metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.215D14, C7⋊C8.3Q8, C4.32(Q8×D7), C4⋊C4.73D14, C28.32(C2×Q8), C7⋊4(C8.5Q8), (C2×C28).274D4, C14.28(C4⋊Q8), C42.C2.3D7, C28.6Q8.7C2, C14.108(C4○D8), (C2×C28).382C23, (C4×C28).112C22, C28.Q8.13C2, C4.Dic14.14C2, C2.8(Dic7⋊Q8), C4⋊Dic7.152C22, C2.27(D4.8D14), (C4×C7⋊C8).10C2, (C2×C14).513(C2×D4), (C2×C7⋊C8).254C22, (C7×C42.C2).2C2, (C2×C4).111(C7⋊D4), (C7×C4⋊C4).120C22, (C2×C4).480(C22×D7), C22.186(C2×C7⋊D4), SmallGroup(448,598)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.215D14
G = < a,b,c,d | a4=b4=1, c14=d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=bc13 >
Subgroups: 316 in 86 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, C2×C4, C14, C14, C42, C4⋊C4, C4⋊C4, C2×C8, Dic7, C28, C28, C2×C14, C4×C8, C4.Q8, C2.D8, C42.C2, C42.C2, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C2×C28, C8.5Q8, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C4×C7⋊C8, C28.Q8, C4.Dic14, C28.6Q8, C7×C42.C2, C42.215D14
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, D14, C4⋊Q8, C4○D8, C7⋊D4, C22×D7, C8.5Q8, Q8×D7, C2×C7⋊D4, Dic7⋊Q8, D4.8D14, C42.215D14
►Regular action on 448 points
Generators in S448
(1 316 15 330)(2 341 16 355)(3 318 17 332)(4 343 18 357)(5 320 19 334)(6 345 20 359)(7 322 21 336)(8 347 22 361)(9 324 23 310)(10 349 24 363)(11 326 25 312)(12 351 26 337)(13 328 27 314)(14 353 28 339)(29 222 43 208)(30 65 44 79)(31 224 45 210)(32 67 46 81)(33 198 47 212)(34 69 48 83)(35 200 49 214)(36 71 50 57)(37 202 51 216)(38 73 52 59)(39 204 53 218)(40 75 54 61)(41 206 55 220)(42 77 56 63)(58 392 72 378)(60 366 74 380)(62 368 76 382)(64 370 78 384)(66 372 80 386)(68 374 82 388)(70 376 84 390)(85 262 99 276)(86 288 100 302)(87 264 101 278)(88 290 102 304)(89 266 103 280)(90 292 104 306)(91 268 105 254)(92 294 106 308)(93 270 107 256)(94 296 108 282)(95 272 109 258)(96 298 110 284)(97 274 111 260)(98 300 112 286)(113 415 127 401)(114 424 128 438)(115 417 129 403)(116 426 130 440)(117 419 131 405)(118 428 132 442)(119 393 133 407)(120 430 134 444)(121 395 135 409)(122 432 136 446)(123 397 137 411)(124 434 138 448)(125 399 139 413)(126 436 140 422)(141 331 155 317)(142 356 156 342)(143 333 157 319)(144 358 158 344)(145 335 159 321)(146 360 160 346)(147 309 161 323)(148 362 162 348)(149 311 163 325)(150 364 164 350)(151 313 165 327)(152 338 166 352)(153 315 167 329)(154 340 168 354)(169 402 183 416)(170 439 184 425)(171 404 185 418)(172 441 186 427)(173 406 187 420)(174 443 188 429)(175 408 189 394)(176 445 190 431)(177 410 191 396)(178 447 192 433)(179 412 193 398)(180 421 194 435)(181 414 195 400)(182 423 196 437)(197 373 211 387)(199 375 213 389)(201 377 215 391)(203 379 217 365)(205 381 219 367)(207 383 221 369)(209 385 223 371)(225 259 239 273)(226 285 240 299)(227 261 241 275)(228 287 242 301)(229 263 243 277)(230 289 244 303)(231 265 245 279)(232 291 246 305)(233 267 247 253)(234 293 248 307)(235 269 249 255)(236 295 250 281)(237 271 251 257)(238 297 252 283)
