direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C7⋊Q16, C28⋊8Q16, C42.212D14, C7⋊4(C4×Q16), Q8.5(C4×D7), (C4×Q8).6D7, C14.75(C4×D4), (Q8×C28).7C2, C4⋊C4.255D14, (C2×C28).259D4, C14.35(C2×Q16), C14.94(C4○D8), C4.42(C4○D28), C28.62(C4○D4), C28.27(C22×C4), (C2×Q8).162D14, (C4×C28).100C22, (C2×C28).349C23, (C4×Dic14).14C2, Dic14.16(C2×C4), C14.Q16.17C2, Q8⋊Dic7.17C2, C28.Q8.18C2, C2.6(D4.8D14), C4⋊Dic7.332C22, (Q8×C14).197C22, (C2×Dic14).266C22, (C4×C7⋊C8).9C2, C7⋊C8.9(C2×C4), C4.27(C2×C4×D7), C2.21(C4×C7⋊D4), C2.3(C2×C7⋊Q16), (C7×Q8).12(C2×C4), (C2×C14).480(C2×D4), (C2×C7⋊C8).247C22, (C2×C7⋊Q16).10C2, C22.81(C2×C7⋊D4), (C2×C4).104(C7⋊D4), (C7×C4⋊C4).286C22, (C2×C4).449(C22×D7), SmallGroup(448,563)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C7⋊Q16
G = < a,b,c,d | a4=b7=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 388 in 110 conjugacy classes, 55 normal (39 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C14, C42, C42, C4⋊C4, C4⋊C4, C2×C8, Q16, C2×Q8, C2×Q8, Dic7, C28, C28, C28, C2×C14, C4×C8, Q8⋊C4, C2.D8, C4×Q8, C4×Q8, C2×Q16, C7⋊C8, C7⋊C8, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C4×Q16, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C7⋊Q16, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, Q8×C14, C4×C7⋊C8, C28.Q8, C14.Q16, Q8⋊Dic7, C4×Dic14, C2×C7⋊Q16, Q8×C28, C4×C7⋊Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, Q16, C22×C4, C2×D4, C4○D4, D14, C4×D4, C2×Q16, C4○D8, C4×D7, C7⋊D4, C22×D7, C4×Q16, C7⋊Q16, C2×C4×D7, C4○D28, C2×C7⋊D4, C4×C7⋊D4, C2×C7⋊Q16, D4.8D14, C4×C7⋊Q16
►Regular action on 448 points
Generators in S448
(1 445 421 252)(2 446 422 253)(3 447 423 254)(4 448 424 255)(5 441 417 256)(6 442 418 249)(7 443 419 250)(8 444 420 251)(9 429 267 228)(10 430 268 229)(11 431 269 230)(12 432 270 231)(13 425 271 232)(14 426 272 225)(15 427 265 226)(16 428 266 227)(17 437 261 159)(18 438 262 160)(19 439 263 153)(20 440 264 154)(21 433 257 155)(22 434 258 156)(23 435 259 157)(24 436 260 158)(25 333 312 280)(26 334 305 273)(27 335 306 274)(28 336 307 275)(29 329 308 276)(30 330 309 277)(31 331 310 278)(32 332 311 279)(33 173 204 89)(34 174 205 90)(35 175 206 91)(36 176 207 92)(37 169 208 93)(38 170 201 94)(39 171 202 95)(40 172 203 96)(41 195 219 97)(42 196 220 98)(43 197 221 99)(44 198 222 100)(45 199 223 101)(46 200 224 102)(47 193 217 103)(48 194 218 104)(49 315 213 105)(50 316 214 106)(51 317 215 107)(52 318 216 108)(53 319 209 109)(54 320 210 110)(55 313 211 111)(56 314 212 112)(57 120 151 374)(58 113 152 375)(59 114 145 376)(60 115 146 369)(61 116 147 370)(62 117 148 371)(63 118 149 372)(64 119 150 373)(65 128 236 368)(66 121 237 361)(67 122 238 362)(68 123 239 363)(69 124 240 364)(70 125 233 365)(71 126 234 366)(72 127 235 367)(73 143 167 390)(74 144 168 391)(75 137 161 392)(76 138 162 385)(77 139 163 386)(78 140 164 387)(79 141 165 388)(80 142 166 389)(81 129 244 384)(82 130 245 377)(83 131 246 378)(84 132 247 379)(85 133 248 380)(86 134 241 381)(87 135 242 382)(88 136 243 383)(177 293 395 339)(178 294 396 340)(179 295 397 341)(180 296 398 342)(181 289 399 343)(182 290 400 344)(183 291 393 337)(184 292 394 338)(185 300 409 353)(186 301 410 354)(187 302 411 355)(188 303 412 356)(189 304 413 357)(190 297 414 358)(191 298 415 359)(192 299 416 360)(281 405 349 328)(282 406 350 321)(283 407 351 322)(284 408 352 323)(285 401 345 324)(286 402 346 325)(287 403 347 326)(288 404 348 327)
