metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C4).17D28, (C2×C28).28D4, (C2×Dic7).3Q8, C22.41(Q8×D7), (C22×C4).12D14, C22.78(C2×D28), C2.7(D14⋊2Q8), C2.5(C4.D28), C14.1(C4.4D4), C14.23(C22⋊Q8), (C22×C28).9C22, C22.86(C4○D28), C2.10(Dic7.Q8), C14.C42.7C2, C14.13(C42.C2), C2.C42.14D7, C23.355(C22×D7), C14.10(C42⋊2C2), C22.84(D4⋊2D7), (C22×C14).284C23, C7⋊2(C23.83C23), C2.9(C23.D14), C14.6(C22.D4), C2.6(C22.D28), (C22×Dic7).9C22, (C2×C14).94(C2×D4), (C2×C4⋊Dic7).8C2, (C2×C14).65(C2×Q8), (C2×Dic7⋊C4).19C2, (C2×C14).128(C4○D4), (C7×C2.C42).10C2, SmallGroup(448,193)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C4).17D28
G = < a,b,c,d | a2=b4=c28=1, d2=ab2, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=ab-1, dcd-1=b2c-1 >
Subgroups: 572 in 134 conjugacy classes, 57 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C4 [×9], C22 [×3], C22 [×4], C7, C2×C4 [×2], C2×C4 [×21], C23, C14 [×3], C14 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], Dic7 [×5], C28 [×4], C2×C14 [×3], C2×C14 [×4], C2.C42, C2.C42 [×4], C2×C4⋊C4 [×2], C2×Dic7 [×2], C2×Dic7 [×11], C2×C28 [×2], C2×C28 [×8], C22×C14, C23.83C23, Dic7⋊C4 [×2], C4⋊Dic7 [×2], C22×Dic7 [×2], C22×Dic7 [×2], C22×C28, C22×C28 [×2], C14.C42 [×4], C7×C2.C42, C2×Dic7⋊C4, C2×C4⋊Dic7, (C2×C4).17D28
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D7, C2×D4, C2×Q8, C4○D4 [×5], D14 [×3], C22⋊Q8, C22.D4, C4.4D4, C42.C2 [×2], C42⋊2C2 [×2], D28 [×2], C22×D7, C23.83C23, C2×D28, C4○D28 [×2], D4⋊2D7 [×3], Q8×D7, C4.D28, C23.D14 [×2], C22.D28, Dic7.Q8 [×2], D14⋊2Q8, (C2×C4).17D28
►Regular action on 448 points
Generators in S448
(1 60)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 71)(13 72)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 84)(26 57)(27 58)(28 59)(29 288)(30 289)(31 290)(32 291)(33 292)(34 293)(35 294)(36 295)(37 296)(38 297)(39 298)(40 299)(41 300)(42 301)(43 302)(44 303)(45 304)(46 305)(47 306)(48 307)(49 308)(50 281)(51 282)(52 283)(53 284)(54 285)(55 286)(56 287)(85 246)(86 247)(87 248)(88 249)(89 250)(90 251)(91 252)(92 225)(93 226)(94 227)(95 228)(96 229)(97 230)(98 231)(99 232)(100 233)(101 234)(102 235)(103 236)(104 237)(105 238)(106 239)(107 240)(108 241)(109 242)(110 243)(111 244)(112 245)(113 384)(114 385)(115 386)(116 387)(117 388)(118 389)(119 390)(120 391)(121 392)(122 365)(123 366)(124 367)(125 368)(126 369)(127 370)(128 371)(129 372)(130 373)(131 374)(132 375)(133 376)(134 377)(135 378)(136 379)(137 380)(138 381)(139 382)(140 383)(141 274)(142 275)(143 276)(144 277)(145 278)(146 279)(147 280)(148 253)(149 254)(150 255)(151 256)(152 257)(153 258)(154 259)(155 260)(156 261)(157 262)(158 263)(159 264)(160 265)(161 266)(162 267)(163 268)(164 269)(165 270)(166 271)(167 272)(168 273)(169 218)(170 219)(171 220)(172 221)(173 222)(174 223)(175 224)(176 197)(177 198)(178 199)(179 200)(180 201)(181 202)(182 203)(183 204)(184 205)(185 206)(186 207)(187 208)(188 209)(189 210)(190 211)(191 212)(192 213)(193 214)(194 215)(195 216)(196 217)(309 395)(310 396)(311 397)(312 398)(313 399)(314 400)(315 401)(316 402)(317 403)(318 404)(319 405)(320 406)(321 407)(322 408)(323 409)(324 410)(325 411)(326 412)(327 413)(328 414)(329 415)(330 416)(331 417)(332 418)(333 419)(334 420)(335 393)(336 394)(337 425)(338 426)(339 427)(340 428)(341 429)(342 430)(343 431)(344 432)(345 433)(346 434)(347 435)(348 436)(349 437)(350 438)(351 439)(352 440)(353 441)(354 442)(355 443)(356 444)(357 445)(358 446)(359 447)(360 448)(361 421)(362 422)(363 423)(364 424)
