metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C28)⋊3C8, C14.5(C4×C8), C14.5(C4⋊C8), C28.42(C4⋊C4), (C2×C28).63Q8, (C2×C42).1D7, (C2×C4).163D28, (C2×C28).487D4, C14.3(C8⋊C4), C2.2(C28⋊C8), C4.45(D14⋊C4), (C22×C28).13C4, (C2×C14).16C42, (C2×C4).53Dic14, (C22×C4).7Dic7, C28.60(C22⋊C4), C14.12(C22⋊C8), C4.29(Dic7⋊C4), (C22×C4).451D14, (C2×C14).21M4(2), C22.16(C4×Dic7), C23.37(C2×Dic7), C2.1(C28.55D4), C22.17(C4⋊Dic7), C7⋊1(C22.7C42), C2.2(C42.D7), C22.7(C4.Dic7), C14.2(C2.C42), C2.1(C14.C42), (C22×C28).548C22, C22.22(C23.D7), C2.5(C4×C7⋊C8), (C2×C7⋊C8)⋊10C4, (C2×C4)⋊2(C7⋊C8), (C2×C4×C28).24C2, C22.10(C2×C7⋊C8), (C2×C14).28(C2×C8), (C2×C4).170(C4×D7), (C22×C7⋊C8).15C2, (C2×C14).34(C4⋊C4), (C2×C28).244(C2×C4), (C2×C4).267(C7⋊D4), (C2×C14).84(C22⋊C4), (C22×C14).122(C2×C4), SmallGroup(448,81)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C28)⋊3C8
G = < a,b,c | a2=b28=c8=1, ab=ba, ac=ca, cbc-1=ab13 >
Subgroups: 292 in 118 conjugacy classes, 75 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C42, C2×C8, C22×C4, C22×C4, C28, C28, C2×C14, C2×C14, C2×C42, C22×C8, C7⋊C8, C2×C28, C2×C28, C2×C28, C22×C14, C22.7C42, C2×C7⋊C8, C2×C7⋊C8, C4×C28, C22×C28, C22×C28, C22×C7⋊C8, C2×C4×C28, (C2×C28)⋊3C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), Dic7, D14, C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C7⋊C8, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C22.7C42, C2×C7⋊C8, C4.Dic7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C4×C7⋊C8, C42.D7, C28⋊C8, C28.55D4, C14.C42, (C2×C28)⋊3C8
►Regular action on 448 points
Generators in S448
(1 177)(2 178)(3 179)(4 180)(5 181)(6 182)(7 183)(8 184)(9 185)(10 186)(11 187)(12 188)(13 189)(14 190)(15 191)(16 192)(17 193)(18 194)(19 195)(20 196)(21 169)(22 170)(23 171)(24 172)(25 173)(26 174)(27 175)(28 176)(29 364)(30 337)(31 338)(32 339)(33 340)(34 341)(35 342)(36 343)(37 344)(38 345)(39 346)(40 347)(41 348)(42 349)(43 350)(44 351)(45 352)(46 353)(47 354)(48 355)(49 356)(50 357)(51 358)(52 359)(53 360)(54 361)(55 362)(56 363)(57 413)(58 414)(59 415)(60 416)(61 417)(62 418)(63 419)(64 420)(65 393)(66 394)(67 395)(68 396)(69 397)(70 398)(71 399)(72 400)(73 401)(74 402)(75 403)(76 404)(77 405)(78 406)(79 407)(80 408)(81 409)(82 410)(83 411)(84 412)(85 378)(86 379)(87 380)(88 381)(89 382)(90 383)(91 384)(92 385)(93 386)(94 387)(95 388)(96 389)(97 390)(98 391)(99 392)(100 365)(101 366)(102 367)(103 368)(104 369)(105 370)(106 371)(107 372)(108 373)(109 374)(110 375)(111 376)(112 377)(113 294)(114 295)(115 296)(116 297)(117 298)(118 299)(119 300)(120 301)(121 302)(122 303)(123 304)(124 305)(125 306)(126 307)(127 308)(128 281)(129 282)(130 283)(131 284)(132 285)(133 286)(134 287)(135 288)(136 289)(137 290)(138 291)(139 292)(140 293)(141 321)(142 322)(143 323)(144 324)(145 325)(146 326)(147 327)(148 328)(149 329)(150 330)(151 331)(152 332)(153 333)(154 334)(155 335)(156 336)(157 309)(158 310)(159 311)(160 312)(161 313)(162 314)(163 315)(164 316)(165 317)(166 318)(167 319)(168 320)(197 