direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C15×Dic7, C7⋊3C60, C21⋊2C20, C105⋊9C4, C35⋊10C12, 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
| C7 — C15×Dic7 |
Generators and relations for C15×Dic7
G = < a,b,c | a15=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >
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 33 302 141 77 209 55 247 263 384 163 192 344 74)(2 34 303 142 78 210 56 248 264 385 164 193 345 75)(3 35 304 143 79 196 57 249 265 386 165 194 331 61)(4 36 305 144 80 197 58 250 266 387 151 195 332 62)(5 37 306 145 81 198 59 251 267 388 152 181 333 63)(6 38 307 146 82 199 60 252 268 389 153 182 334 64)(7 39 308 147 83 200 46 253 269 390 154 183 335 65)(8 40 309 148 84 201 47 254 270 376 155 184 336 66)(9 41 310 149 85 202 48 255 256 377 156 185 337 67)(10 42 311 150 86 203 49 241 257 378 157 186 338 68)(11 43 312 136 87 204 50 242 258 379 158 187 339 69)(12 44 313 137 88 205 51 243 259 380 159 188 340 70)(13 45 314 138 89 206 52 244 260 381 160 189 341 71)(14 31 315 139 90 207 53 245 261 382 161 190 342 72)(15 32 301 140 76 208 54 246 262 383 162 191 343 73)(16 410 397 291 218 327 168 118 93 240 372 285 360 132)(17 411 398 292 219 328 169 119 94 226 373 271 346 133)(18 412 399 293 220 329 170 120 95 227 374 272 347 134)(19 413 400 294 221 330 171 106 96 228 375 273 348 135)(20 414 401 295 222 316 172 107 97 229 361 274 349 121)(21 415 402 296 223 317 173 108 98 230 362 275 350 122)(22 416 403 297 224 318 174 109 99 231 363 276 351 123)(23 417 404 298 225 319 175 110 100 232 364 277 352 124)(24 418 405 299 211 320 176 111 101 233 365 278 353 125)(25 419 391 300 212 321 177 112 102 234 366 279 354 126)(26 420 392 286 213 322 178 113 103 235 367 280 355 127)(27 406 393 287 214 323 179 114 104 236 368 281 356 128)(28 407 394 288 215 324 180 115 105 237 369 282 357 129)(29 408 395 289 216 325 166 116 91 238 370 283 358 130)(30 409 396 290 217 326 167 117 92 239 371 284 359 131)
(1 129 247 180)(2 130 248 166)(3 131 249 167)(4 132 250 168)(5 133 251 169)(6 134 252 170)(7 135 253 171)(8 121 254 172)(9 122 255 173)(10 123 241 174)(11 124 242 175)(12 125 243 176)(13 126 244 177)(14 127 245 178)(15 128 246 179)(16 58 118 62)(17 59 119 63)(18 60 120 64)(19 46 106 65)(20 47 107 66)(21 48 108 67)(22 49 109 68)(23 50 110 69)(24 51 111 70)(25 52 112 71)(26 53 113 72)(27 54 114 73)(28 55 115 74)(29 56 116 75)(30 57 117 61)(31 355 261 322)(32 356 262 323)(33 357 263 324)(34 358 264 325)(35 359 265 326)(36 360 266 327)(37 346 267 328)(38 347 268 329)(39 348 269 330)(40 349 270 316)(41 350 256 317)(42 351 257 318)(43 352 258 319)(44 353 259 320)(45 354 260 321)(76 236 191 393)(77 237 192 394)(78 238 193 395)(79 239 194 396)(80 240 195 397)(81 226 181 398)(82 227 182 399)(83 228 183 400)(84 229 184 401)(85 230 185 402)(86 231 186 403)(87 232 187 404)(88 233 188 405)(89 234 189 391)(90 235 190 392)(91 345 408 210)(92 331 409 196)(93 332 410 197)(94 333 411 198)(95 334 412 199)(96 335 413 200)(97 336 414 201)(98 337 415 202)(99 338 416 203)(100 