(1 239 355 374)(2 107 356 132)(3 241 357 376)(4 109 358 134)(5 243 359 378)(6 111 360 136)(7 245 361 380)(8 85 362 138)(9 247 363 382)(10 87 364 140)(11 249 337 384)(12 89 338 114)(13 251 339 386)(14 91 340 116)(15 225 341 388)(16 93 342 118)(17 227 343 390)(18 95 344 120)(19 229 345 392)(20 97 346 122)(21 231 347 366)(22 99 348 124)(23 233 349 368)(24 101 350 126)(25 235 351 370)(26 103 352 128)(27 237 353 372)(28 105 354 130)(29 329 266 184)(30 416 267 206)(31 331 268 186)(32 418 269 208)(33 333 270 188)(34 420 271 210)(35 335 272 190)(36 394 273 212)(37 309 274 192)(38 396 275 214)(39 311 276 194)(40 398 277 216)(41 313 278 196)(42 400 279 218)(43 315 280 170)(44 402 253 220)(45 317 254 172)(46 404 255 222)(47 319 256 174)(48 406 257 224)(49 321 258 176)(50 408 259 198)(51 323 260 178)(52 410 261 200)(53 325 262 180)(54 412 263 202)(55 327 264 182)(56 414 265 204)(57 236 440 371)(58 104 441 129)(59 238 442 373)(60 106 443 131)(61 240 444 375)(62 108 445 133)(63 242 446 377)(64 110 447 135)(65 244 448 379)(66 112 421 137)(67 246 422 381)(68 86 423 139)(69 248 424 383)(70 88 425 113)(71 250 426 385)(72 90 427 115)(73 252 428 387)(74 92 429 117)(75 226 430 389)(76 94 431 119)(77 228 432 391)(78 96 433 121)(79 230 434 365)(80 98 435 123)(81 232 436 367)(82 100 437 125)(83 234 438 369)(84 102 439 127)(141 213 296 395)(142 193 297 310)(143 215 298 397)(144 195 299 312)(145 217 300 399)(146 169 301 314)(147 219 302 401)(148 171 303 316)(149 221 304 403)(150 173 305 318)(151 223 306 405)(152 175 307 320)(153 197 308 407)(154 177 281 322)(155 199 282 409)(156 179 283 324)(157 201 284 411)(158 181 285 326)(159 203 286 413)(160 183 287 328)(161 205 288 415)(162 185 289 330)(163 207 290 417)(164 187 291 332)(165 209 292 419)(166 189 293 334)(167 211 294 393)(168 191 295 336)
(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)
(1 267 443 289)(2 29 444 161)(3 265 445 287)(4 55 446 159)(5 263 447 285)(6 53 448 157)(7 261 421 283)(8 51 422 155)(9 259 423 281)(10 49 424 153)(11 257 425 307)(12 47 426 151)(13 255 427 305)(14 45 428 149)(15 253 429 303)(16 43 430 147)(17 279 431 301)(18 41 432 145)(19 277 433 299)(20 39 434 143)(21 275 435 297)(22 37 436 141)(23 273 437 295)(24 35 438 167)(25 271 439 293)(26 33 440 165)(27 269 441 291)(28 31 442 163)(30 60 162 355)(32 58 164 353)(34 84 166 351)(36 82 168 349)(38 80 142 347)(40 78 144 345)(42 76 146 343)(44 74 148 341)(46 72 150 339)(48 70 152 337)(50 68 154 363)(52 66 156 361)(54 64 158 359)(56 62 160 357)(57 292 352 270)(59 290 354 268)(61 288 356 266)(63 286 358 264)(65 284 360 262)(67 282 362 260)(69 308 364 258)(71 306 338 256)(73 304 340 254)(75 302 342 280)(77 300 344 278)(79 298 346 276)(81 296 348 274)(83 294 350 272)(85 199 381 323)(86 408 382 177)(87 197 383 321)(88 406 384 175)(89 223 385 319)(90 404 386 173)(91 221 387 317)(92 402 388 171)(93 219 389 315)(94 400 390 169)(95 217 391 313)(96 398 392 195)(97 215 365 311)(98 396 366 193)(99 213 367 309)(100 