(1 162 152 71 432 433 87)(2 88 434 425 72 145 163)(3 164 146 65 426 435 81)(4 82 436 427 66 147 165)(5 166 148 67 428 437 83)(6 84 438 429 68 149 167)(7 168 150 69 430 439 85)(8 86 440 431 70 151 161)(9 363 118 143 249 379 18)(10 19 380 250 144 119 364)(11 365 120 137 251 381 20)(12 21 382 252 138 113 366)(13 367 114 139 253 383 22)(14 23 384 254 140 115 368)(15 361 116 141 255 377 24)(16 17 378 256 142 117 362)(25 104 93 178 322 313 191)(26 192 314 323 179 94 97)(27 98 95 180 324 315 185)(28 186 316 325 181 96 99)(29 100 89 182 326 317 187)(30 188 318 327 183 90 101)(31 102 91 184 328 319 189)(32 190 320 321 177 92 103)(33 290 287 215 302 329 44)(34 45 330 303 216 288 291)(35 292 281 209 304 331 46)(36 47 332 297 210 282 293)(37 294 283 211 298 333 48)(38 41 334 299 212 284 295)(39 296 285 213 300 335 42)(40 43 336 301 214 286 289)(49 353 274 220 202 342 345)(50 346 343 203 221 275 354)(51 355 276 222 204 344 347)(52 348 337 205 223 277 356)(53 357 278 224 206 338 349)(54 350 339 207 217 279 358)(55 359 280 218 208 340 351)(56 352 341 201 219 273 360)(57 75 420 241 154 230 233)(58 234 231 155 242 421 76)(59 77 422 243 156 232 235)(60 236 225 157 244 423 78)(61 79 424 245 158 226 237)(62 238 227 159 246 417 80)(63 73 418 247 160 228 239)(64 240 229 153 248 419 74)(105 409 306 196 171 398 401)(106 402 399 172 197 307 410)(107 411 308 198 173 400 403)(108 404 393 174 199 309 412)(109 413 310 200 175 394 405)(110 406 395 176 193 311 414)(111 415 312 194 169 396 407)(112 408 397 170 195 305 416)(121 370 388 448 130 260 265)(122 266 261 131 441 389 371)(123 372 390 442 132 262 267)(124 268 263 133 443 391 373)(125 374 392 444 134 264 269)(126 270 257 135 445 385 375)(127 376 386 446 136 258 271)(128 272 259 129 447 387 369)
(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)
(1 333 5 329)(2 332 6 336)(3 331 7 335)(4 330 8 334)(9 325 13 321)(10 324 14 328)(11 323 15 327)(12 322 16 326)(17 317 21 313)(18 316 22 320)(19 315 23 319)(20 314 24 318)(25 256 29 252)(26 255 30 251)(27 254 31 250)(28 253 32 249)(33 152 37 148)(34 151 38 147)(35 150 39 146)(36 149 40 145)(41 165 45 161)(42 164 46 168)(43 163 47 167)(44 162 48 166)(49 157 53 153)(50 156 54 160)(51 155 55 159)(52 154 56 158)(57 201 61 205)(58 208 62 204)(59 207 63 203)(60 206 64 202)(65 292 69 296)(66 291 70 295)(67 290 71 294)(68 289 72 293)(73 221 77 217)(74 220 78 224)(75 219 79 223)(76 218 80 222)(81 304 85 300)(82 303 86 299)(83 302 87 298)(84 301 88 297)(89 113 93 117)(90 120 94 116)(91 119 95 115)(92 118 96 114)(97 141 101 137)(98 140 102 144)(99 139 103 143)(100 138 104 142)(105 259 109 263)(106 258 110 262)(107 257 111 261)(108 264 112 260)(121 393 125 397)(122 