direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×Dic15, C15⋊3C28, C105⋊8C4, C70.3S3, C42.3D5, C35⋊5Dic3, C21⋊3Dic5, C210.4C2, C30.1C14, C14.2D15, C6.(C7×D5), C3⋊(C7×Dic5), C10.(S3×C7), C2.(C7×D15), C5⋊2(C7×Dic3), SmallGroup(420,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C7×Dic15 |
Generators and relations for C7×Dic15
G = < a,b,c | a7=b30=1, c2=b15, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C7×Dic15
►Regular action on 420 points
Generators in S420
(1 188 172 135 119 75 46)(2 189 173 136 120 76 47)(3 190 174 137 91 77 48)(4 191 175 138 92 78 49)(5 192 176 139 93 79 50)(6 193 177 140 94 80 51)(7 194 178 141 95 81 52)(8 195 179 142 96 82 53)(9 196 180 143 97 83 54)(10 197 151 144 98 84 55)(11 198 152 145 99 85 56)(12 199 153 146 100 86 57)(13 200 154 147 101 87 58)(14 201 155 148 102 88 59)(15 202 156 149 103 89 60)(16 203 157 150 104 90 31)(17 204 158 121 105 61 32)(18 205 159 122 106 62 33)(19 206 160 123 107 63 34)(20 207 161 124 108 64 35)(21 208 162 125 109 65 36)(22 209 163 126 110 66 37)(23 210 164 127 111 67 38)(24 181 165 128 112 68 39)(25 182 166 129 113 69 40)(26 183 167 130 114 70 41)(27 184 168 131 115 71 42)(28 185 169 132 116 72 43)(29 186 170 133 117 73 44)(30 187 171 134 118 74 45)(211 401 381 336 326 281 241)(212 402 382 337 327 282 242)(213 403 383 338 328 283 243)(214 404 384 339 329 284 244)(215 405 385 340 330 285 245)(216 406 386 341 301 286 246)(217 407 387 342 302 287 247)(218 408 388 343 303 288 248)(219 409 389 344 304 289 249)(220 410 390 345 305 290 250)(221 411 361 346 306 291 251)(222 412 362 347 307 292 252)(223 413 363 348 308 293 253)(224 414 364 349 309 294 254)(225 415 365 350 310 295 255)(226 416 366 351 311 296 256)(227 417 367 352 312 297 257)(228 418 368 353 313 298 258)(229 419 369 354 314 299 259)(230 420 370 355 315 300 260)(231 391 371 356 316 271 261)(232 392 372 357 317 272 262)(233 393 373 358 318 273 263)(234 394 374 359 319 274 264)(235 395 375 360 320 275 265)(236 396 376 331 321 276 266)(237 397 377 332 322 277 267)(238 398 378 333 323 278 268)(239 399 379 334 324 279 269)(240 400 380 335 325 280 270)
(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)
(1 211 16 226)(2 240 17 225)(3 239 18 224)(4 238 19 223)(5 237 20 222)(6 236 21 221)(7 235 22 220)(8 234 23 219)(9 233 24 218)(10 232 25 217)(11 231 26 216)(12 230 27 215)(13 229 28 214)(14 228 29 213)(15 227 30 212)(31 256 46 241)(32 255 47 270)(33 254 48 269)(34 253 49 268)(35 252 50 267)(36 251 51 266)(37 250 52 265)(38 249 53 264)(39 248 54 263)(40 247 55 262)(41 246 56 261)(42 245 57 260)(43 244 58 259)(44 243 59 258)(45 242 60 257)(61 295 76 280)(62 294 77 279)(63 293 78 278)(64 292 79 277)(65 291 80 276)(66 290 81 275)(67 289 82 274)(68 288 83 273)(69 287 84 272)(70 286 85 271)(71 285 86 300)(72 284 87 299)(73 283 88 298)(74 282 89 297)(75 281 90 296)(91 324 106 309)(92 323 107 308)(93 322 108 307)(94 321 109 306)(95 320 110 305)(96 319 111 304)(97 318 112 303)(98 317 113 302)(99 316 114 301)(100 315 115 330)(101 314 116 329)(102 313 117 328)(103 312 118 327)(104 311 119 326)(105 310 120 