394 368 191)(101 211 369 335)(102 420 370 189)(103 209 371 333)(104 418 372 187)(105 207 373 331)(106 416 374 185)(107 205 375 329)(108 414 376 183)(109 203 377 327)(110 412 378 181)(111 201 379 325)(112 410 380 179)(113 224 249 320)(114 405 250 174)(115 222 251 318)(116 403 252 172)(117 220 225 316)(118 401 226 170)(119 218 227 314)(120 399 228 196)(121 216 229 312)(122 397 230 194)(123 214 231 310)(124 395 232 192)(125 212 233 336)(126 393 234 190)(127 210 235 334)(128 419 236 188)(129 208 237 332)(130 417 238 186)(131 206 239 330)(132 415 240 184)(133 204 241 328)(134 413 242 182)(135 202 243 326)(136 411 244 180)(137 200 245 324)(138 409 246 178)(139 198 247 322)(140 407 248 176)
G:=sub<Sym(448)| (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,84)(26,57)(27,58)(28,59)(29,288)(30,289)(31,290)(32,291)(33,292)(34,293)(35,294)(36,295)(37,296)(38,297)(39,298)(40,299)(41,300)(42,301)(43,302)(44,303)(45,304)(46,305)(47,306)(48,307)(49,308)(50,281)(51,282)(52,283)(53,284)(54,285)(55,286)(56,287)(85,246)(86,247)(87,248)(88,249)(89,250)(90,251)(91,252)(92,225)(93,226)(94,227)(95,228)(96,229)(97,230)(98,231)(99,232)(100,233)(101,234)(102,235)(103,236)(104,237)(105,238)(106,239)(107,240)(108,241)(109,242)(110,243)(111,244)(112,245)(113,384)(114,385)(115,386)(116,387)(117,388)(118,389)(119,390)(120,391)(121,392)(122,365)(123,366)(124,367)(125,368)(126,369)(127,370)(128,371)(129,372)(130,373)(131,374)(132,375)(133,376)(134,377)(135,378)(136,379)(137,380)(138,381)(139,382)(140,383)(141,274)(142,275)(143,276)(144,277)(145,278)(146,279)(147,280)(148,253)(149,254)(150,255)(151,256)(152,257)(153,258)(154,259)(155,260)(156,261)(157,262)(158,263)(159,264)(160,265)(161,266)(162,267)(163,268)(164,269)(165,270)(166,271)(167,272)(168,273)(169,218)(170,219)(171,220)(172,221)(173,222)(174,223)(175,224)(176,197)(177,198)(178,199)(179,200)(180,201)(181,202)(182,203)(183,204)(184,205)(185,206)(186,207)(187,208)(188,209)(189,210)(190,211)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217)(309,395)(310,396)(311,397)(312,398)(313,399)(314,400)(315,401)(316,402)(317,403)(318,404)(319,405)(320,406)(321,407)(322,408)(323,409)(324,410)(325,411)(326,412)(327,413)(328,414)(329,415)(330,416)(331,417)(332,418)(333,419)(334,420)(335,393)(336,394)(337,425)(338,426)(339,427)(340,428)(341,429)(342,430)(343,431)(344,432)(345,433)(346,434)(347,435)(348,436)(349,437)(350,438)(351,439)(352,440)(353,441)(354,442)(355,443)(356,444)(357,445)(358,446)(359,447)(360,448)(361,421)(362,422)(363,423)(364,424), (1,239,355,374)(2,107,356,132)(3,241,357,376)(4,109,358,134)(5,243,359,378)(6,111,360,136)(7,245,361,380)(8,85,362,138)(9,247,363,382)(10,87,364,140)(11,249,337,384)(12,89,338,114)(13,251,339,386)(14,91,340,116)(15,225,341,388)(16,93,342,118)(17,227,343,390)(18,95,344,120)(19,229,345,392)(20,97,346,122)(21,231,347,366)(22,99,348,124)(23,233,349,368)(24,101,350,126)(25,235,351,370)(26,103,352,128)(27,237,353,372)(28,105,354,130)(29,329,266,184)(30,416,267,206)(31,331,268,186)(32,418,269,208)(33,333,270,188)(34,420,271,210)(35,335,272,190)(36,394,273,212)(37,309,274,192)(38,396,275,214)(39,311,276,194)(40,398,277,216)(41,313,278,196)(42,400,279,218)(43,315,280,170)(44,402,253,220)(45,317,254,172)(