400 126 396)(123 399 127 395)(124 398 128 394)(129 413 133 409)(130 412 134 416)(131 411 135 415)(132 410 136 414)(169 371 173 375)(170 370 174 374)(171 369 175 373)(172 376 176 372)(177 363 181 367)(178 362 182 366)(179 361 183 365)(180 368 184 364)(185 384 189 380)(186 383 190 379)(187 382 191 378)(188 381 192 377)(193 390 197 386)(194 389 198 385)(195 388 199 392)(196 387 200 391)(209 439 213 435)(210 438 214 434)(211 437 215 433)(212 436 216 440)(225 349 229 345)(226 348 230 352)(227 347 231 351)(228 346 232 350)(233 341 237 337)(234 340 238 344)(235 339 239 343)(236 338 240 342)(241 360 245 356)(242 359 246 355)(243 358 247 354)(244 357 248 353)(265 404 269 408)(266 403 270 407)(267 402 271 406)(268 401 272 405)(273 424 277 420)(274 423 278 419)(275 422 279 418)(276 421 280 417)(281 430 285 426)(282 429 286 425)(283 428 287 432)(284 427 288 431)(305 448 309 444)(306 447 310 443)(307 446 311 442)(308 445 312 441)
G:=sub<Sym(448)| (1,445,421,252)(2,446,422,253)(3,447,423,254)(4,448,424,255)(5,441,417,256)(6,442,418,249)(7,443,419,250)(8,444,420,251)(9,429,267,228)(10,430,268,229)(11,431,269,230)(12,432,270,231)(13,425,271,232)(14,426,272,225)(15,427,265,226)(16,428,266,227)(17,437,261,159)(18,438,262,160)(19,439,263,153)(20,440,264,154)(21,433,257,155)(22,434,258,156)(23,435,259,157)(24,436,260,158)(25,333,312,280)(26,334,305,273)(27,335,306,274)(28,336,307,275)(29,329,308,276)(30,330,309,277)(31,331,310,278)(32,332,311,279)(33,173,204,89)(34,174,205,90)(35,175,206,91)(36,176,207,92)(37,169,208,93)(38,170,201,94)(39,171,202,95)(40,172,203,96)(41,195,219,97)(42,196,220,98)(43,197,221,99)(44,198,222,100)(45,199,223,101)(46,200,224,102)(47,193,217,103)(48,194,218,104)(49,315,213,105)(50,316,214,106)(51,317,215,107)(52,318,216,108)(53,319,209,109)(54,320,210,110)(55,313,211,111)(56,314,212,112)(57,120,151,374)(58,113,152,375)(59,114,145,376)(60,115,146,369)(61,116,147,370)(62,117,148,371)(63,118,149,372)(64,119,150,373)(65,128,236,368)(66,121,237,361)(67,122,238,362)(68,123,239,363)(69,124,240,364)(70,125,233,365)(71,126,234,366)(72,127,235,367)(73,143,167,390)(74,144,168,391)(75,137,161,392)(76,138,162,385)(77,139,163,386)(78,140,164,387)(79,141,165,388)(80,142,166,389)(81,129,244,384)(82,130,245,377)(83,131,246,378)(84,132,247,379)(85,133,248,380)(86,134,241,381)(87,135,242,382)(88,136,243,383)(177,293,395,339)(178,294,396,340)(179,295,397,341)(180,296,398,342)(181,289,399,343)(182,290,400,344)(183,291,393,337)(184,292,394,338)(185,300,409,353)(186,301,410,354)(187,302,411,355)(188,303,412,356)(189,304,413,357)(190,297,414,358)(191,298,415,359)(192,299,416,360)(281,405,349,328)(282,406,350,321)(283,407,351,322)(284,408,352,323)(285,401,345,324)(286,402,346,325)(287,403,347,326)(288,404,348,327), (1,162,152,71,432,433,87)(2,88,434,425,72,145,163)(3,164,146,65,426,435,81)(4,82,436,427,66,147,165)(5,166,148,67,428,437,83)(6,84,438,429,68,149,167)(7,168,150,69,430,439,85)(8,86,440,431,70,151,161)(9,363,118,143,249,379,18)(10,19,380,250,144,119,364)(11,365,120,137,251,381,20)(12