(1 266 168 305)(2 306 141 267)(3 268 142 307)(4 308 143 269)(5 270 144 281)(6 282 145 271)(7 272 146 283)(8 284 147 273)(9 274 148 285)(10 286 149 275)(11 276 150 287)(12 288 151 277)(13 278 152 289)(14 290 153 279)(15 280 154 291)(16 292 155 253)(17 254 156 293)(18 294 157 255)(19 256 158 295)(20 296 159 257)(21 258 160 297)(22 298 161 259)(23 260 162 299)(24 300 163 261)(25 262 164 301)(26 302 165 263)(27 264 166 303)(28 304 167 265)(29 408 370 430)(30 431 371 409)(31 410 372 432)(32 433 373 411)(33 412 374 434)(34 435 375 413)(35 414 376 436)(36 437 377 415)(37 416 378 438)(38 439 379 417)(39 418 380 440)(40 441 381 419)(41 420 382 442)(42 443 383 393)(43 394 384 444)(44 445 385 395)(45 396 386 446)(46 447 387 397)(47 398 388 448)(48 421 389 399)(49 400 390 422)(50 423 391 401)(51 402 392 424)(52 425 365 403)(53 404 366 426)(54 427 367 405)(55 406 368 428)(56 429 369 407)(57 196 201 113)(58 114 202 169)(59 170 203 115)(60 116 204 171)(61 172 205 117)(62 118 206 173)(63 174 207 119)(64 120 208 175)(65 176 209 121)(66 122 210 177)(67 178 211 123)(68 124 212 179)(69 180 213 125)(70 126 214 181)(71 182 215 127)(72 128 216 183)(73 184 217 129)(74 130 218 185)(75 186 219 131)(76 132 220 187)(77 188 221 133)(78 134 222 189)(79 190 223 135)(80 136 224 191)(81 192 197 137)(82 138 198 193)(83 194 199 139)(84 140 200 195)(85 364 242 326)(86 327 243 337)(87 338 244 328)(88 329 245 339)(89 340 246 330)(90 331 247 341)(91 342 248 332)(92 333 249 343)(93 344 250 334)(94 335 251 345)(95 346 252 336)(96 309 225 347)(97 348 226 310)(98 311 227 349)(99 350 228 312)(100 313 229 351)(101 352 230 314)(102 315 231 353)(103 354 232 316)(104 317 233 355)(105 356 234 318)(106 319 235 357)(107 358 236 320)(108 321 237 359)(109 360 238 322)(110 323 239 361)(111 362 240 324)(112 325 241 363)
(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)(421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448)
(1 386 15 372)(2 431 16 445)(3 384 17 370)(4 429 18 443)(5 382 19 368)(6 427 20 441)(7 380 21 366)(8 425 22 439)(9 378 23 392)(10 423 24 437)(11 376 25 390)(12 421 26 435)(13 374 27 388)(14 447 28 433)(29 142 43 156)(30 292 44 306)(31 168 45 154)(32 290 46 304)(33 166 47 152)(34 288 48 302)(35 164 49 150)(36 286 50 300)(37 162 51 148)(38 284 52 298)(39 160 53 146)(40 282 54 296)(41 158 55 144)(42 308 56 294)(57 227 71 241)(58 362 72 348)(59 225 73 239)(60 360 74 346)(61 251 75 237)(62 358 76 344)(63 249 77 235)(64 356 78 342)(65 247 79 233)(66 354 80 340)(67 245 81 231)(68 352 82 338)(69 243 83 229)(70 350 84 364)(85 126 99 140)(86 199 100 213)(87 124 101 138)(88 197 102 211)(89 122 103 136)(90 223 104 209)(91 120 105 134)(92 221 106 207)(93 118 107 132)(94 219 108 205)(95 116 109 130)(96 217 110 203)(97 114 111 128)(98 215 112 201)(113 349 127 363)(115 347 129 361)(117 345 131 359)(119 343 133 357)(121 341 135 355)(123 339 137 353)(125 337 139 351)(141 409 155 395)(143 407 157 393)(145 405 159 419)(147 403 161 417)(149 401 163 415)(151 399 165 413)(153 397 167 411)(169 240 183 226)(170 309 184 323)(171 238 185 252)(172 335 186 321)(173 236 187 250)(174 333 188 319)(175 234 189 248)(176 331 190 317)(177 232 191 246)(178 329 192 315)(179 230 193 244)(180 327 194 313)(181 228 195 242)(182 325 196 311)(198 328 212 314)(200 326 214 312)(202 324 216 310)(204 322 218 336)(206 320 220 334)(208 318 222 332)(210 316 224 330)(253 385 267 371)(254 408 268 394)(255 383 269 369)(256 406 270 420)(257 381 271 367)(258 404 272 418)(259 379 273 365)(260 402 274 416)(261 377 275 391)(262 400 276 414)(263 375 277 389)(264 398 278 412)(265 373 279 387)(266 396 280 410)(281 442 295 428)(283 440 297 426)(285 438 299 424)(287 436 301 422)(289 434 303 448)(291 432 305 446)(293 430 307 444)