46,404,255,222)(47,319,256,174)(48,406,257,224)(49,321,258,176)(50,408,259,198)(51,323,260,178)(52,410,261,200)(53,325,262,180)(54,412,263,202)(55,327,264,182)(56,414,265,204)(57,236,440,371)(58,104,441,129)(59,238,442,373)(60,106,443,131)(61,240,444,375)(62,108,445,133)(63,242,446,377)(64,110,447,135)(65,244,448,379)(66,112,421,137)(67,246,422,381)(68,86,423,139)(69,248,424,383)(70,88,425,113)(71,250,426,385)(72,90,427,115)(73,252,428,387)(74,92,429,117)(75,226,430,389)(76,94,431,119)(77,228,432,391)(78,96,433,121)(79,230,434,365)(80,98,435,123)(81,232,436,367)(82,100,437,125)(83,234,438,369)(84,102,439,127)(141,213,296,395)(142,193,297,310)(143,215,298,397)(144,195,299,312)(145,217,300,399)(146,169,301,314)(147,219,302,401)(148,171,303,316)(149,221,304,403)(150,173,305,318)(151,223,306,405)(152,175,307,320)(153,197,308,407)(154,177,281,322)(155,199,282,409)(156,179,283,324)(157,201,284,411)(158,181,285,326)(159,203,286,413)(160,183,287,328)(161,205,288,415)(162,185,289,330)(163,207,290,417)(164,187,291,332)(165,209,292,419)(166,189,293,334)(167,211,294,393)(168,191,295,336), (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), (1,267,443,289)(2,29,444,161)(3,265,445,287)(4,55,446,159)(5,263,447,285)(6,53,448,157)(7,261,421,283)(8,51,422,155)(9,259,423,281)(10,49,424,153)(11,257,425,307)(12,47,426,151)(13,255,427,305)(14,45,428,149)(15,253,429,303)(16,43,430,147)(17,279,431,301)(18,41,432,145)(19,277,433,299)(20,39,434,143)(21,275,435,297)(22,37,436,141)(23,273,437,295)(24,35,438,167)(25,271,439,293)(26,33,440,165)(27,269,441,291)(28,31,442,163)(30,60,162,355)(32,58,164,353)(34,84,166,351)(36,82,168,349)(38,80,142,347)(40,78,144,345)(42,76,146,343)(44,74,148,341)(46,72,150,339)(48,70,152,337)(50,68,154,363)(52,66,156,361)(54,64,158,359)(56,62,160,357)(57,292,352,270)(59,290,354,268)(61,288,356,266)(63,286,358,264)(65,284,360,262)(67,282,362,260)(69,308,364,258)(71,306,338,256)(73,304,340,254)(75,302,342,280)(77,300,344,278)(79,298,346,276)(81,296,348,274)(83,294,350,272)(85,199,381,323)(86,408,382,177)(87,197,383,321)(88,406,384,175)(89,223,385,319)(90,404,386,173)(91,221,387,317)(92,402,388,171)(93,219,389,315)(94,400,390,169)(95,217,391,313)(96,398,392,195)(97,215,365,311)(98,396,366,193)(99,213,367,309)(100,394,368,191)(101,211,369,335)(102,420,370,189)(103,209,371,333)(104,418,372,187)(105,207,373,331)(106,416,374,185)(107,205,375,329)(108,414,376,183)(109,203,377,327)(110,412,378,181)(111,201,379,325)(112,410,380,179)(113,224,249,320)(114,405,250,174)(115,222,251,318)(116,403,252,172)(117,220,225,316)(118,401,226,170)(119,218,227,314)(120,399,228,196)(121,216,229,312)(122,397,230,194)(123,214,231,310)(124,395,232,192)(125,212,233,336)(126,393,234,190)(127,210,235,334)(128,419,236,188)(129,208,237,332)(130,417,238,186)(131,206,239,330)(132,415,240,184)(133,204,241,328)(134,413,242,182)(135,202,243,326)(136,411,244,180)(137,200,245,324)(138,409,246,178)(139,198,247,322)(140,407,248,176)>;