,21,382,252,138,113,366)(13,367,114,139,253,383,22)(14,23,384,254,140,115,368)(15,361,116,141,255,377,24)(16,17,378,256,142,117,362)(25,104,93,178,322,313,191)(26,192,314,323,179,94,97)(27,98,95,180,324,315,185)(28,186,316,325,181,96,99)(29,100,89,182,326,317,187)(30,188,318,327,183,90,101)(31,102,91,184,328,319,189)(32,190,320,321,177,92,103)(33,290,287,215,302,329,44)(34,45,330,303,216,288,291)(35,292,281,209,304,331,46)(36,47,332,297,210,282,293)(37,294,283,211,298,333,48)(38,41,334,299,212,284,295)(39,296,285,213,300,335,42)(40,43,336,301,214,286,289)(49,353,274,220,202,342,345)(50,346,343,203,221,275,354)(51,355,276,222,204,344,347)(52,348,337,205,223,277,356)(53,357,278,224,206,338,349)(54,350,339,207,217,279,358)(55,359,280,218,208,340,351)(56,352,341,201,219,273,360)(57,75,420,241,154,230,233)(58,234,231,155,242,421,76)(59,77,422,243,156,232,235)(60,236,225,157,244,423,78)(61,79,424,245,158,226,237)(62,238,227,159,246,417,80)(63,73,418,247,160,228,239)(64,240,229,153,248,419,74)(105,409,306,196,171,398,401)(106,402,399,172,197,307,410)(107,411,308,198,173,400,403)(108,404,393,174,199,309,412)(109,413,310,200,175,394,405)(110,406,395,176,193,311,414)(111,415,312,194,169,396,407)(112,408,397,170,195,305,416)(121,370,388,448,130,260,265)(122,266,261,131,441,389,371)(123,372,390,442,132,262,267)(124,268,263,133,443,391,373)(125,374,392,444,134,264,269)(126,270,257,135,445,385,375)(127,376,386,446,136,258,271)(128,272,259,129,447,387,369), (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), (1,333,5,329)(2,332,6,336)(3,331,7,335)(4,330,8,334)(9,325,13,321)(10,324,14,328)(11,323,15,327)(12,322,16,326)(17,317,21,313)(18,316,22,320)(19,315,23,319)(20,314,24,318)(25,256,29,252)(26,255,30,251)(27,254,31,250)(28,253,32,249)(33,152,37,148)(34,151,38,147)(35,150,39,146)(36,149,40,145)(41,165,45,161)(42,164,46,168)(43,163,47,167)(44,162,48,166)(49,157,53,153)(50,156,54,160)(51,155,55,159)(52,154,56,158)(57,201,61,205)(58,208,62,204)(59,207,63,203)(60,206,64,202)(65,292,69,296)(66,291,70,295)(67,290,71,294)(68,289,72,293)(73,221,77,217)(74,220,78,224)(75,219,79,223)(76,218,80,222)(81,304,85,300)(82,303,86,299)(83,302,87,298)(84,301,88,297)(89,113,93,117)(90,120,94,116)(91,119,95,115)(92,118,96,114)(97,141,101,137)(98,140,102,144)(99,139,103,143)(100,138,104,142)(105,259,109,263)(106,258,110,262)(107,257,111,261)(108,264,112,260)(121,393,125,397)(122,400,126,396)(123,399,127,395)(124,398,128,394)(129,413,133,409)(130,412,134,416)(131,411,135,415)(132,410,136,414)(169,371,173,375)(170,370,174,374)(171,369,175,373)(172,376,176,372)(177,363,181,367)(178,362,182,366)(179,361,183,365)(180,368,184,364)(185,384,189,380)(186,383,190,379)(187,382,191,378)(188,381,192,377)(193,390,197,386)(194,389,198,385)(195,388,199,392)(196,387,200,391)(209,439,213,435)(210,438,214,434)(211,437,215,433)(212,436,216,440)(225,349,229,345)(226,348,230,352)(227,347,231,351)(228,346,232,350)(233,341,237,337)(234,340,238,344)(235,339,239,343)(236,338,240,342)(241,360,245,356)(242,359,246,355)(243,358,247,354)(244,357,248,353)(265,404,269,408)(266,403,270,407)(267,402,271,406)(268,401,272,405)(273,424,277,420)(274,423,278,419)(275,422,279,418)(276,421,280,417)(281,430,285,426)(282,429,286,425)(283,428,287,432)(284,427,288,431)(305,448,309,444)(306,447,310,443)(307,446,311,442)(308,445,312,441)>;
