Copied to
clipboard

G = C15×Dic7order 420 = 22·3·5·7

Direct product of C15 and Dic7

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

Aliases: C15×Dic7, C73C60, C212C20, C1059C4, C3510C12, C70.6C6, C30.4D7, C42.2C10, C210.5C2, C14.3C30, C2.(D7×C15), C6.2(C5×D7), C10.2(C3×D7), SmallGroup(420,5)

Series: Derived Chief Lower central Upper central

C1C7 — C15×Dic7
C1C7C14C70C210 — C15×Dic7
C7 — C15×Dic7
C1C30

Generators and relations for C15×Dic7
 G = < a,b,c | a15=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >

7C4
7C12
7C20
7C60

Smallest permutation representation of C15×Dic7
Regular action on 420 points
Generators in S420
(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)
(1 223 125 333 255 152 55 256 148 346 21 284 301 80)(2 224 126 334 241 153 56 257 149 347 22 285 302 81)(3 225 127 335 242 154 57 258 150 348 23 271 303 82)(4 211 128 336 243 155 58 259 136 349 24 272 304 83)(5 212 129 337 244 156 59 260 137 350 25 273 305 84)(6 213 130 338 245 157 60 261 138 351 26 274 306 85)(7 214 131 339 246 158 46 262 139 352 27 275 307 86)(8 215 132 340 247 159 47 263 140 353 28 276 308 87)(9 216 133 341 248 160 48 264 141 354 29 277 309 88)(10 217 134 342 249 161 49 265 142 355 30 278 310 89)(11 218 135 343 250 162 50 266 143 356 16 279 311 90)(12 219 121 344 251 163 51 267 144 357 17 280 312 76)(13 220 122 345 252 164 52 268 145 358 18 281 313 77)(14 221 123 331 253 165 53 269 146 359 19 282 314 78)(15 222 124 332 254 151 54 270 147 360 20 283 315 79)(31 234 175 186 370 117 393 413 297 61 390 100 209 324)(32 235 176 187 371 118 394 414 298 62 376 101 210 325)(33 236 177 188 372 119 395 415 299 63 377 102 196 326)(34 237 178 189 373 120 396 416 300 64 378 103 197 327)(35 238 179 190 374 106 397 417 286 65 379 104 198 328)(36 239 180 191 375 107 398 418 287 66 380 105 199 329)(37 240 166 192 361 108 399 419 288 67 381 91 200 330)(38 226 167 193 362 109 400 420 289 68 382 92 201 316)(39 227 168 194 363 110 401 406 290 69 383 93 202 317)(40 228 169 195 364 111 402 407 291 70 384 94 203 318)(41 229 170 181 365 112 403 408 292 71 385 95 204 319)(42 230 171 182 366 113 404 409 293 72 386 96 205 320)(43 231 172 183 367 114 405 410 294 73 387 97 206 321)(44 232 173 184 368 115 391 411 295 74 388 98 207 322)(45 233 174 185 369 116 392 412 296 75 389 99 208 323)
(1 120 256 197)(2 106 257 198)(3 107 258 199)(4 108 259 200)(5 109 260 201)(6 110 261 202)(7 111 262 203)(8 112 263 204)(9 113 264 205)(10 114 265 206)(11 115 266 207)(12 116 267 208)(13 117 268 209)(14 118 269 210)(15 119 270 196)(16 74 343 173)(17 75 344 174)(18 61 345 175)(19 62 331 176)(20 63 332 177)(21 64 333 178)(22 65 334 179)(23 66 335 180)(24 67 336 166)(25 68 337 167)(26 69 338 168)(27 70 339 169)(28 71 340 170)(29 72 341 171)(30 73 342 172)(31 313 413 164)(32 314 414 165)(33 315 415 151)(34 301 416 152)(35 302 417 153)(36 303 418 154)(37 304 419 155)(38 305 420 156)(39 306 406 157)(40 307 407 158)(41 308 408 159)(42 309 409 160)(43 310 410 161)(44 311 411 162)(45 312 412 163)(46 318 86 402)(47 319 87 403)(48 320 88 404)(49 321 89 405)(50 322 90 391)(51 323 76 392)(52 324 77 393)(53 325 78 394)(54 326 79 395)(55 327 80 396)(56 328 81 397)(57 329 82 398)(58 330 83 399)(59 316 84 400)(60 317 85 401)(91 211 361 136)(92 212 362 137)(93 213 363 138)(94 214 364 139)(95 215 365 140)(96 216 366 141)(97 217 367 142)(98 218 368 143)(99 219 369 144)(100 220 370 145)(101 221 371 146)(102 222 372 147)(103 223 373 148)(104 224 374 149)(105 225 375 150)(121 185 357 389)(122 186 358 390)(123 187 359 376)(124 188 360 377)(125 189 346 378)(126 190 347 379)(127 191 348 380)(128 192 349 381)(129 193 350 382)(130 194 351 383)(131 195 352 384)(132 181 353 385)(133 182 354 386)(134 183 355 387)(135 184 356 388)(226 273 289 244)(227 274 290 245)(228 275 291 246)(229 276 292 247)(230 277 293 248)(231 278 294 249)(232 279 295 250)(233 280 296 251)(234 281 297 252)(235 282 298 253)(236 283 299 254)(237 284 300 255)(238 285 286 241)(239 271 287 242)(240 272 288 243)

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

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

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

150 conjugacy classes

class 1  2 3A3B4A4B5A5B5C5D6A6B7A7B7C10A10B10C10D12A12B12C12D14A14B14C15A···15H20A···20H21A···21F30A···30H35A···35L42A···42F60A···60P70A···70L105A···105X210A···210X
order123344555566777101010101212121214141415···1520···2021···2130···3035···3542···4260···6070···70105···105210···210
size111177111111222111177772221···17···72···21···12···22···27···72···22···22···2

150 irreducible representations

dim11111111111122222222
type+++-
imageC1C2C3C4C5C6C10C12C15C20C30C60D7Dic7C3×D7C5×D7C3×Dic7C5×Dic7D7×C15C15×Dic7
kernelC15×Dic7C210C5×Dic7C105C3×Dic7C70C42C35Dic7C21C14C7C30C15C10C6C5C3C2C1
# reps1122424488816336126122424

Matrix representation of C15×Dic7 in GL3(𝔽421) generated by

40000
02520
00252
,
42000
001
0420403
,
39200
037656
02445
G:=sub<GL(3,GF(421))| [400,0,0,0,252,0,0,0,252],[420,0,0,0,0,420,0,1,403],[392,0,0,0,376,24,0,56,45] >;

C15×Dic7 in GAP, Magma, Sage, TeX

C_{15}\times {\rm Dic}_7
% in TeX

G:=Group("C15xDic7");
// GroupNames label

G:=SmallGroup(420,5);
// by ID

G=gap.SmallGroup(420,5);
# by ID

G:=PCGroup([5,-2,-3,-5,-2,-7,150,9004]);
// Polycyclic

G:=Group<a,b,c|a^15=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C15×Dic7 in TeX

׿
×
𝔽