G:=Group( (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,84)(26,57)(27,58)(28,59)(29,288)(30,289)(31,290)(32,291)(33,292)(34,293)(35,294)(36,295)(37,296)(38,297)(39,298)(40,299)(41,300)(42,301)(43,302)(44,303)(45,304)(46,305)(47,306)(48,307)(49,308)(50,281)(51,282)(52,283)(53,284)(54,285)(55,286)(56,287)(85,246)(86,247)(87,248)(88,249)(89,250)(90,251)(91,252)(92,225)(93,226)(94,227)(95,228)(96,229)(97,230)(98,231)(99,232)(100,233)(101,234)(102,235)(103,236)(104,237)(105,238)(106,239)(107,240)(108,241)(109,242)(110,243)(111,244)(112,245)(113,384)(114,385)(115,386)(116,387)(117,388)(118,389)(119,390)(120,391)(121,392)(122,365)(123,366)(124,367)(125,368)(126,369)(127,370)(128,371)(129,372)(130,373)(131,374)(132,375)(133,376)(134,377)(135,378)(136,379)(137,380)(138,381)(139,382)(140,383)(141,274)(142,275)(143,276)(144,277)(145,278)(146,279)(147,280)(148,253)(149,254)(150,255)(151,256)(152,257)(153,258)(154,259)(155,260)(156,261)(157,262)(158,263)(159,264)(160,265)(161,266)(162,267)(163,268)(164,269)(165,270)(166,271)(167,272)(168,273)(169,218)(170,219)(171,220)(172,221)(173,222)(174,223)(175,224)(176,197)(177,198)(178,199)(179,200)(180,201)(181,202)(182,203)(183,204)(184,205)(185,206)(186,207)(187,208)(188,209)(189,210)(190,211)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217)(309,395)(310,396)(311,397)(312,398)(313,399)(314,400)(315,401)(316,402)(317,403)(318,404)(319,405)(320,406)(321,407)(322,408)(323,409)(324,410)(325,411)(326,412)(327,413)(328,414)(329,415)(330,416)(331,417)(332,418)(333,419)(334,420)(335,393)(336,394)(337,425)(338,426)(339,427)(340,428)(341,429)(342,430)(343,431)(344,432)(345,433)(346,434)(347,435)(348,436)(349,437)(350,438)(351,439)(352,440)(353,441)(354,442)(355,443)(356,444)(357,445)(358,446)(359,447)(360,448)(361,421)(362,422)(363,423)(364,424), (1,239,355,374)(2,107,356,132)(3,241,357,376)(4,109,358,134)(5,243,359,378)(6,111,360,136)(7,245,361,380)(8,85,362,138)(9,247,363,382)(10,87,364,140)(11,249,337,384)(12,89,338,114)(13,251,339,386)(14,91,340,116)(15,225,341,388)(16,93,342,118)(17,227,343,390)(18,95,344,120)(19,229,345,392)(20,97,346,122)(21,231,347,366)(22,99,348,124)(23,233,349,368)(24,101,350,126)(25,235,351,370)(26,103,352,128)(27,237,353,372)(28,105,354,130)(29,329,266,184)(30,416,267,206)(31,331,268,186)(32,418,269,208)(33,333,270,188)(34,420,271,210)(35,335,272,190)(36,394,273,212)(37,309,274,192)(38,396,275,214)(39,311,276,194)(40,398,277,216)(41,313,278,196)(42,400,279,218)(43,315,280,170)(44,402,253,220)(45,317,254,172)(46,404,255,222)(47,319,256,174)(48,406,257,224)(49,321,258,176)(50,408,259,198)(51,323,260,178)(52,410,261,200)(53,325,262,180)(54,412,263,202)(55,327,264,182)(56,414,265,204)(57,236,440,371)(58,104,441,129)(59,238,442,373)(60,106,443,131)(61,240,444,375)(62,108,445,133)(63,242,446,377)(64,110,447,135)(65,244,448,379)(66,112,421,137)(67,246,422,381)(68,86,423,139)(69,248,424,383)(70,88,425,113)(71,250,426,385)(72,90,427,115)(73,252,428,387)(74,92,429,117)(75,226,430,389)(76,94,431,119)(77,228,432,391)(78,96,433,121)(79,230,434,365)(80,98,435,123)(81,232,436,367)(82,100,437,125)(83,234,438,369)(84,102,439,127)(141,213,296,395)(142,193,297,310)(143,215,298,397)(144,195,299,312)(145,217,300,399)(146,169,301,314)(147,219,302,401)(148,171,303,316)(149,221,304,403)(150,173,305,318)(151,223,306,405)(152,175,307,320)(153,197,308,407)(154,177,281,322)(155,199,282,409)(156,179,283,324)(157,201,284,411)(158,181,285,326)(159,203,286,413)(160,183,287,328)(161,205,288,415