325)(121 350 136 335)(122 349 137 334)(123 348 138 333)(124 347 139 332)(125 346 140 331)(126 345 141 360)(127 344 142 359)(128 343 143 358)(129 342 144 357)(130 341 145 356)(131 340 146 355)(132 339 147 354)(133 338 148 353)(134 337 149 352)(135 336 150 351)(151 372 166 387)(152 371 167 386)(153 370 168 385)(154 369 169 384)(155 368 170 383)(156 367 171 382)(157 366 172 381)(158 365 173 380)(159 364 174 379)(160 363 175 378)(161 362 176 377)(162 361 177 376)(163 390 178 375)(164 389 179 374)(165 388 180 373)(181 408 196 393)(182 407 197 392)(183 406 198 391)(184 405 199 420)(185 404 200 419)(186 403 201 418)(187 402 202 417)(188 401 203 416)(189 400 204 415)(190 399 205 414)(191 398 206 413)(192 397 207 412)(193 396 208 411)(194 395 209 410)(195 394 210 409)
G:=sub<Sym(420)| (1,188,172,135,119,75,46)(2,189,173,136,120,76,47)(3,190,174,137,91,77,48)(4,191,175,138,92,78,49)(5,192,176,139,93,79,50)(6,193,177,140,94,80,51)(7,194,178,141,95,81,52)(8,195,179,142,96,82,53)(9,196,180,143,97,83,54)(10,197,151,144,98,84,55)(11,198,152,145,99,85,56)(12,199,153,146,100,86,57)(13,200,154,147,101,87,58)(14,201,155,148,102,88,59)(15,202,156,149,103,89,60)(16,203,157,150,104,90,31)(17,204,158,121,105,61,32)(18,205,159,122,106,62,33)(19,206,160,123,107,63,34)(20,207,161,124,108,64,35)(21,208,162,125,109,65,36)(22,209,163,126,110,66,37)(23,210,164,127,111,67,38)(24,181,165,128,112,68,39)(25,182,166,129,113,69,40)(26,183,167,130,114,70,41)(27,184,168,131,115,71,42)(28,185,169,132,116,72,43)(29,186,170,133,117,73,44)(30,187,171,134,118,74,45)(211,401,381,336,326,281,241)(212,402,382,337,327,282,242)(213,403,383,338,328,283,243)(214,404,384,339,329,284,244)(215,405,385,340,330,285,245)(216,406,386,341,301,286,246)(217,407,387,342,302,287,247)(218,408,388,343,303,288,248)(219,409,389,344,304,289,249)(220,410,390,345,305,290,250)(221,411,361,346,306,291,251)(222,412,362,347,307,292,252)(223,413,363,348,308,293,253)(224,414,364,349,309,294,254)(225,415,365,350,310,295,255)(226,416,366,351,311,296,256)(227,417,367,352,312,297,257)(228,418,368,353,313,298,258)(229,419,369,354,314,299,259)(230,420,370,355,315,300,260)(231,391,371,356,316,271,261)(232,392,372,357,317,272,262)(233,393,373,358,318,273,263)(234,394,374,359,319,274,264)(235,395,375,360,320,275,265)(236,396,376,331,321,276,266)(237,397,377,332,322,277,267)(238,398,378,333,323,278,268)(239,399,379,334,324,279,269)(240,400,380,335,325,280,270), (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), (1,211,16,226)(2,240,17,225)(3,239,18,224)(4,238,19,223)(5,237,20,222)(6,236,21,221)(7,235,22,220)(8,234,23,219)(9,233,24,218)(10,232,25,217)(11,231,26,216)(12,230,27,215)(13,229,28,214)(14,228,29,213)(15,227,30,212)(31,256,46,241)(32,255,47,270)(33,254,48,269)(34,253,49,268)(35,252,50,267)(36,251,51,266)(37,250,52,265)(38,249,53,264)(39,248,54,263)(40,247,55,262)(41,246,56,261)(42,245,57,260)(43,244,58,259)(44,243,59,258)(45,242,60,257)(61,295,76,280)(62,294,77,279)(63,293,78,278)(64,292,79,277)(65,291,80,276)(66,290,81,275)(67,289,82,274)(68,288,83,273)(69,287,84,272)(70,286,85,271)(71,285,86,