G:=sub<Sym(448)| (1,316,15,330)(2,341,16,355)(3,318,17,332)(4,343,18,357)(5,320,19,334)(6,345,20,359)(7,322,21,336)(8,347,22,361)(9,324,23,310)(10,349,24,363)(11,326,25,312)(12,351,26,337)(13,328,27,314)(14,353,28,339)(29,222,43,208)(30,65,44,79)(31,224,45,210)(32,67,46,81)(33,198,47,212)(34,69,48,83)(35,200,49,214)(36,71,50,57)(37,202,51,216)(38,73,52,59)(39,204,53,218)(40,75,54,61)(41,206,55,220)(42,77,56,63)(58,392,72,378)(60,366,74,380)(62,368,76,382)(64,370,78,384)(66,372,80,386)(68,374,82,388)(70,376,84,390)(85,262,99,276)(86,288,100,302)(87,264,101,278)(88,290,102,304)(89,266,103,280)(90,292,104,306)(91,268,105,254)(92,294,106,308)(93,270,107,256)(94,296,108,282)(95,272,109,258)(96,298,110,284)(97,274,111,260)(98,300,112,286)(113,415,127,401)(114,424,128,438)(115,417,129,403)(116,426,130,440)(117,419,131,405)(118,428,132,442)(119,393,133,407)(120,430,134,444)(121,395,135,409)(122,432,136,446)(123,397,137,411)(124,434,138,448)(125,399,139,413)(126,436,140,422)(141,331,155,317)(142,356,156,342)(143,333,157,319)(144,358,158,344)(145,335,159,321)(146,360,160,346)(147,309,161,323)(148,362,162,348)(149,311,163,325)(150,364,164,350)(151,313,165,327)(152,338,166,352)(153,315,167,329)(154,340,168,354)(169,402,183,416)(170,439,184,425)(171,404,185,418)(172,441,186,427)(173,406,187,420)(174,443,188,429)(175,408,189,394)(176,445,190,431)(177,410,191,396)(178,447,192,433)(179,412,193,398)(180,421,194,435)(181,414,195,400)(182,423,196,437)(197,373,211,387)(199,375,213,389)(201,377,215,391)(203,379,217,365)(205,381,219,367)(207,383,221,369)(209,385,223,371)(225,259,239,273)(226,285,240,299)(227,261,241,275)(228,287,242,301)(229,263,243,277)(230,289,244,303)(231,265,245,279)(232,291,246,305)(233,267,247,253)(234,293,248,307)(235,269,249,255)(236,295,250,281)(237,271,251,257)(238,297,252,283), (1,266,168,305)(2,306,141,267)(3,268,142,307)(4,308,143,269)(5,270,144,281)(6,282,145,271)(7,272,146,283)(8,284,147,273)(9,274,148,285)(10,286,149,275)(11,276,150,287)(12,288,151,277)(13,278,152,289)(14,290,153,279)(15,280,154,291)(16,292,155,253)(17,254,156,293)(18,294,157,255)(19,256,158,295)(20,296,159,257)(21,258,160,297)(22,298,161,259)(23,260,162,299)(24,300,163,261)(25,262,164,301)(26,302,165,263)(27,264,166,303)(28,304,167,265)(29,408,370,430)(30,431,371,409)(31,410,372,432)(32,433,373,411)(33,412,374,434)(34,435,375,413)(35,414,376,436)(36,437,377,415)(37,416,378,438)(38,439,379,417)(39,418,380,440)(40,441,381,419)(41,420,382,442)(42,443,383,393)(43,394,384,444)(44,445,385,395)(45,396,386,446)(46,447,387,397)(47,398,388,448)(48,421,389,399)(49,400,390,422)(50,423,391,401)(51,402,392,424)(52,425,365,403)(53,404,366,426)(54,427,367,405)(55,406,368,428)(56,429,369,407)(57,196,201,113)(58,114,202,169)(59,170,203,115)(60,116