metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.210D14, C28.11M4(2), C7⋊C8⋊11Q8, C7⋊5(C8⋊4Q8), (C4×Q8).5D7, C4.58(Q8×D7), C4⋊C4.9Dic7, (Q8×C28).6C2, (Q8×C14).9C4, C14.27(C4×Q8), C2.5(Q8×Dic7), C28.116(C2×Q8), (C2×Q8).5Dic7, C28⋊C8.18C2, C14.42(C8○D4), (C4×C28).95C22, C4.3(C4.Dic7), C28.339(C4○D4), (C2×C28).852C23, C4.59(Q8⋊2D7), C14.42(C2×M4(2)), C2.8(Q8.Dic7), C42.D7.3C2, C22.47(C22×Dic7), (C4×C7⋊C8).8C2, (C7×C4⋊C4).14C4, (C2×C28).166(C2×C4), (C2×C7⋊C8).202C22, (C2×C4).45(C2×Dic7), C2.10(C2×C4.Dic7), (C2×C4).794(C22×D7), (C2×C14).189(C22×C4), SmallGroup(448,558)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.210D14
G = < a,b,c,d | a4=b4=1, c14=a2, d2=a2b-1, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=b2c13 >
Subgroups: 228 in 94 conjugacy classes, 61 normal (33 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C28, C28, C28, C2×C14, C4×C8, C8⋊C4, C4⋊C8, C4×Q8, C7⋊C8, C7⋊C8, C2×C28, C2×C28, C7×Q8, C8⋊4Q8, C2×C7⋊C8, C2×C7⋊C8, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, Q8×C14, C4×C7⋊C8, C42.D7, C28⋊C8, C28⋊C8, Q8×C28, C42.210D14
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D7, M4(2), C22×C4, C2×Q8, C4○D4, Dic7, D14, C4×Q8, C2×M4(2), C8○D4, C2×Dic7, C22×D7, C8⋊4Q8, C4.Dic7, Q8×D7, Q8⋊2D7, C22×Dic7, C2×C4.Dic7, Q8×Dic7, Q8.Dic7, C42.210D14
►Regular action on 448 points
Generators in S448
(1 247 15 233)(2 234 16 248)(3 249 17 235)(4 236 18 250)(5 251 19 237)(6 238 20 252)(7 225 21 239)(8 240 22 226)(9 227 23 241)(10 242 24 228)(11 229 25 243)(12 244 26 230)(13 231 27 245)(14 246 28 232)(29 349 43 363)(30 364 44 350)(31 351 45 337)(32 338 46 352)(33 353 47 339)(34 340 48 354)(35 355 49 341)(36 342 50 356)(37 357 51 343)(38 344 52 358)(39 359 53 345)(40 346 54 360)(41 361 55 347)(42 348 56 362)(57 289 71 303)(58 304 72 290)(59 291 73 305)(60 306 74 292)(61 293 75 307)(62 308 76 294)(63 295 77 281)(64 282 78 296)(65 297 79 283)(66 284 80 298)(67 299 81 285)(68 286 82 300)(69 301 83 287)(70 288 84 302)(85 155 99 141)(86 142 100 156)(87 157 101 143)(88 144 102 158)(89 159 103 145)(90 146 104 160)(91 161 105 147)(92 148 106 162)(93 163 107 149)(94 150 108 164)(95 165 109 151)(96 152 110 166)(97 167 111 153)(98 154 112 168)(113 409 127 395)(114 396 128 410)(115 411 129 397)(116 398 130 412)(117 413 131 399)(118 400 132 414)(119 415 133 401)(120 402 134 416)(121 417 135 403)(122 404 136 418)(123 419 137 405)(124 406 138 420)(125 393 139 407)(126 408 140 394)(169 372 183 386)(170 387 184 373)(171 374 185 388)(172 389 186 375)(173 376 187 390)(174 391 188 377)(175 378 189 392)(176 365 190 379)(177 380 191 366)(178 367 192 381)(179 382 193 368)(180 369 194 383)(181 384 195 370)(182 371 196 385)(197 259 211 273)(198 274 212 260)(199 261 213 275)(200 276 214 262)(201 263 215 277)(202 278 216 264)(203 265 217 279)(204 280 218 266)(205 267 219 253)(206 254 220 268)(207 269 221 255)(208 256 222 270)(209 271 223 257)(210 258 224 272)(309 430 323 444)(310 445 324 431)(311 432 325 446)(312 447 326 433)(313 434 327 448)(314 421 328 435)(315 436 329 422)(316 423 330 437)(317 438 331 424)(318 425 332 439)(319 440 333 426)(320 427 334 441)(321 442 335 428)(322 429 336 443)