279)(198 280)(199 253)(200 254)(201 255)(202 256)(203 257)(204 258)(205 259)(206 260)(207 261)(208 262)(209 263)(210 264)(211 265)(212 266)(213 267)(214 268)(215 269)(216 270)(217 271)(218 272)(219 273)(220 274)(221 275)(222 276)(223 277)(224 278)(225 439)(226 440)(227 441)(228 442)(229 443)(230 444)(231 445)(232 446)(233 447)(234 448)(235 421)(236 422)(237 423)(238 424)(239 425)(240 426)(241 427)(242 428)(243 429)(244 430)(245 431)(246 432)(247 433)(248 434)(249 435)(250 436)(251 437)(252 438)
(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 67 378 324 248 136 353 209)(2 408 379 157 249 302 354 276)(3 65 380 322 250 134 355 207)(4 406 381 155 251 300 356 274)(5 63 382 320 252 132 357 205)(6 404 383 153 225 298 358 272)(7 61 384 318 226 130 359 203)(8 402 385 151 227 296 360 270)(9 59 386 316 228 128 361 201)(10 400 387 149 229 294 362 268)(11 57 388 314 230 126 363 199)(12 398 389 147 231 292 364 266)(13 83 390 312 232 124 337 197)(14 396 391 145 233 290 338 264)(15 81 392 310 234 122 339 223)(16 394 365 143 235 288 340 262)(17 79 366 336 236 120 341 221)(18 420 367 141 237 286 342 260)(19 77 368 334 238 118 343 219)(20 418 369 167 239 284 344 258)(21 75 370 332 240 116 345 217)(22 416 371 165 241 282 346 256)(23 73 372 330 242 114 347 215)(24 414 373 163 243 308 348 254)(25 71 374 328 244 140 349 213)(26 412 375 161 245 306 350 280)(27 69 376 326 246 138 351 211)(28 410 377 159 247 304 352 278)(29 212 188 70 96 327 445 139)(30 279 189 411 97 160 446 305)(31 210 190 68 98 325 447 137)(32 277 191 409 99 158 448 303)(33 208 192 66 100 323 421 135)(34 275 193 407 101 156 422 301)(35 206 194 64 102 321 423 133)(36 273 195 405 103 154 424 299)(37 204 196 62 104 319 425 131)(38 271 169 403 105 152 426 297)(39 202 170 60 106 317 427 129)(40 269 171 401 107 150 428 295)(41 200 172 58 108 315 429 127)(42 267 173 399 109 148 430 293)(43 198 174 84 110 313 431 125)(44 265 175 397 111 146 432 291)(45 224 176 82 112 311 433 123)(46 263 177 395 85 144 434 289)(47 222 178 80 86 309 435 121)(48 261 179 393 87 142 436 287)(49 220 180 78 88 335 437 119)(50 259 181 419 89 168 438 285)(51 218 182 76 90 333 439 117)(52 257 183 417 91 166 440 283)(53 216 184 74 92 331 441 115)(54 255 185 415 93 164 442 281)(55 214 186 72 94 329 443 113)(56 253 187 413 95 162 444 307)
G:=sub<Sym(448)| (1,177)(2,178)(3,179)(4,180)(5,181)(6,182)(7,183)(8,184)(9,185)(10,186)(11,187)(12,188)(13,189)(14,190)(15,191)(16,192)(17,193)(18,194)(19,195)(20,196)(21,169)(22,170)(23,171)(24,172)(25,173)(26,174)(27,175)(28,176)(29,364)(30,337)(31,338)(32,339)(33,340)(34,341)(35,342)(36,343)(37,344)(38,345)(39,346)(40,347)(41,348)(42,349)(43,350)(44,351)(45,352)(46,353)(47,354)(48,355)(49,356)(50,357)(51,358)(52,359)(53,360)(54,361)(55,362)(56,363)(57,413)(58,414)(59,415)(60,416)(61,417)(62,418)(63,419)(64,420)(65,393)(66,394)(67,395)(68,396)(69,397)(70,398)(71,399)(72,400)(73,401)(74,402)(75,403)(76,404)(77,405)(78,406)(79,407)(80,408)(81,409)(82,410)(83,411)(84,412)(85,378)(86,379