,204,171)(61,172,205,117)(62,118,206,173)(63,174,207,119)(64,120,208,175)(65,176,209,121)(66,122,210,177)(67,178,211,123)(68,124,212,179)(69,180,213,125)(70,126,214,181)(71,182,215,127)(72,128,216,183)(73,184,217,129)(74,130,218,185)(75,186,219,131)(76,132,220,187)(77,188,221,133)(78,134,222,189)(79,190,223,135)(80,136,224,191)(81,192,197,137)(82,138,198,193)(83,194,199,139)(84,140,200,195)(85,364,242,326)(86,327,243,337)(87,338,244,328)(88,329,245,339)(89,340,246,330)(90,331,247,341)(91,342,248,332)(92,333,249,343)(93,344,250,334)(94,335,251,345)(95,346,252,336)(96,309,225,347)(97,348,226,310)(98,311,227,349)(99,350,228,312)(100,313,229,351)(101,352,230,314)(102,315,231,353)(103,354,232,316)(104,317,233,355)(105,356,234,318)(106,319,235,357)(107,358,236,320)(108,321,237,359)(109,360,238,322)(110,323,239,361)(111,362,240,324)(112,325,241,363), (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)(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448), (1,386,15,372)(2,431,16,445)(3,384,17,370)(4,429,18,443)(5,382,19,368)(6,427,20,441)(7,380,21,366)(8,425,22,439)(9,378,23,392)(10,423,24,437)(11,376,25,390)(12,421,26,435)(13,374,27,388)(14,447,28,433)(29,142,43,156)(30,292,44,306)(31,168,45,154)(32,290,46,304)(33,166,47,152)(34,288,48,302)(35,164,49,150)(36,286,50,300)(37,162,51,148)(38,284,52,298)(39,160,53,146)(40,282,54,296)(41,158,55,144)(42,308,56,294)(57,227,71,241)(58,362,72,348)(59,225,73,239)(60,360,74,346)(61,251,75,237)(62,358,76,344)(63,249,77,235)(64,356,78,342)(65,247,79,233)(66,354,80,340)(67,245,81,231)(68,352,82,338)(69,243,83,229)(70,350,84,364)(85,126,99,140)(86,199,100,213)(87,124,101,138)(88,197,102,211)(89,122,103,136)(90,223,104,209)(91,120,105,134)(92,221,106,207)(93,118,107,132)(94,219,108,205)(95,116,109,130)(96,217,110,203)(97,114,111,128)(98,215,112,201)(113,349,127,363)(115,347,129,361)(117,345,131,359)(119,343,133,357)(121,341,135,355)(123,339,137,353)(125,337,139,351)(141,409,155,395)(143,407,157,393)(145,405,159,419)(147,403,161,417)(149,401,163,415)(151,399,165,413)(153,397,167,411)(169,240,183,226)(170,309,184,323)(171,238,185,252)(172,335,186,321)(173,236,187,250)(174,333,188,319)(175,234,189,248)(176,331,190,317)(177,232,191,246)(178,329,192,315)(179,230,193,244)(180,327,194,313)(181,228,195,242)(182,325,196,311)(198,328,212,314)(200,326,214,312)(202,324,216,310)(204,322,218,336)(206,320,220,334)(208,318,222,332)(210,316,224,330)(253,385,267,371)(254,408,268,394)(255,383,269,369)(256,406,270,420)(257,381,271,367)(258,404,272,418)(259,379,273,365)(260,402,274,416)(261,377,275,391)(262,400,276,414)(263,375,277,389)(264,398,278,412)(265,373,279,387)(266,396,280,410)(281,442,295,428)(283,440,297,426)(285,438,299,424)(287,436,301,422)(289,434,303,448)(291,432,305,446)(293,430,307,444)>;