(1 431 296 175)(2 432 297 176)(3 433 298 177)(4 434 299 178)(5 435 300 179)(6 436 301 180)(7 437 302 181)(8 438 303 182)(9 439 304 183)(10 440 305 184)(11 441 306 185)(12 442 307 186)(13 443 308 187)(14 444 281 188)(15 445 282 189)(16 446 283 190)(17 447 284 191)(18 448 285 192)(19 421 286 193)(20 422 287 194)(21 423 288 195)(22 424 289 196)(23 425 290 169)(24 426 291 170)(25 427 292 171)(26 428 293 172)(27 429 294 173)(28 430 295 174)(29 416 279 88)(30 417 280 89)(31 418 253 90)(32 419 254 91)(33 420 255 92)(34 393 256 93)(35 394 257 94)(36 395 258 95)(37 396 259 96)(38 397 260 97)(39 398 261 98)(40 399 262 99)(41 400 263 100)(42 401 264 101)(43 402 265 102)(44 403 266 103)(45 404 267 104)(46 405 268 105)(47 406 269 106)(48 407 270 107)(49 408 271 108)(50 409 272 109)(51 410 273 110)(52 411 274 111)(53 412 275 112)(54 413 276 85)(55 414 277 86)(56 415 278 87)(57 371 240 331)(58 372 241 332)(59 373 242 333)(60 374 243 334)(61 375 244 335)(62 376 245 336)(63 377 246 309)(64 378 247 310)(65 379 248 311)(66 380 249 312)(67 381 250 313)(68 382 251 314)(69 383 252 315)(70 384 225 316)(71 385 226 317)(72 386 227 318)(73 387 228 319)(74 388 229 320)(75 389 230 321)(76 390 231 322)(77 391 232 323)(78 392 233 324)(79 365 234 325)(80 366 235 326)(81 367 236 327)(82 368 237 328)(83 369 238 329)(84 370 239 330)(113 224 165 342)(114 197 166 343)(115 198 167 344)(116 199 168 345)(117 200 141 346)(118 201 142 347)(119 202 143 348)(120 203 144 349)(121 204 145 350)(122 205 146 351)(123 206 147 352)(124 207 148 353)(125 208 149 354)(126 209 150 355)(127 210 151 356)(128 211 152 357)(129 212 153 358)(130 213 154 359)(131 214 155 360)(132 215 156 361)(133 216 157 362)(134 217 158 363)(135 218 159 364)(136 219 160 337)(137 220 161 338)(138 221 162 339)(139 222 163 340)(140 223 164 341)
(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)
(1 276 189 399 296 54 445 99)(2 39 190 112 297 261 446 412)(3 274 191 397 298 52 447 97)(4 37 192 110 299 259 448 410)(5 272 193 395 300 50 421 95)(6 35 194 108 301 257 422 408)(7 270 195 393 302 48 423 93)(8 33 196 106 303 255 424 406)(9 268 169 419 304 46 425 91)(10 31 170 104 305 253 426 404)(11 266 171 417 306 44 427 89)(12 29 172 102 307 279 428 402)(13 264 173 415 308 42 429 87)(14 55 174 100 281 277 430 400)(15 262 175 413 282 40 431 85)(16 53 176 98 283 275 432 398)(17 260 177 411 284 38 433 111)(18 51 178 96 285 273 434 396)(19 258 179 409 286 36 435 109)(20 49 180 94 287 271 436 394)(21 256 181 407 288 34 437 107)(22 47 182 92 289 269 438 420)(23 254 183 405 290 32 439 105)(24 45 184 90 291 267 440 418)(25 280 185 403 292 30 441 103)(26 43 186 88 293 265 442 416)(27 278 187 401 294 56 443 101)(28 41 188 86 295 263 444 414)(57 221 317 124 240 339 385 148)(58 352 318 161 241 206 386 137)(59 219 319 122 242 337 387 146)(60 350 320 159 243 204 388 135)(61 217 321 120 244 363 389 144)(62 348 322 157 245 202 390 133)(63 215 323 118 246 361 391 142)(64 346 324 155 247 200 392 131)(65 213 325 116 248 359 365 168)(66 344 326 153 249 198 