300)(72,284,87,299)(73,283,88,298)(74,282,89,297)(75,281,90,296)(91,324,106,309)(92,323,107,308)(93,322,108,307)(94,321,109,306)(95,320,110,305)(96,319,111,304)(97,318,112,303)(98,317,113,302)(99,316,114,301)(100,315,115,330)(101,314,116,329)(102,313,117,328)(103,312,118,327)(104,311,119,326)(105,310,120,325)(121,350,136,335)(122,349,137,334)(123,348,138,333)(124,347,139,332)(125,346,140,331)(126,345,141,360)(127,344,142,359)(128,343,143,358)(129,342,144,357)(130,341,145,356)(131,340,146,355)(132,339,147,354)(133,338,148,353)(134,337,149,352)(135,336,150,351)(151,372,166,387)(152,371,167,386)(153,370,168,385)(154,369,169,384)(155,368,170,383)(156,367,171,382)(157,366,172,381)(158,365,173,380)(159,364,174,379)(160,363,175,378)(161,362,176,377)(162,361,177,376)(163,390,178,375)(164,389,179,374)(165,388,180,373)(181,408,196,393)(182,407,197,392)(183,406,198,391)(184,405,199,420)(185,404,200,419)(186,403,201,418)(187,402,202,417)(188,401,203,416)(189,400,204,415)(190,399,205,414)(191,398,206,413)(192,397,207,412)(193,396,208,411)(194,395,209,410)(195,394,210,409)>;
G:=Group( (1,188,172,135,119,75,46)(2,189,173,136,120,76,47)(3,190,174,137,91,77,48)(4,191,175,138,92,78,49)(5,192,176,139,93,79,50)(6,193,177,140,94,80,51)(7,194,178,141,95,81,52)(8,195,179,142,96,82,53)(9,196,180,143,97,83,54)(10,197,151,144,98,84,55)(11,198,152,145,99,85,56)(12,199,153,146,100,86,57)(13,200,154,147,101,87,58)(14,201,155,148,102,88,59)(15,202,156,149,103,89,60)(16,203,157,150,104,90,31)(17,204,158,121,105,61,32)(18,205,159,122,106,62,33)(19,206,160,123,107,63,34)(20,207,161,124,108,64,35)(21,208,162,125,109,65,36)(22,209,163,126,110,66,37)(23,210,164,127,111,67,38)(24,181,165,128,112,68,39)(25,182,166,129,113,69,40)(26,183,167,130,114,70,41)(27,184,168,131,115,71,42)(28,185,169,132,116,72,43)(29,186,170,133,117,73,44)(30,187,171,134,118,74,45)(211,401,381,336,326,281,241)(212,402,382,337,327,282,242)(213,403,383,338,328,283,243)(214,404,384,339,329,284,244)(215,405,385,340,330,285,245)(216,406,386,341,301,286,246)(217,407,387,342,302,287,247)(218,408,388,343,303,288,248)(219,409,389,344,304,289,249)(220,410,390,345,305,290,250)(221,411,361,346,306,291,251)(222,412,362,347,307,292,252)(223,413,363,348,308,293,253)(224,414,364,349,309,294,254)(225,415,365,350,310,295,255)(226,416,366,351,311,296,256)(227,417,367,352,312,297,257)(228,418,368,353,313,298,258)(229,419,369,354,314,299,259)(230,420,370,355,315,300,260)(231,391,371,356,316,271,261)(232,392,372,357,317,272,262)(233,393,373,358,318,273,263)(234,394,374,359,319,274,264)(235,395,375,360,320,275,265)(236,396,376,331,321,276,266)(237,397,377,332,322,277,267)(238,398,378,333,323,278,268)(239,399,379,334,324,279,269)(240,400,380,335,325,280,270), (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), (1,211,16,226)(2,240,17,225)(3,239,18,224)(4,238,19,223)(5,237,20,222)(6,236,21,221)(7,235,22,220)(8,234,23,219)(9,233,24,218)(10,232,25,217)(11,231,26,216)(12,230,27,215)(13,229,28,214)(14,228,29,213)(15,227,30,212)(31,256,46,241)(32,255,47,270)(33,254,48,269)(34,253,49,268)(35,252,50,267)(36,251,51