G:=Group( (1,445,421,252)(2,446,422,253)(3,447,423,254)(4,448,424,255)(5,441,417,256)(6,442,418,249)(7,443,419,250)(8,444,420,251)(9,429,267,228)(10,430,268,229)(11,431,269,230)(12,432,270,231)(13,425,271,232)(14,426,272,225)(15,427,265,226)(16,428,266,227)(17,437,261,159)(18,438,262,160)(19,439,263,153)(20,440,264,154)(21,433,257,155)(22,434,258,156)(23,435,259,157)(24,436,260,158)(25,333,312,280)(26,334,305,273)(27,335,306,274)(28,336,307,275)(29,329,308,276)(30,330,309,277)(31,331,310,278)(32,332,311,279)(33,173,204,89)(34,174,205,90)(35,175,206,91)(36,176,207,92)(37,169,208,93)(38,170,201,94)(39,171,202,95)(40,172,203,96)(41,195,219,97)(42,196,220,98)(43,197,221,99)(44,198,222,100)(45,199,223,101)(46,200,224,102)(47,193,217,103)(48,194,218,104)(49,315,213,105)(50,316,214,106)(51,317,215,107)(52,318,216,108)(53,319,209,109)(54,320,210,110)(55,313,211,111)(56,314,212,112)(57,120,151,374)(58,113,152,375)(59,114,145,376)(60,115,146,369)(61,116,147,370)(62,117,148,371)(63,118,149,372)(64,119,150,373)(65,128,236,368)(66,121,237,361)(67,122,238,362)(68,123,239,363)(69,124,240,364)(70,125,233,365)(71,126,234,366)(72,127,235,367)(73,143,167,390)(74,144,168,391)(75,137,161,392)(76,138,162,385)(77,139,163,386)(78,140,164,387)(79,141,165,388)(80,142,166,389)(81,129,244,384)(82,130,245,377)(83,131,246,378)(84,132,247,379)(85,133,248,380)(86,134,241,381)(87,135,242,382)(88,136,243,383)(177,293,395,339)(178,294,396,340)(179,295,397,341)(180,296,398,342)(181,289,399,343)(182,290,400,344)(183,291,393,337)(184,292,394,338)(185,300,409,353)(186,301,410,354)(187,302,411,355)(188,303,412,356)(189,304,413,357)(190,297,414,358)(191,298,415,359)(192,299,416,360)(281,405,349,328)(282,406,350,321)(283,407,351,322)(284,408,352,323)(285,401,345,324)(286,402,346,325)(287,403,347,326)(288,404,348,327), (1,162,152,71,432,433,87)(2,88,434,425,72,145,163)(3,164,146,65,426,435,81)(4,82,436,427,66,147,165)(5,166,148,67,428,437,83)(6,84,438,429,68,149,167)(7,168,150,69,430,439,85)(8,86,440,431,70,151,161)(9,363,118,143,249,379,18)(10,19,380,250,144,119,364)(11,365,120,137,251,381,20)(12,21,382,252,138,113,366)(13,367,114,139,253,383,22)(14,23,384,254,140,115,368)(15,361,116,141,255,377,24)(16,17,378,256,142,117,362)(25,104,93,178,322,313,191)(26,192,314,323,179,94,97)(27,98,95,180,324,315,185)(28,186,316,325,181,96,99)(29,100,89,182,326,317,187)(30,188,318,327,183,90,101)(31,102,91,184,328,319,189)(32,190,320,321,177,92,103)(33,290,287,215,302,329,44)(34,45,330,303,216,288,291)(35,292,281,209,304,331,46)(36,47,332,297,210,282,293)(37,294,283,211,298,333,48)(38,41,334,299,212,284,295)(39,296,285,213,300,335,42)(40,43,336,301,214,286,289)(49,353,274,220,202,342,345)(50,346,343,203,221,275,354)(51,355,276,222,204,344,347)(52,348,