366 129)(67 211 327 114 250 357 367 166)(68 342 328 151 251 224 368 127)(69 209 329 140 252 355 369 164)(70 340 330 149 225 222 370 125)(71 207 331 138 226 353 371 162)(72 338 332 147 227 220 372 123)(73 205 333 136 228 351 373 160)(74 364 334 145 229 218 374 121)(75 203 335 134 230 349 375 158)(76 362 336 143 231 216 376 119)(77 201 309 132 232 347 377 156)(78 360 310 141 233 214 378 117)(79 199 311 130 234 345 379 154)(80 358 312 167 235 212 380 115)(81 197 313 128 236 343 381 152)(82 356 314 165 237 210 382 113)(83 223 315 126 238 341 383 150)(84 354 316 163 239 208 384 139)
G:=sub<Sym(448)| (1,247,15,233)(2,234,16,248)(3,249,17,235)(4,236,18,250)(5,251,19,237)(6,238,20,252)(7,225,21,239)(8,240,22,226)(9,227,23,241)(10,242,24,228)(11,229,25,243)(12,244,26,230)(13,231,27,245)(14,246,28,232)(29,349,43,363)(30,364,44,350)(31,351,45,337)(32,338,46,352)(33,353,47,339)(34,340,48,354)(35,355,49,341)(36,342,50,356)(37,357,51,343)(38,344,52,358)(39,359,53,345)(40,346,54,360)(41,361,55,347)(42,348,56,362)(57,289,71,303)(58,304,72,290)(59,291,73,305)(60,306,74,292)(61,293,75,307)(62,308,76,294)(63,295,77,281)(64,282,78,296)(65,297,79,283)(66,284,80,298)(67,299,81,285)(68,286,82,300)(69,301,83,287)(70,288,84,302)(85,155,99,141)(86,142,100,156)(87,157,101,143)(88,144,102,158)(89,159,103,145)(90,146,104,160)(91,161,105,147)(92,148,106,162)(93,163,107,149)(94,150,108,164)(95,165,109,151)(96,152,110,166)(97,167,111,153)(98,154,112,168)(113,409,127,395)(114,396,128,410)(115,411,129,397)(116,398,130,412)(117,413,131,399)(118,400,132,414)(119,415,133,401)(120,402,134,416)(121,417,135,403)(122,404,136,418)(123,419,137,405)(124,406,138,420)(125,393,139,407)(126,408,140,394)(169,372,183,386)(170,387,184,373)(171,374,185,388)(172,389,186,375)(173,376,187,390)(174,391,188,377)(175,378,189,392)(176,365,190,379)(177,380,191,366)(178,367,192,381)(179,382,193,368)(180,369,194,383)(181,384,195,370)(182,371,196,385)(197,259,211,273)(198,274,212,260)(199,261,213,275)(200,276,214,262)(201,263,215,277)(202,278,216,264)(203,265,217,279)(204,280,218,266)(205,267,219,253)(206,254,220,268)(207,269,221,255)(208,256,222,270)(209,271,223,257)(210,258,224,272)(309,430,323,444)(310,445,324,431)(311,432,325,446)(312,447,326,433)(313,434,327,448)(314,421,328,435)(315,436,329,422)(316,423,330,437)(317,438,331,424)(318,425,332,439)(319,440,333,426)(320,427,334,441)(321,442,335,428)(322,429,336,443), (1,431,296,175)(2,432,297,176)(3,433,298,177)(4,434,299,178)(5,435,300,179)(6,436,301,180)(7,437,302,181)(8,438,303,182)(9,439,304,183)(10,440,305,184)(11,441,306,185)(12,442,307,186)(13,443,308,187)(14,444,281,188)(15,445,282,189)(16,446,283,190)(17,447,284,191)(18,448,285,192)(19,421,286,193)(20,422,287,194)(21,423,288,195)(22,424,289,196)(23,425,290,169)(24,426,291,170)(25,427,292,171)(26,428,293,172)(27,429,294,173)(28,430,295,174)(29,416,279,88)(30,417,280,89)(31,418,253,90)(32,419,254,91)(33,420,255,92)(34,393,256,93)(35,394,257,94)(36,395,258,95)(37,396,259,96)(38,397,260,97)(39,398,261,98)(40,399,262,99)(41,400,263,100)(42,401,264,101)(43,402,265,102)(44,403,266,