)(162,185,289,330)(163,207,290,417)(164,187,291,332)(165,209,292,419)(166,189,293,334)(167,211,294,393)(168,191,295,336), (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), (1,267,443,289)(2,29,444,161)(3,265,445,287)(4,55,446,159)(5,263,447,285)(6,53,448,157)(7,261,421,283)(8,51,422,155)(9,259,423,281)(10,49,424,153)(11,257,425,307)(12,47,426,151)(13,255,427,305)(14,45,428,149)(15,253,429,303)(16,43,430,147)(17,279,431,301)(18,41,432,145)(19,277,433,299)(20,39,434,143)(21,275,435,297)(22,37,436,141)(23,273,437,295)(24,35,438,167)(25,271,439,293)(26,33,440,165)(27,269,441,291)(28,31,442,163)(30,60,162,355)(32,58,164,353)(34,84,166,351)(36,82,168,349)(38,80,142,347)(40,78,144,345)(42,76,146,343)(44,74,148,341)(46,72,150,339)(48,70,152,337)(50,68,154,363)(52,66,156,361)(54,64,158,359)(56,62,160,357)(57,292,352,270)(59,290,354,268)(61,288,356,266)(63,286,358,264)(65,284,360,262)(67,282,362,260)(69,308,364,258)(71,306,338,256)(73,304,340,254)(75,302,342,280)(77,300,344,278)(79,298,346,276)(81,296,348,274)(83,294,350,272)(85,199,381,323)(86,408,382,177)(87,197,383,321)(88,406,384,175)(89,223,385,319)(90,404,386,173)(91,221,387,317)(92,402,388,171)(93,219,389,315)(94,400,390,169)(95,217,391,313)(96,398,392,195)(97,215,365,311)(98,396,366,193)(99,213,367,309)(100,394,368,191)(101,211,369,335)(102,420,370,189)(103,209,371,333)(104,418,372,187)(105,207,373,331)(106,416,374,185)(107,205,375,329)(108,414,376,183)(109,203,377,327)(110,412,378,181)(111,201,379,325)(112,410,380,179)(113,224,249,320)(114,405,250,174)(115,222,251,318)(116,403,252,172)(117,220,225,316)(118,401,226,170)(119,218,227,314)(120,399,228,196)(121,216,229,312)(122,397,230,194)(123,214,231,310)(124,395,232,192)(125,212,233,336)(126,393,234,190)(127,210,235,334)(128,419,236,188)(129,208,237,332)(130,417,238,186)(131,206,239,330)(132,415,240,184)(133,204,241,328)(134,413,242,182)(135,202,243,326)(136,411,244,180)(137,200,245,324)(138,409,246,178)(139,198,247,322)(140,407,248,176) );
G=PermutationGroup([(1,60),(2,61),(3,62),(4,63),(5,64),(6,65),(7,66),(8,67),(9,68),(10,69),(11,70),(12,71),(13,72),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(21,80),(22,81),(23,82),(24,83),(25,84),(26,57),(27,58),(28,59),(29,288),(30,289),(31,290),(32,291),(33,292),(34,293),(35,294),(36,295),(37,296),(38,297),(39,298),(40,299),(41,300),(42,301),(43,302),(44,303),(45,304),(46,305),(47,306),(48,307),(49,308),(50,281),(51,282),(52,283),(53,284),(54,285),(55,286),(56,287),(85,246),(86,247),(87,248),(88,249),(89,250),(90,251),(91,252),(92,225),(93,226),(94,227),(95,228),(96,229),(97,230),(98,231),(99,232),(100,233),(101,234),(102,235),(103,236),(104,237),(105,238),(106,239),(107,240),(108,241),(109,242),(110,243),(111,244),(112,245),(113,384),(114,385),(115,386),(116,387),(117,388),(118,389),(119,390),(120,391),(121,392),(122,365),(123,366),(124,367),(125,368),(126,369),(127,370),(128,371),(129,372),(130,373),(131,374),(132,375),(133,376),(134,377),(135,378),(136,379),(137,380),(138,381),(139,382),(140,383),(141,274),(142,275),(143,276),(144,277),(145,278),(146,279),(147,280),(148,253),(149,254),(150,255),(151,256),(152,257),(153,258),(154,259),(155,260),(156,261),(157,262),(158,263),(159,264),(160,265),(161,266),