337,205,223,277,356)(53,357,278,224,206,338,349)(54,350,339,207,217,279,358)(55,359,280,218,208,340,351)(56,352,341,201,219,273,360)(57,75,420,241,154,230,233)(58,234,231,155,242,421,76)(59,77,422,243,156,232,235)(60,236,225,157,244,423,78)(61,79,424,245,158,226,237)(62,238,227,159,246,417,80)(63,73,418,247,160,228,239)(64,240,229,153,248,419,74)(105,409,306,196,171,398,401)(106,402,399,172,197,307,410)(107,411,308,198,173,400,403)(108,404,393,174,199,309,412)(109,413,310,200,175,394,405)(110,406,395,176,193,311,414)(111,415,312,194,169,396,407)(112,408,397,170,195,305,416)(121,370,388,448,130,260,265)(122,266,261,131,441,389,371)(123,372,390,442,132,262,267)(124,268,263,133,443,391,373)(125,374,392,444,134,264,269)(126,270,257,135,445,385,375)(127,376,386,446,136,258,271)(128,272,259,129,447,387,369), (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), (1,333,5,329)(2,332,6,336)(3,331,7,335)(4,330,8,334)(9,325,13,321)(10,324,14,328)(11,323,15,327)(12,322,16,326)(17,317,21,313)(18,316,22,320)(19,315,23,319)(20,314,24,318)(25,256,29,252)(26,255,30,251)(27,254,31,250)(28,253,32,249)(33,152,37,148)(34,151,38,147)(35,150,39,146)(36,149,40,145)(41,165,45,161)(42,164,46,168)(43,163,47,167)(44,162,48,166)(49,157,53,153)(50,156,54,160)(51,155,55,159)(52,154,56,158)(57,201,61,205)(58,208,62,204)(59,207,63,203)(60,206,64,202)(65,292,69,296)(66,291,70,295)(67,290,71,294)(68,289,72,293)(73,221,77,217)(74,220,78,224)(75,219,79,223)(76,218,80,222)(81,304,85,300)(82,303,86,299)(83,302,87,298)(84,301,88,297)(89,113,93,117)(90,120,94,116)(91,119,95,115)(92,118,96,114)(97,141,101,137)(98,140,102,144)(99,139,103,143)(100,138,104,142)(105,259,109,263)(106,258,110,262)(107,257,111,261)(108,264,112,260)(121,393,125,397)(122,400,126,396)(123,399,127,395)(124,398,128,394)(129,413,133,409)(130,412,134,416)(131,411,135,415)(132,410,136,414)(169,371,173,375)(170,370,174,374)(171,369,175,373)(172,376,176,372)(177,363,181,367)(178,362,182,366)(179,361,183,365)(180,368,184,364)(185,384,189,380)(186,383,190,379)(187,382,191,378)(188,381,192,377)(193,390,197,386)(194,389,198,385)(195,388,199,392)(196,387,200,391)(209,439,213,435)(210,438,214,434)(211,437,215,433)(212,436,216,440)(225,349,229,345)(226,348,230,352)(227,347,231,351)(228,346,232,350)(233,341,237,337)(234,340,238,344)(235,339,239,343)(236,338,240,342)(241,360,245,356)(242,359,246,355)(243,358,247,354)(244,357,248,353)(265,404,269,408)(266,403,270,407)(267,402,271,406)(268,401,272,405)(273,424,277,420)(274,423,278,419)(275,422,279,418)(276,421,280,417)(281,430,285,426)(282,429,286,425)(283,428,287,432)(284,427,288,431)(305,448,309,444)(306,447,310,443)(307,446,311,442)(308,445,312,441) );