339 417 204)(101 340 418 205)(102 341 419 206)(103 342 420 207)(104 343 406 208)(105 344 407 209)(136 364 158 298)(137 365 159 299)(138 366 160 300)(139 367 161 286)(140 368 162 287)(141 369 163 288)(142 370 164 289)(143 371 165 290)(144 372 151 291)(145 373 152 292)(146 374 153 293)(147 375 154 294)(148 361 155 295)(149 362 156 296)(150 363 157 297)(211 313 278 380)(212 314 279 381)(213 315 280 382)(214 301 281 383)(215 302 282 384)(216 303 283 385)(217 304 284 386)(218 305 285 387)(219 306 271 388)(220 307 272 389)(221 308 273 390)(222 309 274 376)(223 310 275 377)(224 311 276 378)(225 312 277 379)
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), (1,33,302,141,77,209,55,247,263,384,163,192,344,74)(2,34,303,142,78,210,56,248,264,385,164,193,345,75)(3,35,304,143,79,196,57,249,265,386,165,194,331,61)(4,36,305,144,80,197,58,250,266,387,151,195,332,62)(5,37,306,145,81,198,59,251,267,388,152,181,333,63)(6,38,307,146,82,199,60,252,268,389,153,182,334,64)(7,39,308,147,83,200,46,253,269,390,154,183,335,65)(8,40,309,148,84,201,47,254,270,376,155,184,336,66)(9,41,310,149,85,202,48,255,256,377,156,185,337,67)(10,42,311,150,86,203,49,241,257,378,157,186,338,68)(11,43,312,136,87,204,50,242,258,379,158,187,339,69)(12,44,313,137,88,205,51,243,259,380,159,188,340,70)(13,45,314,138,89,206,52,244,260,381,160,189,341,71)(14,31,315,139,90,207,53,245,261,382,161,190,342,72)(15,32,301,140,76,208,54,246,262,383,162,191,343,73)(16,410,397,291,218,327,168,118,93,240,372,285,360,132)(17,411,398,292,219,328,169,119,94,226,373,271,346,133)(18,412,399,293,220,329,170,120,95,227,374,272,347,134)(19,413,400,294,221,330,171,106,96,228,375,273,348,135)(20,414,401,295,222,316,172,107,97,229,361,274,349,121)(21,415,402,296,223,317,173,108,98,230,362,275,350,122)(22,416,403,297,224,318,174,109,99,231,363,276,351,123)(23,417,404,298,225,319,175,110,100,232,364,277,352,124)(24,418,405,299,211,320,176,111,101,233,365,278,353,125)(25,419,391,300,212,321,177,112,102,234,366,279,354,126)(26,420,392,286,213,322,178,113,103,235,367,280,355,127)(27,406,393,287,214,323,179,114,104,236,368,281,356,128)(28,407,394,288,215,324,180,115,105,237,369,282,357,129)(29,408,395,289,216,325,166,116,91,238,370,283,358,130)(30,409,396,290,217,326,167,117,92,239,371,284,359,131), (1,129,247,180)(2,130,248,166)(3,131,249,167)(4,132,250,168)(5,133,251,169)(6,134,252,170)(7,135,253,171)(8,121,254,172)(9,122,255,173)(10,123,241,174)(11,124,242,175)(12,125,243,176)(13,126,244,177)(14,127,245,178)(15,128,246,179)(16,58,118,62)(17,59,119,63)(18,60,120,64)(19,46,106,65)(20,47,107,66)(21,48,108,67)(22,49,109,68)(23,50,110,69)(24,51,111,70)(25,52,112,71)(26,53,113,72)(27,54,114,73)(28,55,115,74)(29,56,116,75)(30,57,117,61)(31,355,261,322)(32,356,262,323)(33,357,263,324)(34,358,264,325)(35,359,265,326)(36,360,266,327)(37,346,267,328)(38,347,268,329)(39,348,269,330)(40,349,270,316)(41,350,256,317)(42,351,257,318)(43,352,258,319)(44,353,259,320)(45,354,260,321)(76,236,191,393)(77,237,192,394)(78,238,193,395)(79,239,194,396)(80,240