G:=Group( (1,316,15,330)(2,341,16,355)(3,318,17,332)(4,343,18,357)(5,320,19,334)(6,345,20,359)(7,322,21,336)(8,347,22,361)(9,324,23,310)(10,349,24,363)(11,326,25,312)(12,351,26,337)(13,328,27,314)(14,353,28,339)(29,222,43,208)(30,65,44,79)(31,224,45,210)(32,67,46,81)(33,198,47,212)(34,69,48,83)(35,200,49,214)(36,71,50,57)(37,202,51,216)(38,73,52,59)(39,204,53,218)(40,75,54,61)(41,206,55,220)(42,77,56,63)(58,392,72,378)(60,366,74,380)(62,368,76,382)(64,370,78,384)(66,372,80,386)(68,374,82,388)(70,376,84,390)(85,262,99,276)(86,288,100,302)(87,264,101,278)(88,290,102,304)(89,266,103,280)(90,292,104,306)(91,268,105,254)(92,294,106,308)(93,270,107,256)(94,296,108,282)(95,272,109,258)(96,298,110,284)(97,274,111,260)(98,300,112,286)(113,415,127,401)(114,424,128,438)(115,417,129,403)(116,426,130,440)(117,419,131,405)(118,428,132,442)(119,393,133,407)(120,430,134,444)(121,395,135,409)(122,432,136,446)(123,397,137,411)(124,434,138,448)(125,399,139,413)(126,436,140,422)(141,331,155,317)(142,356,156,342)(143,333,157,319)(144,358,158,344)(145,335,159,321)(146,360,160,346)(147,309,161,323)(148,362,162,348)(149,311,163,325)(150,364,164,350)(151,313,165,327)(152,338,166,352)(153,315,167,329)(154,340,168,354)(169,402,183,416)(170,439,184,425)(171,404,185,418)(172,441,186,427)(173,406,187,420)(174,443,188,429)(175,408,189,394)(176,445,190,431)(177,410,191,396)(178,447,192,433)(179,412,193,398)(180,421,194,435)(181,414,195,400)(182,423,196,437)(197,373,211,387)(199,375,213,389)(201,377,215,391)(203,379,217,365)(205,381,219,367)(207,383,221,369)(209,385,223,371)(225,259,239,273)(226,285,240,299)(227,261,241,275)(228,287,242,301)(229,263,243,277)(230,289,244,303)(231,265,245,279)(232,291,246,305)(233,267,247,253)(234,293,248,307)(235,269,249,255)(236,295,250,281)(237,271,251,257)(238,297,252,283), (1,266,168,305)(2,306,141,267)(3,268,142,307)(4,308,143,269)(5,270,144,281)(6,282,145,271)(7,272,146,283)(8,284,147,273)(9,274,148,285)(10,286,149,275)(11,276,150,287)(12,288,151,277)(13,278,152,289)(14,290,153,279)(15,280,154,291)(16,292,155,253)(17,254,156,293)(18,294,157,255)(19,256,158,295)(20,296,159,257)(21,258,160,297)(22,298,161,259)(23,260,162,299)(24,300,163,261)(25,262,164,301)(26,302,165,263)(27,264,166,303)(28,304,167,265)(29,408,370,430)(30,431,371,409)(31,410,372,432)(32,433,373,411)(33,412,374,434)(34,435,375,413)(35,414,376,436)(36,437,377,415)(37,416,378,438)(38,439,379,417)(39,418,380,440)(40,441,381,419)(41,420,382,442)(42,443,383,393)(43,394,384,444)(44,445,385,395)(45,396,386,446)(46,447,387,397)(47,398,388,448)(48,421,389,399)(49,400,390,422)(50,423,391,401)(51,402,392,424)(52,425,365,403)(53,404,366,426)(54,427,367,405)(55,406,368,428)(56,429,369,407)(57,196,201,113)(58,114,202,169)(59,170,203,115)(60,116,204,171)(61,172,205,117)(62,118,206,173)(63,174,207,119)(64,120,208,175)(65,176,209,121)(66,122,210,177)(67,178,211,123)(68,124,212,179)(69,180,213,125)(70,126,214,181)(71,182,215,127)(72,128,216,183)(73,184,217,129)(74,130,218,185)(75,186,219,131)(76,132,220,187)(77,188,221,133)(78,134,222,189)(79,190,223,135)(80,136,224,191)(81,192,197,137)(82,138,198,193)(83,194,199,139)(84,140,200,195)(85,364,242,326)(86,327,243,337)(87,338,244,328)(88,329,245,339)(89,340,246,330)(90,331,247,341)(91,342,248,332)(92,333,249,343)(93,344,250,334)(94,335,251,345)(95,346,252,336)(96,309,225,347)(97,348,226,310)(98,311,227,349)(99,350,228,312)(100,313,229,351)(101,352,230,314)(102,315,231,353)(103,354,232,316)(104,317,233,355)(105,356,234,318)(106,319,235,357)(107,358,236,320)(108,321,237