103)(45,404,267,104)(46,405,268,105)(47,406,269,106)(48,407,270,107)(49,408,271,108)(50,409,272,109)(51,410,273,110)(52,411,274,111)(53,412,275,112)(54,413,276,85)(55,414,277,86)(56,415,278,87)(57,371,240,331)(58,372,241,332)(59,373,242,333)(60,374,243,334)(61,375,244,335)(62,376,245,336)(63,377,246,309)(64,378,247,310)(65,379,248,311)(66,380,249,312)(67,381,250,313)(68,382,251,314)(69,383,252,315)(70,384,225,316)(71,385,226,317)(72,386,227,318)(73,387,228,319)(74,388,229,320)(75,389,230,321)(76,390,231,322)(77,391,232,323)(78,392,233,324)(79,365,234,325)(80,366,235,326)(81,367,236,327)(82,368,237,328)(83,369,238,329)(84,370,239,330)(113,224,165,342)(114,197,166,343)(115,198,167,344)(116,199,168,345)(117,200,141,346)(118,201,142,347)(119,202,143,348)(120,203,144,349)(121,204,145,350)(122,205,146,351)(123,206,147,352)(124,207,148,353)(125,208,149,354)(126,209,150,355)(127,210,151,356)(128,211,152,357)(129,212,153,358)(130,213,154,359)(131,214,155,360)(132,215,156,361)(133,216,157,362)(134,217,158,363)(135,218,159,364)(136,219,160,337)(137,220,161,338)(138,221,162,339)(139,222,163,340)(140,223,164,341), (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), (1,276,189,399,296,54,445,99)(2,39,190,112,297,261,446,412)(3,274,191,397,298,52,447,97)(4,37,192,110,299,259,448,410)(5,272,193,395,300,50,421,95)(6,35,194,108,301,257,422,408)(7,270,195,393,302,48,423,93)(8,33,196,106,303,255,424,406)(9,268,169,419,304,46,425,91)(10,31,170,104,305,253,426,404)(11,266,171,417,306,44,427,89)(12,29,172,102,307,279,428,402)(13,264,173,415,308,42,429,87)(14,55,174,100,281,277,430,400)(15,262,175,413,282,40,431,85)(16,53,176,98,283,275,432,398)(17,260,177,411,284,38,433,111)(18,51,178,96,285,273,434,396)(19,258,179,409,286,36,435,109)(20,49,180,94,287,271,436,394)(21,256,181,407,288,34,437,107)(22,47,182,92,289,269,438,420)(23,254,183,405,290,32,439,105)(24,45,184,90,291,267,440,418)(25,280,185,403,292,30,441,103)(26,43,186,88,293,265,442,416)(27,278,187,401,294,56,443,101)(28,41,188,86,295,263,444,414)(57,221,317,124,240,339,385,148)(58,352,318,161,241,206,386,137)(59,219,319,122,242,337,387,146)(60,350,320,159,243,204,388,135)(61,217,321,120,244,363,389,144)(62,348,322,157,245,202,390,133)(63,215,323,118,246,361,391,142)(64,346,324,155,247,200,392,131)(65,213,325,116,248,359,365,168)(66,344,326,153,249,198,366,129)(67,211,327,114,250,357,367,166)(68,342,328,151,251,224,368,127)(69,209,329,140,252,355,369,164)(70,340,330,149,225,222,370,125)(71,207,331,138,226,353,371,162)(72,338,332,147,227,220,372,123)(73,205,333,136,228,351,373,160)(74,364,334,145,229,218,374,121)(75,203,335,134,230,349,375,158)(76,362,336,143,231,216,376,119)(77,201,309,132,232,347,377,156)(78,360,310,141,233,214,378,117)(79,199,311,130,234,345,379,154)(80,358,312,167,235,212,380,115)(81,197,313,128,236,343,381,152)(82,356,314,165,237,210,382,113)(83,223,315,126,238,341,383,150)(84,354,316,163,239,208,384,139)>;