,195,397)(81,226,181,398)(82,227,182,399)(83,228,183,400)(84,229,184,401)(85,230,185,402)(86,231,186,403)(87,232,187,404)(88,233,188,405)(89,234,189,391)(90,235,190,392)(91,345,408,210)(92,331,409,196)(93,332,410,197)(94,333,411,198)(95,334,412,199)(96,335,413,200)(97,336,414,201)(98,337,415,202)(99,338,416,203)(100,339,417,204)(101,340,418,205)(102,341,419,206)(103,342,420,207)(104,343,406,208)(105,344,407,209)(136,364,158,298)(137,365,159,299)(138,366,160,300)(139,367,161,286)(140,368,162,287)(141,369,163,288)(142,370,164,289)(143,371,165,290)(144,372,151,291)(145,373,152,292)(146,374,153,293)(147,375,154,294)(148,361,155,295)(149,362,156,296)(150,363,157,297)(211,313,278,380)(212,314,279,381)(213,315,280,382)(214,301,281,383)(215,302,282,384)(216,303,283,385)(217,304,284,386)(218,305,285,387)(219,306,271,388)(220,307,272,389)(221,308,273,390)(222,309,274,376)(223,310,275,377)(224,311,276,378)(225,312,277,379)>;
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), (1,33,302,141,77,209,55,247,263,384,163,192,344,74)(2,34,303,142,78,210,56,248,264,385,164,193,345,75)(3,35,304,143,79,196,57,249,265,386,165,194,331,61)(4,36,305,144,80,197,58,250,266,387,151,195,332,62)(5,37,306,145,81,198,59,251,267,388,152,181,333,63)(6,38,307,146,82,199,60,252,268,389,153,182,334,64)(7,39,308,147,83,200,46,253,269,390,154,183,335,65)(8,40,309,148,84,201,47,254,270,376,155,184,336,66)(9,41,310,149,85,202,48,255,256,377,156,185,337,67)(10,42,311,150,86,203,49,241,257,378,157,186,338,68)(11,43,312,136,87,204,50,242,258,379,158,187,339,69)(12,44,313,137,88,205,51,243,259,380,159,188,340,70)(13,45,314,138,89,206,52,244,260,381,160,189,341,71)(14,31,315,139,90,207,53,245,261,382,161,190,342,72)(15,32,301,140,76,208,54,246,262,383,162,191,343,73)(16,410,397,291,218,327,168,118,93,240,372,285,360,132)(17,411,398,292,219,328,169,119,94,226,373,271,346,133)(18,412,399,293,220,329,170,120,95,227,374,272,347,134)(19,413,400,294,221,330,171,106,96,228,375,273,348,135)(20,414,401,295,222,316,172,107,97,229,361,274,349,121)(21,415,402,296,223,317,173,108,98,230,362,275,350,122)(22,416,403,297,224,318,174,109,99,231,363,276,351,123)(23,417,404,298,225,319,175,110,100,232,364,277,352,124)(24,418,405,299,211,320,176,111,101,233,365,278,353,125)(25,419,391,300,212,321,177,112,102,234,366,279,354,126)(26,420,392,286,213,322,178,113,103,235,367,280,355,127)(27,406,393,287,214,323,179,114,104,236,368,281,356,128)(28,407,394,288,215,324,180,115,105,237,369,282,357,129)(29,408,395,289,216,325,166,116,91,238,370,283,358,130)(30,409,396,290,217,326,167,117,92,239,371,284,359,131), (1,129,247,180)(2,130,248,166)(3,131,249,167)(4,132,250,168)(5,133,251,169)(6,134,252,170)(7,135,253,171)(8,121,254,172)(9,122,255,173)(10,123,241,174)(11,124,242,175)(12,125,243,176)(13,126,244,177)(14,127,245,178)(15,128,246,179)(16,58,118,62)(17,59,119,63)(18,60,120,64)(19,46,106,65)(20,47,107,66)(21,48,108,67)(22,49,109,68)(23,50,110,69)(24,51,111,70)(25,52,112,71)(26,53,113,72)(27,54,114,73)(28,55,115,74)(29,56,116,75