,359)(109,360,238,322)(110,323,239,361)(111,362,240,324)(112,325,241,363), (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)(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448), (1,386,15,372)(2,431,16,445)(3,384,17,370)(4,429,18,443)(5,382,19,368)(6,427,20,441)(7,380,21,366)(8,425,22,439)(9,378,23,392)(10,423,24,437)(11,376,25,390)(12,421,26,435)(13,374,27,388)(14,447,28,433)(29,142,43,156)(30,292,44,306)(31,168,45,154)(32,290,46,304)(33,166,47,152)(34,288,48,302)(35,164,49,150)(36,286,50,300)(37,162,51,148)(38,284,52,298)(39,160,53,146)(40,282,54,296)(41,158,55,144)(42,308,56,294)(57,227,71,241)(58,362,72,348)(59,225,73,239)(60,360,74,346)(61,251,75,237)(62,358,76,344)(63,249,77,235)(64,356,78,342)(65,247,79,233)(66,354,80,340)(67,245,81,231)(68,352,82,338)(69,243,83,229)(70,350,84,364)(85,126,99,140)(86,199,100,213)(87,124,101,138)(88,197,102,211)(89,122,103,136)(90,223,104,209)(91,120,105,134)(92,221,106,207)(93,118,107,132)(94,219,108,205)(95,116,109,130)(96,217,110,203)(97,114,111,128)(98,215,112,201)(113,349,127,363)(115,347,129,361)(117,345,131,359)(119,343,133,357)(121,341,135,355)(123,339,137,353)(125,337,139,351)(141,409,155,395)(143,407,157,393)(145,405,159,419)(147,403,161,417)(149,401,163,415)(151,399,165,413)(153,397,167,411)(169,240,183,226)(170,309,184,323)(171,238,185,252)(172,335,186,321)(173,236,187,250)(174,333,188,319)(175,234,189,248)(176,331,190,317)(177,232,191,246)(178,329,192,315)(179,230,193,244)(180,327,194,313)(181,228,195,242)(182,325,196,311)(198,328,212,314)(200,326,214,312)(202,324,216,310)(204,322,218,336)(206,320,220,334)(208,318,222,332)(210,316,224,330)(253,385,267,371)(254,408,268,394)(255,383,269,369)(256,406,270,420)(257,381,271,367)(258,404,272,418)(259,379,273,365)(260,402,274,416)(261,377,275,391)(262,400,276,414)(263,375,277,389)(264,398,278,412)(265,373,279,387)(266,396,280,410)(281,442,295,428)(283,440,297,426)(285,438,299,424)(287,436,301,422)(289,434,303,448)(291,432,305,446)(293,430,307,444) );
G=PermutationGroup([[(1,316,15,330),(2,341,16,355),(3,318,17,332),(4,343,18,357),(5,320,19,334),(6,345,20,359),(7,322,21,336),(8,347,22,361),(9,324,23,310),(10,349,24,363),(11,326,25,312),(12,351,26,337),(13,328,27,314),(14,353,28,339),(29,222,43,208),(30,65,44,79),(31,224,45,210),(32,67,46,81),(33,198,47,212),(34,69,48,83),(35,200,49,214),(36,71,50,57),(37,202,51,216),(38,73,52,59),(39,204,53,218),(40,75,54,61),(41,206,55,220),(42,77,56,63),(58,392,72,378),(60,366,74,380),(62,368,76,382),(64,370,78,384),(66,372,80,386),(68,374,82,388),(70,376,84,390),(85,262,99,276),(86,288,100,302),(87,264,101,278),(88,290,102,304),(89,266,103,280),(90,292,104,306),(91,268,105,254),(92,294,106,308),(93,270,107,256),(94,296,108,282),(95,272,109,258),(96,298,110,284),(97,274,111,260),(98,300,112,286),(113,415,127,401),(114,424,128,438),(115,417,129,403),(116,426,130,440),(117,419,131,405),(118,428,132,442),(119,393,133,407),(120,430,134,444),(121,395,135,409),(122,432,136,446),(123,397,137,411),(124,434,138,448),(125,399,139,413),(126,436,140,422),(141,331,155,317),(142,356,156,342),(143,333,157,319),(144,358,158,344),(145,335,159,321),(146,360,160,346),(147,309