G:=Group( (1,247,15,233)(2,234,16,248)(3,249,17,235)(4,236,18,250)(5,251,19,237)(6,238,20,252)(7,225,21,239)(8,240,22,226)(9,227,23,241)(10,242,24,228)(11,229,25,243)(12,244,26,230)(13,231,27,245)(14,246,28,232)(29,349,43,363)(30,364,44,350)(31,351,45,337)(32,338,46,352)(33,353,47,339)(34,340,48,354)(35,355,49,341)(36,342,50,356)(37,357,51,343)(38,344,52,358)(39,359,53,345)(40,346,54,360)(41,361,55,347)(42,348,56,362)(57,289,71,303)(58,304,72,290)(59,291,73,305)(60,306,74,292)(61,293,75,307)(62,308,76,294)(63,295,77,281)(64,282,78,296)(65,297,79,283)(66,284,80,298)(67,299,81,285)(68,286,82,300)(69,301,83,287)(70,288,84,302)(85,155,99,141)(86,142,100,156)(87,157,101,143)(88,144,102,158)(89,159,103,145)(90,146,104,160)(91,161,105,147)(92,148,106,162)(93,163,107,149)(94,150,108,164)(95,165,109,151)(96,152,110,166)(97,167,111,153)(98,154,112,168)(113,409,127,395)(114,396,128,410)(115,411,129,397)(116,398,130,412)(117,413,131,399)(118,400,132,414)(119,415,133,401)(120,402,134,416)(121,417,135,403)(122,404,136,418)(123,419,137,405)(124,406,138,420)(125,393,139,407)(126,408,140,394)(169,372,183,386)(170,387,184,373)(171,374,185,388)(172,389,186,375)(173,376,187,390)(174,391,188,377)(175,378,189,392)(176,365,190,379)(177,380,191,366)(178,367,192,381)(179,382,193,368)(180,369,194,383)(181,384,195,370)(182,371,196,385)(197,259,211,273)(198,274,212,260)(199,261,213,275)(200,276,214,262)(201,263,215,277)(202,278,216,264)(203,265,217,279)(204,280,218,266)(205,267,219,253)(206,254,220,268)(207,269,221,255)(208,256,222,270)(209,271,223,257)(210,258,224,272)(309,430,323,444)(310,445,324,431)(311,432,325,446)(312,447,326,433)(313,434,327,448)(314,421,328,435)(315,436,329,422)(316,423,330,437)(317,438,331,424)(318,425,332,439)(319,440,333,426)(320,427,334,441)(321,442,335,428)(322,429,336,443), (1,431,296,175)(2,432,297,176)(3,433,298,177)(4,434,299,178)(5,435,300,179)(6,436,301,180)(7,437,302,181)(8,438,303,182)(9,439,304,183)(10,440,305,184)(11,441,306,185)(12,442,307,186)(13,443,308,187)(14,444,281,188)(15,445,282,189)(16,446,283,190)(17,447,284,191)(18,448,285,192)(19,421,286,193)(20,422,287,194)(21,423,288,195)(22,424,289,196)(23,425,290,169)(24,426,291,170)(25,427,292,171)(26,428,293,172)(27,429,294,173)(28,430,295,174)(29,416,279,88)(30,417,280,89)(31,418,253,90)(32,419,254,91)(33,420,255,92)(34,393,256,93)(35,394,257,94)(36,395,258,95)(37,396,259,96)(38,397,260,97)(39,398,261,98)(40,399,262,99)(41,400,263,100)(42,401,264,101)(43,402,265,102)(44,403,266,103)(45,404,267,104)(46,405,268,105)(47,406,269,106)(48,407,270,107)(49,408,271,108)(50,409,272,109)(51,410,273,110)(52,411,274,111)(53,412,275,112)(54,413,276,85)(55,414,277,86)(56,415,278,87)(57,371,240,331)(58,372,241,332)(59,373,242,333)(60,374,243,334)(61,375,244,335)(62,376,245,336)(63,377,246,309)(64,378,247,310)(65,379,248,311)(66,380,249,312)(67,381,250,313)(68,382,251,314)(69,383,252,315)(70,384,225,316)(71,385,226,317)(72,386,227,318)(73,387,228,319)(74,388,229,320)(75,389,230,321)(76,390,231,322)(77,391,232,323)(78,392,233,324)(79,365,234,325)(80,366,235,326)(81,367,236,327)(82,368,237,328)(83,369,238,329)(84,370,239,330)(113,224,165,342)(114,197,166,343)(115,198,167,344)(116,199,168,345)(117,200,141,346)(118,201,142,347)(119,202