)(30,57,117,61)(31,355,261,322)(32,356,262,323)(33,357,263,324)(34,358,264,325)(35,359,265,326)(36,360,266,327)(37,346,267,328)(38,347,268,329)(39,348,269,330)(40,349,270,316)(41,350,256,317)(42,351,257,318)(43,352,258,319)(44,353,259,320)(45,354,260,321)(76,236,191,393)(77,237,192,394)(78,238,193,395)(79,239,194,396)(80,240,195,397)(81,226,181,398)(82,227,182,399)(83,228,183,400)(84,229,184,401)(85,230,185,402)(86,231,186,403)(87,232,187,404)(88,233,188,405)(89,234,189,391)(90,235,190,392)(91,345,408,210)(92,331,409,196)(93,332,410,197)(94,333,411,198)(95,334,412,199)(96,335,413,200)(97,336,414,201)(98,337,415,202)(99,338,416,203)(100,339,417,204)(101,340,418,205)(102,341,419,206)(103,342,420,207)(104,343,406,208)(105,344,407,209)(136,364,158,298)(137,365,159,299)(138,366,160,300)(139,367,161,286)(140,368,162,287)(141,369,163,288)(142,370,164,289)(143,371,165,290)(144,372,151,291)(145,373,152,292)(146,374,153,293)(147,375,154,294)(148,361,155,295)(149,362,156,296)(150,363,157,297)(211,313,278,380)(212,314,279,381)(213,315,280,382)(214,301,281,383)(215,302,282,384)(216,303,283,385)(217,304,284,386)(218,305,285,387)(219,306,271,388)(220,307,272,389)(221,308,273,390)(222,309,274,376)(223,310,275,377)(224,311,276,378)(225,312,277,379) );
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)], [(1,33,302,141,77,209,55,247,263,384,163,192,344,74),(2,34,303,142,78,210,56,248,264,385,164,193,345,75),(3,35,304,143,79,196,57,249,265,386,165,194,331,61),(4,36,305,144,80,197,58,250,266,387,151,195,332,62),(5,37,306,145,81,198,59,251,267,388,152,181,333,63),(6,38,307,146,82,199,60,252,268,389,153,182,334,64),(7,39,308,147,83,200,46,253,269,390,154,183,335,65),(8,40,309,148,84,201,47,254,270,376,155,184,336,66),(9,41,310,149,85,202,48,255,256,377,156,185,337,67),(10,42,311,150,86,203,49,241,257,378,157,186,338,68),(11,43,312,136,87,204,50,242,258,379,158,187,339,69),(12,44,313,137,88,205,51,243,259,380,159,188,340,70),(13,45,314,138,89,206,52,244,260,381,160,189,341,71),(14,31,315,139,90,207,53,245,261,382,161,190,342,72),(15,32,301,140,76,208,54,246,262,383,162,191,343,73),(16,410,397,291,218,327,168,118,93,240,372,285,360,132),(17,411,398,292,219,328,169,119,94,226,373,271,346,133),(18,412,399,293,220,329,170,120,95,227,374,272,347,134),(19,413,400,294,221,330,171,106,96,228,375,273,348,135),(20,414,401,295,222,316,172,107,97,229,361,274,349,121),(21,415,402,296,223,317,173,108,98,230,362,275,350,122),(22,416,403,297,224,318,174,109,99,231,363,276,351,123),(23,417,404,298,225,319,175,110,100,232,364,277,352,124),(24,418,405,299,211,320,176,111,101,233,365,278,353,125),(25,419,391,300,212,321,177,112,102,234,366,279,354,126),(26,420,392,286,213,322,178,113,103,235,367,280,355,127),(27,406,393,287,214,323,179,114,104,236,368,281,356,128),(28,407,394,288,215,324,180,115,105,237,369,282,357,129),(29,408,395,289,216,325,166,116,91,238,370,283,358,130),(30,409,396,290,217,326,167,117,92,239,371,284,359,131)], [(1,129,247,180),(2,130,248,166),(3,131,249,167),(4,132,250,168),(5,133