,266)(37,250,52,265)(38,249,53,264)(39,248,54,263)(40,247,55,262)(41,246,56,261)(42,245,57,260)(43,244,58,259)(44,243,59,258)(45,242,60,257)(61,295,76,280)(62,294,77,279)(63,293,78,278)(64,292,79,277)(65,291,80,276)(66,290,81,275)(67,289,82,274)(68,288,83,273)(69,287,84,272)(70,286,85,271)(71,285,86,300)(72,284,87,299)(73,283,88,298)(74,282,89,297)(75,281,90,296)(91,324,106,309)(92,323,107,308)(93,322,108,307)(94,321,109,306)(95,320,110,305)(96,319,111,304)(97,318,112,303)(98,317,113,302)(99,316,114,301)(100,315,115,330)(101,314,116,329)(102,313,117,328)(103,312,118,327)(104,311,119,326)(105,310,120,325)(121,350,136,335)(122,349,137,334)(123,348,138,333)(124,347,139,332)(125,346,140,331)(126,345,141,360)(127,344,142,359)(128,343,143,358)(129,342,144,357)(130,341,145,356)(131,340,146,355)(132,339,147,354)(133,338,148,353)(134,337,149,352)(135,336,150,351)(151,372,166,387)(152,371,167,386)(153,370,168,385)(154,369,169,384)(155,368,170,383)(156,367,171,382)(157,366,172,381)(158,365,173,380)(159,364,174,379)(160,363,175,378)(161,362,176,377)(162,361,177,376)(163,390,178,375)(164,389,179,374)(165,388,180,373)(181,408,196,393)(182,407,197,392)(183,406,198,391)(184,405,199,420)(185,404,200,419)(186,403,201,418)(187,402,202,417)(188,401,203,416)(189,400,204,415)(190,399,205,414)(191,398,206,413)(192,397,207,412)(193,396,208,411)(194,395,209,410)(195,394,210,409) );
G=PermutationGroup([[(1,188,172,135,119,75,46),(2,189,173,136,120,76,47),(3,190,174,137,91,77,48),(4,191,175,138,92,78,49),(5,192,176,139,93,79,50),(6,193,177,140,94,80,51),(7,194,178,141,95,81,52),(8,195,179,142,96,82,53),(9,196,180,143,97,83,54),(10,197,151,144,98,84,55),(11,198,152,145,99,85,56),(12,199,153,146,100,86,57),(13,200,154,147,101,87,58),(14,201,155,148,102,88,59),(15,202,156,149,103,89,60),(16,203,157,150,104,90,31),(17,204,158,121,105,61,32),(18,205,159,122,106,62,33),(19,206,160,123,107,63,34),(20,207,161,124,108,64,35),(21,208,162,125,109,65,36),(22,209,163,126,110,66,37),(23,210,164,127,111,67,38),(24,181,165,128,112,68,39),(25,182,166,129,113,69,40),(26,183,167,130,114,70,41),(27,184,168,131,115,71,42),(28,185,169,132,116,72,43),(29,186,170,133,117,73,44),(30,187,171,134,118,74,45),(211,401,381,336,326,281,241),(212,402,382,337,327,282,242),(213,403,383,338,328,283,243),(214,404,384,339,329,284,244),(215,405,385,340,330,285,245),(216,406,386,341,301,286,246),(217,407,387,342,302,287,247),(218,408,388,343,303,288,248),(219,409,389,344,304,289,249),(220,410,390,345,305,290,250),(221,411,361,346,306,291,251),(222,412,362,347,307,292,252),(223,413,363,348,308,293,253),(224,414,364,349,309,294,254),(225,415,365,350,310,295,255),(226,416,366,351,311,296,256),(227,417,367,352,312,297,257),(228,418,368,353,313,298,258),(229,419,369,354,314,299,259),(230,420,370,355,315,300,260),(231,391,371,356,316,271,261),(232,392,372,357,317,272,262),(233,393,373,358,318,273,263),(234,394,374,359,319,274,264),(235,395,375,360,320,275,265),(236,396,376,331,321,276,266),(237,397,377,332,322,277,267),(238,398,378,333,323,278,268),(239,399,379,334,324,279,269),(240,400,380,335