)(87,380)(88,381)(89,382)(90,383)(91,384)(92,385)(93,386)(94,387)(95,388)(96,389)(97,390)(98,391)(99,392)(100,365)(101,366)(102,367)(103,368)(104,369)(105,370)(106,371)(107,372)(108,373)(109,374)(110,375)(111,376)(112,377)(113,294)(114,295)(115,296)(116,297)(117,298)(118,299)(119,300)(120,301)(121,302)(122,303)(123,304)(124,305)(125,306)(126,307)(127,308)(128,281)(129,282)(130,283)(131,284)(132,285)(133,286)(134,287)(135,288)(136,289)(137,290)(138,291)(139,292)(140,293)(141,321)(142,322)(143,323)(144,324)(145,325)(146,326)(147,327)(148,328)(149,329)(150,330)(151,331)(152,332)(153,333)(154,334)(155,335)(156,336)(157,309)(158,310)(159,311)(160,312)(161,313)(162,314)(163,315)(164,316)(165,317)(166,318)(167,319)(168,320)(197,279)(198,280)(199,253)(200,254)(201,255)(202,256)(203,257)(204,258)(205,259)(206,260)(207,261)(208,262)(209,263)(210,264)(211,265)(212,266)(213,267)(214,268)(215,269)(216,270)(217,271)(218,272)(219,273)(220,274)(221,275)(222,276)(223,277)(224,278)(225,439)(226,440)(227,441)(228,442)(229,443)(230,444)(231,445)(232,446)(233,447)(234,448)(235,421)(236,422)(237,423)(238,424)(239,425)(240,426)(241,427)(242,428)(243,429)(244,430)(245,431)(246,432)(247,433)(248,434)(249,435)(250,436)(251,437)(252,438), (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), (1,67,378,324,248,136,353,209)(2,408,379,157,249,302,354,276)(3,65,380,322,250,134,355,207)(4,406,381,155,251,300,356,274)(5,63,382,320,252,132,357,205)(6,404,383,153,225,298,358,272)(7,61,384,318,226,130,359,203)(8,402,385,151,227,296,360,270)(9,59,386,316,228,128,361,201)(10,400,387,149,229,294,362,268)(11,57,388,314,230,126,363,199)(12,398,389,147,231,292,364,266)(13,83,390,312,232,124,337,197)(14,396,391,145,233,290,338,264)(15,81,392,310,234,122,339,223)(16,394,365,143,235,288,340,262)(17,79,366,336,236,120,341,221)(18,420,367,141,237,286,342,260)(19,77,368,334,238,118,343,219)(20,418,369,167,239,284,344,258)(21,75,370,332,240,116,345,217)(22,416,371,165,241,282,346,256)(23,73,372,330,242,114,347,215)(24,414,373,163,243,308,348,254)(25,71,374,328,244,140,349,213)(26,412,375,161,245,306,350,280)(27,69,376,326,246,138,351,211)(28,410,377,159,247,304,352,278)(29,212,188,70,96,327,445,139)(30,279,189,411,97,160,446,305)(31,210,190,68,98,325,447,137)(32,277,191,409,99,158,448,303)(33,208,192,66,100,323,421,135)(34,275,193,407,101,156,422,301)(35,206,194,64,102,321,423,133)(36,273,195,405,103,154,424,299)(37,204,196,62,104,319,425,131)(38,271,169,403,105,152,426,297)(39,202,170,60,106,317,427,129)(40,269,171,401,107,150,428,295)(41,200,172,58,108,315,429,127)(42,267,173,399,109,148,430,293)(43,198,174,84,110,313,431,125)(44,265,175,397,111,146,432,291)(45,224,176,82,112,311,433,123)(46,263,177,395,85,144,434,289)(47,222,178,80,86,309,435,121)(48,261,179,393,87,142,436,287)(49,220,180,78,88,335,437,119)(50,259,181,419,89,168,438,285)(51,218,182,76,90,333,439,117)(52,257,183,417,91,166,440,283)(53,216,184,74,92,331,441,115)(54,255,185,415,93,164,442,281)(55,214,186,72,94,329,443,113)(56,253,187,413,95,162,444,307)>;
G:=Group( (1,177)(2,178)(3,179)(4,180)(5,181)(6,182)(7,183)(8,184)(9,185)(10,186)(11,187)(12,188)(13,189)(14,190)(15,191)(16,192)(17,193)(18,194)(19,195)(20,196)(21,169)(22,170)(23,171)(24,172)(25,173)(26,174)(27,175)(28,176)(29,364)(30,337)(31,338)(32,339)(33,340)(34,341)(35,342)(36,343)(37,344)(38,345)(39,346)(40,347)(41,348)(42,349)(43,350)(44,351)(45,352)(46,353)(47,354)(48,355)(49,356)(50,357)(51,358)(52,359)(53,360)(54,361)(55,362)(56,363)(57,413)(58,414)(59,415)(60,416)(61,417)(62,418)(63,419)(64,420)(65,393)(66,394)(67,395)(68,396)(69,397)(70,398)(71,399)(72,400)(73,401)(74,402)(75,403)(76,404)(77,405)(78,406)(79,407)(80,408)(81,409)(82,410)(83,411)(84,412)(85,378)(86,379)(87,380)(88,381)(89,382)(90,383)(91,384)(92,385)(93,386)(94,387)(95,388)(96,389)(97,390)(98,391)(99,392)(100,365)(101,366)(102,367)(103,368)(104,369)(105,370)(106,371)(107,372)(108,373)(109,374)(110,375)(111,376)(112,377)(113,294)(114,295)(115,296)(116,297)(117,298)(118,299)(119,300)(120,301)(121,302)(122,303)(123,304)(124,305)(125,306)(126,307)(127,308)(128,281)(129,282)(130,283)(131,284)(132,285)(133,286)(134,287)(135,288)(136,289)(137,290)(138,291)(139,292)(140,293)(141,321)(142,322)(143,323)(144,324)(145,325)(146,326)(147,327)(148,328)(149,329)(150,330)(151,331)(152,332)(153,333)(154,334)(155,335)(156,336)(157,309)(158,310)(159,311)(160,312)(161,313)(162,314)(163,315)(164,316)(165,317)(166,318)(167,319)(168,320)(197,279)(198,280)(199,253)(200,254)(201,255)(202,256)(203,257)(204,258)(205,259)(206,260)(207,261)(208,262)(209,263)(210,264)(211,265)(212,266)(213,267)(214,268)(215,269)(216,270)(217,271)(218,272)(219,273)(220,274)(221,275)(222,276)(223,277)(224,278)(225,439)(226,440)(227,441)(228,442)(229,443)(230,444)(231,445)(232,446)(233,447)(234,448)(235,421)(236,422)(237,423)(238,424)(239,425)(240,426)(241,427)(242,428)(243,429)(244,430)(245,431)(246,432)(247,433)(248,434)(249,435)(250,436)(251,437)(252,438), (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), (1,67,378,324,248,136,353,209)(2,408,379,157,249,302,354,276)(3,65,380,322,250,134,355,207)(4,406,381,155,251,300,356,274)(5,63,382,320,252,132,357,205)(6,404,383,153,225,298,358,272)(7,61,384,318,226,130,359,203)(8,402,385,151,227,296,360,270)(9,59,386,316,228,128,361,201)(10,400,387,149,229,294,362,268)(11,57,388,314,230,126,363,199)(12,398,389,147,231,292,364,266)(13,83,390,312,232,124,337,197)(14,396,391,145,233,290,338,264)(15,81,392,310,234,122,339,223)(16,394,365,143,235,288,340,262)(17,79,366,336,236,120,341,221)(18,420,367,141,237,286,342,260)(19,77,368,334,238,118,343,219)(20,418,369,167,239,284,344,258)(21,75,370,332,240,116,345,217)(22,416,371,165,241,282,346,256)(23,73,372,330,242,114,347,215)(24,414,373,163,243,308,348,254)(25,71,374,328,244,140,349,213)(26,412,375,161,245,306,350,280)(27,69,376,326,246,138,351,211)(28,410,377,159,247,304,352,278)(29,212,188,70,96,327,445,139)(30,279,189,411,97,160,446,305)(31,210,190,68,98,325,447,137)(32,277,191,409,99,158,448,303)(33,208,192,66,100,323,421,135)(34,275,193,407,101,156,422,301)(35,206,194,64,102,321,423,133)(36,273,195,405,103,154,424,299)(37,204,196,62,104,319,425,131)(38,271,169,403,105,152,426,297)(39,202,170,60,106,317,427,129)(40,269,171,401,107,150,428,295)(41,200,172,58,108,315,429,127)(42,267,173,399,109,148,430,293)(43,198,174,84,110,313,431,125)(44,265,175,397,111,146,432,291)(45,224,176,82,112,311,433,123)(46,263,177,395,85,144,434,289)(47,222,178,80,86,309,435,121)(48,261,179,393,87,142,436,287)(49,220,180,78,88,335,437,119)(50,259,181,419,89,168,438,285)(51,218,182,76,90,333,439,117)(52,257,183,417,91,166,440,283)(53,216,184,74,92,331,441,115)(54,255,185,415,93,164,442,281)(55,214,186,72,94,329,443,113)(56,253,187,413,95,162,444,307) );
G=PermutationGroup([[(1,177),(2,178),(3,179),(4,180),(5,181),(6,182),(7,183),(8,184),(9,185),(10,186),(11,187),(12,188),(13,189),(14,190),(15,191),(16,192),(17,193),(18,194),(19,195),(20,196),(21,169),(22,170),(23,171),(24,172),(25,173),(26,174),(27,175),(28,176),(29,364),(30,337),(31,338),(32,339),(33,340),(34,341),(35,342),(36,343),(37,344),(38,345),(39,346),(40,347),(41,348),(42,349),(43,350),(44,351),(45,352),(46,353),(47,354),(48,355),(49,356),(50,357),(51,358),(52,359),(53,360),(54,361),(55,362),(56,363),(57,413),(58,414),(59,415),(60,416),(61,417),(62,418),(63,419),(64,420),(65,393),(66,394),(67,395),(68,396),(69,397),(70,398),(71,399),(72,400),(73,401),(74,402),(75,403),(76,404),(77,405),(78,406),(79,407),(80,408),(81,409),(82,410),(83,411),(84,412),(85,378),(86,379),(87,380),(88,381),(89,382),(90,383),(91,384),(92,385),(93,386),(94,387),(95,388),(96,389),(97,390),(98,391),(99,392),(100,365),(101,366),(102,367),(103,368),(104,369),(105,370),(106,371),(107,372),(108,373),(109,374),(110,375),(111,376),(112,377),(113,294),(114,295),(115,296),(116,297),(117,298),(118,299),(119,300),(120,301),(121,302),(122,303),(123,304),(124,305),(125,306),(126,307),(127,308),(128,281),(129,282),(130,283),(131,284),(132,285),(133,286),(134,287),(135,288),(136,289),(137,290),(138,291),(139,292),(140,293),(141,321),(142,322),(143,323),(144,324),(145,325),(146,326),(147,327),(148,328),(149,329),(150,330),(151,331),(152,332),(153,333),(154,334),(155,335),(156,336),(157,309),(158,310),(159,311),(160,312),(161,313),(162,314),(163,315),(164,316),(165,317),(166,318),(167,319),(168,320),(197,279),(198,280),(199,253),(200,254),(201,255),(202,256),(203,257),(204,258),(205,259),(206,260),(207,261),(208,262),(209,263),(210,264),(211,265),(212,266),(213,267),(214,268),(215,269),(216,270),(217,271),(218,272),(219,273),(220,274),(221,275),(222,276),(223,277),(224,278),(225,439),(226,440),(227,441),(228,442),(229,443),(230,444),(231,445),(232,446),(233,447),(234,448),(235,421),(236,422),(237,423),(238,424),(239,425),(240,426),(241,427),(242,428),(243,429),(244,430),(245,431),(246,432),(247,433),(248,434),(249,435),(250,436),(251,437),(252,438)], [(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)], [(1,67,378,324,248,136,353,209),(2,408,379,157,249,302,354,276),(3,65,380,322,250,134,355,207),(4,406,381,155,251,300,356,274),(5,63,382,320,252,132,357,205),(6,404,383,153,225,298,358,272),(7,61,384,318,226,130,359,203),(8,402,385,151,227,296,360,270),(9,59,386,316,228,128,361,201),(10,400,387,149,229,294,362,268),(11,57,388,314,230,126,363,199),(12,398,389,147,231,292,364,266),(13,83,390,312,232,124,337,197),(14,396,391,145,233,290,338,264),(15,81,392,310,234,122,339,223),(16,394,365,143,235,288,340,262),(17,79,366,336,236,120,341,221),(18,420,367,141,237,286,342,260),(19,77,368,334,238,118,343,219),(20,418,369,167,239,284,344,258),(21,75,370,332,240,116,345,217),(22,416,371,165,241,282,346,256),(23,73,372,330,242,114,347,215),(24,414,373,163,243,308,348,254),(25,71,374,328,244,140,349,213),(26,412,375,161,245,306,350,280),(27,69,376,326,246,138,351,211),(28,410,377,159,247,304,352,278),(29,212,188,70,96,327,445,139),(30,279,189,411,97,160,446,305),(31,210,190,68,98,325,447,137),(32,277,191,409,99,158,448,303),(33,208,192,66,100,323,421,135),(34,275,193,407,101,156,422,301),(35,206,194,64,102,321,423,133),(36,273,195,405,103,154,424,299),(37,204,196,62,104,319,425,131),(38,271,169,403,105,152,426,297),(39,202,170,60,106,317,427,129),(40,269,171,401,107,150,428,295),(41,200,172,58,108,315,429,127),(42,267,173,399,109,148,430,293),(43,198,174,84,110,313,431,125),(44,265,175,397,111,146,432,291),(45,224,176,82,112,311,433,123),(46,263,177,395,85,144,434,289),(47,222,178,80,86,309,435,121),(48,261,179,393,87,142,436,287),(49,220,180,78,88,335,437,119),(50,259,181,419,89,168,438,285),(51,218,182,76,90,333,439,117),(52,257,183,417,91,166,440,283),(53,216,184,74,92,331,441,115),(54,255,185,415,93,164,442,281),(55,214,186,72,94,329,443,113),(56,253,187,413,95,162,444,307)]])
136 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 7A | 7B | 7C | 8A | ··· | 8P | 14A | ··· | 14U | 28A | ··· | 28BT |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | D7 | M4(2) | Dic7 | D14 | C7⋊C8 | Dic14 | C4×D7 | D28 | C7⋊D4 | C4.Dic7 |
| kernel | (C2×C28)⋊3C8 | C22×C7⋊C8 | C2×C4×C28 | C2×C7⋊C8 | C22×C28 | C2×C28 | C2×C28 | C2×C28 | C2×C42 | C2×C14 | C22×C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 8 | 4 | 16 | 3 | 1 | 3 | 4 | 6 | 3 | 24 | 6 | 12 | 6 | 12 | 24 |
Matrix representation of (C2×C28)⋊3C8 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 84 | 25 |
| 0 | 0 | 88 | 6 |
| 95 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 23 | 90 |
| 0 | 0 | 77 | 90 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,98,0,0,0,0,84,88,0,0,25,6],[95,0,0,0,0,15,0,0,0,0,23,77,0,0,90,90] >;
(C2×C28)⋊3C8 in GAP, Magma, Sage, TeX
(C_2\times C_{28})\rtimes_3C_8 % in TeX
G:=Group("(C2xC28):3C8"); // GroupNames label
G:=SmallGroup(448,81);
// by ID
G=gap.SmallGroup(448,81);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,184,18822]);
// Polycyclic
G:=Group<a,b,c|a^2=b^28=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^13>;
// generators/relations