,251,169),(6,134,252,170),(7,135,253,171),(8,121,254,172),(9,122,255,173),(10,123,241,174),(11,124,242,175),(12,125,243,176),(13,126,244,177),(14,127,245,178),(15,128,246,179),(16,58,118,62),(17,59,119,63),(18,60,120,64),(19,46,106,65),(20,47,107,66),(21,48,108,67),(22,49,109,68),(23,50,110,69),(24,51,111,70),(25,52,112,71),(26,53,113,72),(27,54,114,73),(28,55,115,74),(29,56,116,75),(30,57,117,61),(31,355,261,322),(32,356,262,323),(33,357,263,324),(34,358,264,325),(35,359,265,326),(36,360,266,327),(37,346,267,328),(38,347,268,329),(39,348,269,330),(40,349,270,316),(41,350,256,317),(42,351,257,318),(43,352,258,319),(44,353,259,320),(45,354,260,321),(76,236,191,393),(77,237,192,394),(78,238,193,395),(79,239,194,396),(80,240,195,397),(81,226,181,398),(82,227,182,399),(83,228,183,400),(84,229,184,401),(85,230,185,402),(86,231,186,403),(87,232,187,404),(88,233,188,405),(89,234,189,391),(90,235,190,392),(91,345,408,210),(92,331,409,196),(93,332,410,197),(94,333,411,198),(95,334,412,199),(96,335,413,200),(97,336,414,201),(98,337,415,202),(99,338,416,203),(100,339,417,204),(101,340,418,205),(102,341,419,206),(103,342,420,207),(104,343,406,208),(105,344,407,209),(136,364,158,298),(137,365,159,299),(138,366,160,300),(139,367,161,286),(140,368,162,287),(141,369,163,288),(142,370,164,289),(143,371,165,290),(144,372,151,291),(145,373,152,292),(146,374,153,293),(147,375,154,294),(148,361,155,295),(149,362,156,296),(150,363,157,297),(211,313,278,380),(212,314,279,381),(213,315,280,382),(214,301,281,383),(215,302,282,384),(216,303,283,385),(217,304,284,386),(218,305,285,387),(219,306,271,388),(220,307,272,389),(221,308,273,390),(222,309,274,376),(223,310,275,377),(224,311,276,378),(225,312,277,379)]])
150 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 7A | 7B | 7C | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 14A | 14B | 14C | 15A | ··· | 15H | 20A | ··· | 20H | 21A | ··· | 21F | 30A | ··· | 30H | 35A | ··· | 35L | 42A | ··· | 42F | 60A | ··· | 60P | 70A | ··· | 70L | 105A | ··· | 105X | 210A | ··· | 210X |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 7 | 7 | 7 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 15 | ··· | 15 | 20 | ··· | 20 | 21 | ··· | 21 | 30 | ··· | 30 | 35 | ··· | 35 | 42 | ··· | 42 | 60 | ··· | 60 | 70 | ··· | 70 | 105 | ··· | 105 | 210 | ··· | 210 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||||||||
| image | C1 | C2 | C3 | C4 | C5 | C6 | C10 | C12 | C15 | C20 | C30 | C60 | D7 | Dic7 | C3×D7 | C5×D7 | C3×Dic7 | C5×Dic7 | D7×C15 | C15×Dic7 |
| kernel | C15×Dic7 | C210 | C5×Dic7 | C105 | C3×Dic7 | C70 | C42 | C35 | Dic7 | C21 | C14 | C7 | C30 | C15 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 16 | 3 | 3 | 6 | 12 | 6 | 12 | 24 | 24 |
Matrix representation of C15×Dic7 ►in GL3(𝔽421) generated by
| 400 | 0 | 0 |
| 0 | 252 | 0 |
| 0 | 0 | 252 |
| 420 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 420 | 403 |
| 392 | 0 | 0 |
| 0 | 376 | 56 |
| 0 | 24 | 45 |
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