,143,348)(120,203,144,349)(121,204,145,350)(122,205,146,351)(123,206,147,352)(124,207,148,353)(125,208,149,354)(126,209,150,355)(127,210,151,356)(128,211,152,357)(129,212,153,358)(130,213,154,359)(131,214,155,360)(132,215,156,361)(133,216,157,362)(134,217,158,363)(135,218,159,364)(136,219,160,337)(137,220,161,338)(138,221,162,339)(139,222,163,340)(140,223,164,341), (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), (1,276,189,399,296,54,445,99)(2,39,190,112,297,261,446,412)(3,274,191,397,298,52,447,97)(4,37,192,110,299,259,448,410)(5,272,193,395,300,50,421,95)(6,35,194,108,301,257,422,408)(7,270,195,393,302,48,423,93)(8,33,196,106,303,255,424,406)(9,268,169,419,304,46,425,91)(10,31,170,104,305,253,426,404)(11,266,171,417,306,44,427,89)(12,29,172,102,307,279,428,402)(13,264,173,415,308,42,429,87)(14,55,174,100,281,277,430,400)(15,262,175,413,282,40,431,85)(16,53,176,98,283,275,432,398)(17,260,177,411,284,38,433,111)(18,51,178,96,285,273,434,396)(19,258,179,409,286,36,435,109)(20,49,180,94,287,271,436,394)(21,256,181,407,288,34,437,107)(22,47,182,92,289,269,438,420)(23,254,183,405,290,32,439,105)(24,45,184,90,291,267,440,418)(25,280,185,403,292,30,441,103)(26,43,186,88,293,265,442,416)(27,278,187,401,294,56,443,101)(28,41,188,86,295,263,444,414)(57,221,317,124,240,339,385,148)(58,352,318,161,241,206,386,137)(59,219,319,122,242,337,387,146)(60,350,320,159,243,204,388,135)(61,217,321,120,244,363,389,144)(62,348,322,157,245,202,390,133)(63,215,323,118,246,361,391,142)(64,346,324,155,247,200,392,131)(65,213,325,116,248,359,365,168)(66,344,326,153,249,198,366,129)(67,211,327,114,250,357,367,166)(68,342,328,151,251,224,368,127)(69,209,329,140,252,355,369,164)(70,340,330,149,225,222,370,125)(71,207,331,138,226,353,371,162)(72,338,332,147,227,220,372,123)(73,205,333,136,228,351,373,160)(74,364,334,145,229,218,374,121)(75,203,335,134,230,349,375,158)(76,362,336,143,231,216,376,119)(77,201,309,132,232,347,377,156)(78,360,310,141,233,214,378,117)(79,199,311,130,234,345,379,154)(80,358,312,167,235,212,380,115)(81,197,313,128,236,343,381,152)(82,356,314,165,237,210,382,113)(83,223,315,126,238,341,383,150)(84,354,316,163,239,208,384,139) );
G=PermutationGroup([[(1,247,15,233),(2,234,16,248),(3,249,17,235),(4,236,18,250),(5,251,19,237),(6,238,20,252),(7,225,21,239),(8,240,22,226),(9,227,23,241),(10,242,24,228),(11,229,25,243),(12,244,26,230),(13,231,27,245),(14,246,28,232),(29,349,43,363),(30,364,44,350),(31,351,45,337),(32,338,46,352),(33,353,47,339),(34,340,48,354),(35,355,49,341),(36,342,50,356),(37,357,51,343),(38,344,52,358),(39,359,53,345),(40,346,54,360),(41,361,55,347),(42,348,56,362),(57,289,71,303),(58,304,72,290),(59,291,73,305),(60,306,74,292),(61,293,75,307),(62,308,76,294),(63,295,77,281),(64,282,78,296),(65,297,79,283),(66,284,80,298),(67,299,81,285),(68,286,82,300),(69,301,83,287),(70,288,84,302),(85,155,99,141),(86,142,100,156),(87,157,101,143),(88,144,102,158),(89,159,103,145),(90,146,104,160),(91,161,105,147),(92,148,106,162),(93,163,107,149),(94,150,108,164),(95,165,109,151),(96,152,110,166),(97,167,111,153),(98,154