(162,267),(163,268),(164,269),(165,270),(166,271),(167,272),(168,273),(169,218),(170,219),(171,220),(172,221),(173,222),(174,223),(175,224),(176,197),(177,198),(178,199),(179,200),(180,201),(181,202),(182,203),(183,204),(184,205),(185,206),(186,207),(187,208),(188,209),(189,210),(190,211),(191,212),(192,213),(193,214),(194,215),(195,216),(196,217),(309,395),(310,396),(311,397),(312,398),(313,399),(314,400),(315,401),(316,402),(317,403),(318,404),(319,405),(320,406),(321,407),(322,408),(323,409),(324,410),(325,411),(326,412),(327,413),(328,414),(329,415),(330,416),(331,417),(332,418),(333,419),(334,420),(335,393),(336,394),(337,425),(338,426),(339,427),(340,428),(341,429),(342,430),(343,431),(344,432),(345,433),(346,434),(347,435),(348,436),(349,437),(350,438),(351,439),(352,440),(353,441),(354,442),(355,443),(356,444),(357,445),(358,446),(359,447),(360,448),(361,421),(362,422),(363,423),(364,424)], [(1,239,355,374),(2,107,356,132),(3,241,357,376),(4,109,358,134),(5,243,359,378),(6,111,360,136),(7,245,361,380),(8,85,362,138),(9,247,363,382),(10,87,364,140),(11,249,337,384),(12,89,338,114),(13,251,339,386),(14,91,340,116),(15,225,341,388),(16,93,342,118),(17,227,343,390),(18,95,344,120),(19,229,345,392),(20,97,346,122),(21,231,347,366),(22,99,348,124),(23,233,349,368),(24,101,350,126),(25,235,351,370),(26,103,352,128),(27,237,353,372),(28,105,354,130),(29,329,266,184),(30,416,267,206),(31,331,268,186),(32,418,269,208),(33,333,270,188),(34,420,271,210),(35,335,272,190),(36,394,273,212),(37,309,274,192),(38,396,275,214),(39,311,276,194),(40,398,277,216),(41,313,278,196),(42,400,279,218),(43,315,280,170),(44,402,253,220),(45,317,254,172),(46,404,255,222),(47,319,256,174),(48,406,257,224),(49,321,258,176),(50,408,259,198),(51,323,260,178),(52,410,261,200),(53,325,262,180),(54,412,263,202),(55,327,264,182),(56,414,265,204),(57,236,440,371),(58,104,441,129),(59,238,442,373),(60,106,443,131),(61,240,444,375),(62,108,445,133),(63,242,446,377),(64,110,447,135),(65,244,448,379),(66,112,421,137),(67,246,422,381),(68,86,423,139),(69,248,424,383),(70,88,425,113),(71,250,426,385),(72,90,427,115),(73,252,428,387),(74,92,429,117),(75,226,430,389),(76,94,431,119),(77,228,432,391),(78,96,433,121),(79,230,434,365),(80,98,435,123),(81,232,436,367),(82,100,437,125),(83,234,438,369),(84,102,439,127),(141,213,296,395),(142,193,297,310),(143,215,298,397),(144,195,299,312),(145,217,300,399),(146,169,301,314),(147,219,302,401),(148,171,303,316),(149,221,304,403),(150,173,305,318),(151,223,306,405),(152,175,307,320),(153,197,308,407),(154,177,281,322),(155,199,282,409),(156,179,283,324),(157,201,284,411),(158,181,285,326),(159,203,286,413),(160,183,287,328),(161,205,288,415),(162,185,289,330),(163,207,290,417),(164,187,291,332),(165,209,292,419),(166,189,293,334),(167,211,294,393),(168,191,295,336)], [(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)], [(1,267,443,289),(2,29,444,161),(3,265,445,287),(4,55,446,159),(5,263,447,285),(6,53,448,157),(7,261,421,283),(8,51,422,155),(9,259,423,281),(10,49,424,153),(11,257,425,307),(12,47,426,151),(13,255,427,305),(14,45,428,149),(15,253,429,303),(16,43,430,147),(17,279,431,301),(18,41,432,145),(19,277,433,299),(20,39,434,143),(21,275,435,297