,161,323),(148,362,162,348),(149,311,163,325),(150,364,164,350),(151,313,165,327),(152,338,166,352),(153,315,167,329),(154,340,168,354),(169,402,183,416),(170,439,184,425),(171,404,185,418),(172,441,186,427),(173,406,187,420),(174,443,188,429),(175,408,189,394),(176,445,190,431),(177,410,191,396),(178,447,192,433),(179,412,193,398),(180,421,194,435),(181,414,195,400),(182,423,196,437),(197,373,211,387),(199,375,213,389),(201,377,215,391),(203,379,217,365),(205,381,219,367),(207,383,221,369),(209,385,223,371),(225,259,239,273),(226,285,240,299),(227,261,241,275),(228,287,242,301),(229,263,243,277),(230,289,244,303),(231,265,245,279),(232,291,246,305),(233,267,247,253),(234,293,248,307),(235,269,249,255),(236,295,250,281),(237,271,251,257),(238,297,252,283)], [(1,266,168,305),(2,306,141,267),(3,268,142,307),(4,308,143,269),(5,270,144,281),(6,282,145,271),(7,272,146,283),(8,284,147,273),(9,274,148,285),(10,286,149,275),(11,276,150,287),(12,288,151,277),(13,278,152,289),(14,290,153,279),(15,280,154,291),(16,292,155,253),(17,254,156,293),(18,294,157,255),(19,256,158,295),(20,296,159,257),(21,258,160,297),(22,298,161,259),(23,260,162,299),(24,300,163,261),(25,262,164,301),(26,302,165,263),(27,264,166,303),(28,304,167,265),(29,408,370,430),(30,431,371,409),(31,410,372,432),(32,433,373,411),(33,412,374,434),(34,435,375,413),(35,414,376,436),(36,437,377,415),(37,416,378,438),(38,439,379,417),(39,418,380,440),(40,441,381,419),(41,420,382,442),(42,443,383,393),(43,394,384,444),(44,445,385,395),(45,396,386,446),(46,447,387,397),(47,398,388,448),(48,421,389,399),(49,400,390,422),(50,423,391,401),(51,402,392,424),(52,425,365,403),(53,404,366,426),(54,427,367,405),(55,406,368,428),(56,429,369,407),(57,196,201,113),(58,114,202,169),(59,170,203,115),(60,116,204,171),(61,172,205,117),(62,118,206,173),(63,174,207,119),(64,120,208,175),(65,176,209,121),(66,122,210,177),(67,178,211,123),(68,124,212,179),(69,180,213,125),(70,126,214,181),(71,182,215,127),(72,128,216,183),(73,184,217,129),(74,130,218,185),(75,186,219,131),(76,132,220,187),(77,188,221,133),(78,134,222,189),(79,190,223,135),(80,136,224,191),(81,192,197,137),(82,138,198,193),(83,194,199,139),(84,140,200,195),(85,364,242,326),(86,327,243,337),(87,338,244,328),(88,329,245,339),(89,340,246,330),(90,331,247,341),(91,342,248,332),(92,333,249,343),(93,344,250,334),(94,335,251,345),(95,346,252,336),(96,309,225,347),(97,348,226,310),(98,311,227,349),(99,350,228,312),(100,313,229,351),(101,352,230,314),(102,315,231,353),(103,354,232,316),(104,317,233,355),(105,356,234,318),(106,319,235,357),(107,358,236,320),(108,321,237,359),(109,360,238,322),(110,323,239,361),(111,362,240,324),(112,325,241,363)], [(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),(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448)], [(1,386,15,372),(2,431,16,445),(3,384,17,370),(4,429,18,443),(5,382,19,368),(6,427,20,441),(7,380,21,366),(8,425,22,439),(9,378,23,392),(10,423,24,437),(11,376,25,390),(12,421,26,435),(13,374,27,388),(14,447,28,433),(29,142,43,156),(30,292,44,306),(31,168,45,154),(32,290,46,304),(33,166,47,152),(34,288,48,302),(35,164,49,150),(36,286,50,300),(37,162,