,112,168),(113,409,127,395),(114,396,128,410),(115,411,129,397),(116,398,130,412),(117,413,131,399),(118,400,132,414),(119,415,133,401),(120,402,134,416),(121,417,135,403),(122,404,136,418),(123,419,137,405),(124,406,138,420),(125,393,139,407),(126,408,140,394),(169,372,183,386),(170,387,184,373),(171,374,185,388),(172,389,186,375),(173,376,187,390),(174,391,188,377),(175,378,189,392),(176,365,190,379),(177,380,191,366),(178,367,192,381),(179,382,193,368),(180,369,194,383),(181,384,195,370),(182,371,196,385),(197,259,211,273),(198,274,212,260),(199,261,213,275),(200,276,214,262),(201,263,215,277),(202,278,216,264),(203,265,217,279),(204,280,218,266),(205,267,219,253),(206,254,220,268),(207,269,221,255),(208,256,222,270),(209,271,223,257),(210,258,224,272),(309,430,323,444),(310,445,324,431),(311,432,325,446),(312,447,326,433),(313,434,327,448),(314,421,328,435),(315,436,329,422),(316,423,330,437),(317,438,331,424),(318,425,332,439),(319,440,333,426),(320,427,334,441),(321,442,335,428),(322,429,336,443)], [(1,431,296,175),(2,432,297,176),(3,433,298,177),(4,434,299,178),(5,435,300,179),(6,436,301,180),(7,437,302,181),(8,438,303,182),(9,439,304,183),(10,440,305,184),(11,441,306,185),(12,442,307,186),(13,443,308,187),(14,444,281,188),(15,445,282,189),(16,446,283,190),(17,447,284,191),(18,448,285,192),(19,421,286,193),(20,422,287,194),(21,423,288,195),(22,424,289,196),(23,425,290,169),(24,426,291,170),(25,427,292,171),(26,428,293,172),(27,429,294,173),(28,430,295,174),(29,416,279,88),(30,417,280,89),(31,418,253,90),(32,419,254,91),(33,420,255,92),(34,393,256,93),(35,394,257,94),(36,395,258,95),(37,396,259,96),(38,397,260,97),(39,398,261,98),(40,399,262,99),(41,400,263,100),(42,401,264,101),(43,402,265,102),(44,403,266,103),(45,404,267,104),(46,405,268,105),(47,406,269,106),(48,407,270,107),(49,408,271,108),(50,409,272,109),(51,410,273,110),(52,411,274,111),(53,412,275,112),(54,413,276,85),(55,414,277,86),(56,415,278,87),(57,371,240,331),(58,372,241,332),(59,373,242,333),(60,374,243,334),(61,375,244,335),(62,376,245,336),(63,377,246,309),(64,378,247,310),(65,379,248,311),(66,380,249,312),(67,381,250,313),(68,382,251,314),(69,383,252,315),(70,384,225,316),(71,385,226,317),(72,386,227,318),(73,387,228,319),(74,388,229,320),(75,389,230,321),(76,390,231,322),(77,391,232,323),(78,392,233,324),(79,365,234,325),(80,366,235,326),(81,367,236,327),(82,368,237,328),(83,369,238,329),(84,370,239,330),(113,224,165,342),(114,197,166,343),(115,198,167,344),(116,199,168,345),(117,200,141,346),(118,201,142,347),(119,202,143,348),(120,203,144,349),(121,204,145,350),(122,205,146,351),(123,206,147,352),(124,207,148,353),(125,208,149,354),(126,209,150,355),(127,210,151,356),(128,211,152,357),(129,212,153,358),(130,213,154,359),(131,214,155,360),(132,215,156,361),(133,216,157,362),(134,217,158,363),(135,218,159,364),(136,219,160,337),(137,220,161,338),(138,221,162,339),(139,222,163,340),(140,223,164,341)], [(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)], [(1,276,189,399,296,54,445,99),(2,39,190,112,297,261,446,412),(3,274,191,397,298,52,447