),(22,37,436,141),(23,273,437,295),(24,35,438,167),(25,271,439,293),(26,33,440,165),(27,269,441,291),(28,31,442,163),(30,60,162,355),(32,58,164,353),(34,84,166,351),(36,82,168,349),(38,80,142,347),(40,78,144,345),(42,76,146,343),(44,74,148,341),(46,72,150,339),(48,70,152,337),(50,68,154,363),(52,66,156,361),(54,64,158,359),(56,62,160,357),(57,292,352,270),(59,290,354,268),(61,288,356,266),(63,286,358,264),(65,284,360,262),(67,282,362,260),(69,308,364,258),(71,306,338,256),(73,304,340,254),(75,302,342,280),(77,300,344,278),(79,298,346,276),(81,296,348,274),(83,294,350,272),(85,199,381,323),(86,408,382,177),(87,197,383,321),(88,406,384,175),(89,223,385,319),(90,404,386,173),(91,221,387,317),(92,402,388,171),(93,219,389,315),(94,400,390,169),(95,217,391,313),(96,398,392,195),(97,215,365,311),(98,396,366,193),(99,213,367,309),(100,394,368,191),(101,211,369,335),(102,420,370,189),(103,209,371,333),(104,418,372,187),(105,207,373,331),(106,416,374,185),(107,205,375,329),(108,414,376,183),(109,203,377,327),(110,412,378,181),(111,201,379,325),(112,410,380,179),(113,224,249,320),(114,405,250,174),(115,222,251,318),(116,403,252,172),(117,220,225,316),(118,401,226,170),(119,218,227,314),(120,399,228,196),(121,216,229,312),(122,397,230,194),(123,214,231,310),(124,395,232,192),(125,212,233,336),(126,393,234,190),(127,210,235,334),(128,419,236,188),(129,208,237,332),(130,417,238,186),(131,206,239,330),(132,415,240,184),(133,204,241,328),(134,413,242,182),(135,202,243,326),(136,411,244,180),(137,200,245,324),(138,409,246,178),(139,198,247,322),(140,407,248,176)])
82 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4F | 4G | ··· | 4N | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D4 | D7 | C4○D4 | D14 | D28 | C4○D28 | D4⋊2D7 | Q8×D7 |
| kernel | (C2×C4).17D28 | C14.C42 | C7×C2.C42 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C2×Dic7 | C2×C28 | C2.C42 | C2×C14 | C22×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 3 | 10 | 9 | 12 | 24 | 9 | 3 |
Matrix representation of (C2×C4).17D28 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 12 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 11 |
| 0 | 0 | 0 | 0 | 18 | 2 |
| 1 | 27 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 5 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 7 |
| 0 | 0 | 0 | 0 | 22 | 12 |
| 17 | 24 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 3 | 0 | 0 |
| 0 | 0 | 18 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 12 |
| 0 | 0 | 0 | 0 | 20 | 7 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,5,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,18,0,0,0,0,11,2],[1,0,0,0,0,0,27,28,0,0,0,0,0,0,7,4,0,0,0,0,5,3,0,0,0,0,0,0,20,22,0,0,0,0,7,12],[17,0,0,0,0,0,24,12,0,0,0,0,0,0,18,18,0,0,0,0,3,11,0,0,0,0,0,0,22,20,0,0,0,0,12,7] >;
(C2×C4).17D28 in GAP, Magma, Sage, TeX
(C_2\times C_4)._{17}D_{28} % in TeX
G:=Group("(C2xC4).17D28"); // GroupNames label
G:=SmallGroup(448,193);
// by ID
G=gap.SmallGroup(448,193);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,64,254,387,268,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^28=1,d^2=a*b^2,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=a*b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations