),(124,345,139,360),(125,344,140,359),(126,343,141,358),(127,342,142,357),(128,341,143,356),(129,340,144,355),(130,339,145,354),(131,338,146,353),(132,337,147,352),(133,336,148,351),(134,335,149,350),(135,334,150,349),(151,382,166,367),(152,381,167,366),(153,380,168,365),(154,379,169,364),(155,378,170,363),(156,377,171,362),(157,376,172,361),(158,375,173,390),(159,374,174,389),(160,373,175,388),(161,372,176,387),(162,371,177,386),(163,370,178,385),(164,369,179,384),(165,368,180,383),(181,405,196,420),(182,404,197,419),(183,403,198,418),(184,402,199,417),(185,401,200,416),(186,400,201,415),(187,399,202,414),(188,398,203,413),(189,397,204,412),(190,396,205,411),(191,395,206,410),(192,394,207,409),(193,393,208,408),(194,392,209,407),(195,391,210,406)])

126 conjugacy classes

class 1  2  3 4A4B5A5B 6 7A···7F10A10B14A···14F15A15B15C15D21A···21F28A···28L30A30B30C30D35A···35L42A···42F70A···70L105A···105X210A···210X
order123445567···7101014···141515151521···2128···283030303035···3542···4270···70105···105210···210
size11215152221···1221···122222···215···1522222···22···22···22···22···2

126 irreducible representations

dim111111222222222222
type++++--+-
imageC1C2C4C7C14C28S3D5Dic3Dic5D15S3×C7Dic15C7×D5C7×Dic3C7×Dic5C7×D15C7×Dic15
kernelC7×Dic15C210C105Dic15C30C15C70C42C35C21C14C10C7C6C5C3C2C1
# reps11266121212464126122424

Matrix representation of C7×Dic15 in GL2(𝔽29) generated by

70
07
,
110
1014
,
1725
012
G:=sub<GL(2,GF(29))| [7,0,0,7],[1,10,10,14],[17,0,25,12] >;

C7×Dic15 in GAP, Magma, Sage, TeX

C_7\times {\rm Dic}_{15}
% in TeX

G:=Group("C7xDic15");
// GroupNames label

G:=SmallGroup(420,10);
// by ID

G=gap.SmallGroup(420,10);
# by ID

G:=PCGroup([5,-2,-7,-2,-3,-5,70,1123,8404]);
// Polycyclic

G:=Group<a,b,c|a^7=b^30=1,c^2=b^15,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C7×Dic15 in TeX

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