,97),(4,37,192,110,299,259,448,410),(5,272,193,395,300,50,421,95),(6,35,194,108,301,257,422,408),(7,270,195,393,302,48,423,93),(8,33,196,106,303,255,424,406),(9,268,169,419,304,46,425,91),(10,31,170,104,305,253,426,404),(11,266,171,417,306,44,427,89),(12,29,172,102,307,279,428,402),(13,264,173,415,308,42,429,87),(14,55,174,100,281,277,430,400),(15,262,175,413,282,40,431,85),(16,53,176,98,283,275,432,398),(17,260,177,411,284,38,433,111),(18,51,178,96,285,273,434,396),(19,258,179,409,286,36,435,109),(20,49,180,94,287,271,436,394),(21,256,181,407,288,34,437,107),(22,47,182,92,289,269,438,420),(23,254,183,405,290,32,439,105),(24,45,184,90,291,267,440,418),(25,280,185,403,292,30,441,103),(26,43,186,88,293,265,442,416),(27,278,187,401,294,56,443,101),(28,41,188,86,295,263,444,414),(57,221,317,124,240,339,385,148),(58,352,318,161,241,206,386,137),(59,219,319,122,242,337,387,146),(60,350,320,159,243,204,388,135),(61,217,321,120,244,363,389,144),(62,348,322,157,245,202,390,133),(63,215,323,118,246,361,391,142),(64,346,324,155,247,200,392,131),(65,213,325,116,248,359,365,168),(66,344,326,153,249,198,366,129),(67,211,327,114,250,357,367,166),(68,342,328,151,251,224,368,127),(69,209,329,140,252,355,369,164),(70,340,330,149,225,222,370,125),(71,207,331,138,226,353,371,162),(72,338,332,147,227,220,372,123),(73,205,333,136,228,351,373,160),(74,364,334,145,229,218,374,121),(75,203,335,134,230,349,375,158),(76,362,336,143,231,216,376,119),(77,201,309,132,232,347,377,156),(78,360,310,141,233,214,378,117),(79,199,311,130,234,345,379,154),(80,358,312,167,235,212,380,115),(81,197,313,128,236,343,381,152),(82,356,314,165,237,210,382,113),(83,223,315,126,238,341,383,150),(84,354,316,163,239,208,384,139)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | - | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | Q8 | D7 | M4(2) | C4○D4 | D14 | Dic7 | Dic7 | C8○D4 | C4.Dic7 | Q8×D7 | Q8⋊2D7 | Q8.Dic7 |
| kernel | C42.210D14 | C4×C7⋊C8 | C42.D7 | C28⋊C8 | Q8×C28 | C7×C4⋊C4 | Q8×C14 | C7⋊C8 | C4×Q8 | C28 | C28 | C42 | C4⋊C4 | C2×Q8 | C14 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 3 | 1 | 6 | 2 | 2 | 3 | 4 | 2 | 9 | 9 | 3 | 4 | 24 | 3 | 3 | 6 |
Matrix representation of C42.210D14 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 25 | 47 |
| 0 | 0 | 54 | 88 |
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 53 | 0 | 0 |
| 60 | 85 | 0 | 0 |
| 0 | 0 | 96 | 87 |
| 0 | 0 | 72 | 17 |
| 102 | 93 | 0 | 0 |
| 107 | 11 | 0 | 0 |
| 0 | 0 | 17 | 26 |
| 0 | 0 | 41 | 96 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,25,54,0,0,47,88],[98,0,0,0,0,98,0,0,0,0,1,0,0,0,0,1],[60,60,0,0,53,85,0,0,0,0,96,72,0,0,87,17],[102,107,0,0,93,11,0,0,0,0,17,41,0,0,26,96] >;
C42.210D14 in GAP, Magma, Sage, TeX
C_4^2._{210}D_{14} % in TeX
G:=Group("C4^2.210D14"); // GroupNames label
G:=SmallGroup(448,558);
// by ID
G=gap.SmallGroup(448,558);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,120,758,219,100,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=a^2,d^2=a^2*b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^13>;
// generators/relations