,325,280,270)], [(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)], [(1,211,16,226),(2,240,17,225),(3,239,18,224),(4,238,19,223),(5,237,20,222),(6,236,21,221),(7,235,22,220),(8,234,23,219),(9,233,24,218),(10,232,25,217),(11,231,26,216),(12,230,27,215),(13,229,28,214),(14,228,29,213),(15,227,30,212),(31,256,46,241),(32,255,47,270),(33,254,48,269),(34,253,49,268),(35,252,50,267),(36,251,51,266),(37,250,52,265),(38,249,53,264),(39,248,54,263),(40,247,55,262),(41,246,56,261),(42,245,57,260),(43,244,58,259),(44,243,59,258),(45,242,60,257),(61,295,76,280),(62,294,77,279),(63,293,78,278),(64,292,79,277),(65,291,80,276),(66,290,81,275),(67,289,82,274),(68,288,83,273),(69,287,84,272),(70,286,85,271),(71,285,86,300),(72,284,87,299),(73,283,88,298),(74,282,89,297),(75,281,90,296),(91,324,106,309),(92,323,107,308),(93,322,108,307),(94,321,109,306),(95,320,110,305),(96,319,111,304),(97,318,112,303),(98,317,113,302),(99,316,114,301),(100,315,115,330),(101,314,116,329),(102,313,117,328),(103,312,118,327),(104,311,119,326),(105,310,120,325),(121,350,136,335),(122,349,137,334),(123,348,138,333),(124,347,139,332),(125,346,140,331),(126,345,141,360),(127,344,142,359),(128,343,143,358),(129,342,144,357),(130,341,145,356),(131,340,146,355),(132,339,147,354),(133,338,148,353),(134,337,149,352),(135,336,150,351),(151,372,166,387),(152,371,167,386),(153,370,168,385),(154,369,169,384),(155,368,170,383),(156,367,171,382),(157,366,172,381),(158,365,173,380),(159,364,174,379),(160,363,175,378),(161,362,176,377),(162,361,177,376),(163,390,178,375),(164,389,179,374),(165,388,180,373),(181,408,196,393),(182,407,197,392),(183,406,198,391),(184,405,199,420),(185,404,200,419),(186,403,201,418),(187,402,202,417),(188,401,203,416),(189,400,204,415),(190,399,205,414),(191,398,206,413),(192,397,207,412),(193,396,208,411),(194,395,209,410),(195,394,210,409)]])
126 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 6 | 7A | ··· | 7F | 10A | 10B | 14A | ··· | 14F | 15A | 15B | 15C | 15D | 21A | ··· | 21F | 28A | ··· | 28L | 30A | 30B | 30C | 30D | 35A | ··· | 35L | 42A | ··· | 42F | 70A | ··· | 70L | 105A | ··· | 105X | 210A | ··· | 210X |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 7 | ··· | 7 | 10 | 10 | 14 | ··· | 14 | 15 | 15 | 15 | 15 | 21 | ··· | 21 | 28 | ··· | 28 | 30 | 30 | 30 | 30 | 35 | ··· | 35 | 42 | ··· | 42 | 70 | ··· | 70 | 105 | ··· | 105 | 210 | ··· | 210 |
| size | 1 | 1 | 2 | 15 | 15 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 15 | ··· | 15 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | - | ||||||||||
| image | C1 | C2 | C4 | C7 | C14 | C28 | S3 | D5 | Dic3 | Dic5 | D15 | S3×C7 | Dic15 | C7×D5 | C7×Dic3 | C7×Dic5 | C7×D15 | C7×Dic15 |
| kernel | C7×Dic15 | C210 | C105 | Dic15 | C30 | C15 | C70 | C42 | C35 | C21 | C14 | C10 | C7 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 1 | 2 | 1 | 2 | 4 | 6 | 4 | 12 | 6 | 12 | 24 | 24 |
Matrix representation of C7×Dic15 ►in GL2(𝔽29) generated by
| 7 | 0 |
| 0 | 7 |
| 1 | 10 |
| 10 | 14 |
| 17 | 25 |
| 0 | 12 |
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