51,148),(38,284,52,298),(39,160,53,146),(40,282,54,296),(41,158,55,144),(42,308,56,294),(57,227,71,241),(58,362,72,348),(59,225,73,239),(60,360,74,346),(61,251,75,237),(62,358,76,344),(63,249,77,235),(64,356,78,342),(65,247,79,233),(66,354,80,340),(67,245,81,231),(68,352,82,338),(69,243,83,229),(70,350,84,364),(85,126,99,140),(86,199,100,213),(87,124,101,138),(88,197,102,211),(89,122,103,136),(90,223,104,209),(91,120,105,134),(92,221,106,207),(93,118,107,132),(94,219,108,205),(95,116,109,130),(96,217,110,203),(97,114,111,128),(98,215,112,201),(113,349,127,363),(115,347,129,361),(117,345,131,359),(119,343,133,357),(121,341,135,355),(123,339,137,353),(125,337,139,351),(141,409,155,395),(143,407,157,393),(145,405,159,419),(147,403,161,417),(149,401,163,415),(151,399,165,413),(153,397,167,411),(169,240,183,226),(170,309,184,323),(171,238,185,252),(172,335,186,321),(173,236,187,250),(174,333,188,319),(175,234,189,248),(176,331,190,317),(177,232,191,246),(178,329,192,315),(179,230,193,244),(180,327,194,313),(181,228,195,242),(182,325,196,311),(198,328,212,314),(200,326,214,312),(202,324,216,310),(204,322,218,336),(206,320,220,334),(208,318,222,332),(210,316,224,330),(253,385,267,371),(254,408,268,394),(255,383,269,369),(256,406,270,420),(257,381,271,367),(258,404,272,418),(259,379,273,365),(260,402,274,416),(261,377,275,391),(262,400,276,414),(263,375,277,389),(264,398,278,412),(265,373,279,387),(266,396,280,410),(281,442,295,428),(283,440,297,426),(285,438,299,424),(287,436,301,422),(289,434,303,448),(291,432,305,446),(293,430,307,444)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 56 | 56 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D7 | D14 | D14 | C4○D8 | C7⋊D4 | Q8×D7 | D4.8D14 |
| kernel | C42.215D14 | C4×C7⋊C8 | C28.Q8 | C4.Dic14 | C28.6Q8 | C7×C42.C2 | C7⋊C8 | C2×C28 | C42.C2 | C42 | C4⋊C4 | C14 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 2 | 3 | 3 | 6 | 8 | 12 | 6 | 12 |
Matrix representation of C42.215D14 ►in GL6(𝔽113)
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 |
| 0 | 0 | 0 | 0 | 98 | 0 |
| 1 | 111 | 0 | 0 | 0 | 0 |
| 1 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 26 | 87 | 0 | 0 | 0 | 0 |
| 13 | 87 | 0 | 0 | 0 | 0 |
| 0 | 0 | 54 | 59 | 0 | 0 |
| 0 | 0 | 54 | 82 | 0 | 0 |
| 0 | 0 | 0 | 0 | 86 | 96 |
| 0 | 0 | 0 | 0 | 96 | 27 |
| 15 | 83 | 0 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 82 | 0 | 0 |
| 0 | 0 | 76 | 53 | 0 | 0 |
| 0 | 0 | 0 | 0 | 84 | 8 |
| 0 | 0 | 0 | 0 | 8 | 29 |
G:=sub<GL(6,GF(113))| [15,0,0,0,0,0,0,15,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,98,0,0,0,0,15,0],[1,1,0,0,0,0,111,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[26,13,0,0,0,0,87,87,0,0,0,0,0,0,54,54,0,0,0,0,59,82,0,0,0,0,0,0,86,96,0,0,0,0,96,27],[15,0,0,0,0,0,83,98,0,0,0,0,0,0,60,76,0,0,0,0,82,53,0,0,0,0,0,0,84,8,0,0,0,0,8,29] >;
C42.215D14 in GAP, Magma, Sage, TeX
C_4^2._{215}D_{14} % in TeX
G:=Group("C4^2.215D14"); // GroupNames label
G:=SmallGroup(448,598);
// by ID
G=gap.SmallGroup(448,598);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,64,422,471,58,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^13>;
// generators/relations