G=PermutationGroup([[(1,445,421,252),(2,446,422,253),(3,447,423,254),(4,448,424,255),(5,441,417,256),(6,442,418,249),(7,443,419,250),(8,444,420,251),(9,429,267,228),(10,430,268,229),(11,431,269,230),(12,432,270,231),(13,425,271,232),(14,426,272,225),(15,427,265,226),(16,428,266,227),(17,437,261,159),(18,438,262,160),(19,439,263,153),(20,440,264,154),(21,433,257,155),(22,434,258,156),(23,435,259,157),(24,436,260,158),(25,333,312,280),(26,334,305,273),(27,335,306,274),(28,336,307,275),(29,329,308,276),(30,330,309,277),(31,331,310,278),(32,332,311,279),(33,173,204,89),(34,174,205,90),(35,175,206,91),(36,176,207,92),(37,169,208,93),(38,170,201,94),(39,171,202,95),(40,172,203,96),(41,195,219,97),(42,196,220,98),(43,197,221,99),(44,198,222,100),(45,199,223,101),(46,200,224,102),(47,193,217,103),(48,194,218,104),(49,315,213,105),(50,316,214,106),(51,317,215,107),(52,318,216,108),(53,319,209,109),(54,320,210,110),(55,313,211,111),(56,314,212,112),(57,120,151,374),(58,113,152,375),(59,114,145,376),(60,115,146,369),(61,116,147,370),(62,117,148,371),(63,118,149,372),(64,119,150,373),(65,128,236,368),(66,121,237,361),(67,122,238,362),(68,123,239,363),(69,124,240,364),(70,125,233,365),(71,126,234,366),(72,127,235,367),(73,143,167,390),(74,144,168,391),(75,137,161,392),(76,138,162,385),(77,139,163,386),(78,140,164,387),(79,141,165,388),(80,142,166,389),(81,129,244,384),(82,130,245,377),(83,131,246,378),(84,132,247,379),(85,133,248,380),(86,134,241,381),(87,135,242,382),(88,136,243,383),(177,293,395,339),(178,294,396,340),(179,295,397,341),(180,296,398,342),(181,289,399,343),(182,290,400,344),(183,291,393,337),(184,292,394,338),(185,300,409,353),(186,301,410,354),(187,302,411,355),(188,303,412,356),(189,304,413,357),(190,297,414,358),(191,298,415,359),(192,299,416,360),(281,405,349,328),(282,406,350,321),(283,407,351,322),(284,408,352,323),(285,401,345,324),(286,402,346,325),(287,403,347,326),(288,404,348,327)], [(1,162,152,71,432,433,87),(2,88,434,425,72,145,163),(3,164,146,65,426,435,81),(4,82,436,427,66,147,165),(5,166,148,67,428,437,83),(6,84,438,429,68,149,167),(7,168,150,69,430,439,85),(8,86,440,431,70,151,161),(9,363,118,143,249,379,18),(10,19,380,250,144,119,364),(11,365,120,137,251,381,20),(12,21,382,252,138,113,366),(13,367,114,139,253,383,22),(14,23,384,254,140,115,368),(15,361,116,141,255,377,24),(16,17,378,256,142,117,362),(25,104,93,178,322,313,191),(26,192,314,323,179,94,97),(27,98,95,180,324,315,185),(28,186,316,325,181,96,99),(29,100,89,182,326,317,187),(30,188,318,327,183,90,101),(31,102,91,184,328,319,189),(32,190,320,321,177,92,103),(33,290,287,215,302,329,44),(34,45,330,303,216,288,291),(35,292,281,209,304,331,46),(36,47,332,297,210,282,293),(37,294,283,211,298,333,48),(38,41,334,299,212,284,295),(39,296,285,213,300,335,42),(40,43,336,301,214,286,289),(49,353,274,220,202,342,345),(50,346,343,203,221,275,354),(51,355,276,222,204,344,347),(52,348,337,205,223,277,356),(53,357,278,224,206,338,349),(54,350,339,207,217,279,358),(55,359,280,218,208,340,351),(56,352,341,201,219,273,360),(57,75,420,241,154,230,233),(58,234,231,155,242,421,76),(59,77,422,243,156,232,235),(60,236,225,157,244,423,78),(61,79,424,245,158,226,237),(62,238,227,159,246,417,80),(63,73,418,247,160,228,239),(64,240,229,153,248,419,74),(105,409,306,196,171,398,401),(106,402,399,172,197,307,410),(107,411,308,198,173,400,403),(108,404,393,174,199,309,412),(109,413,310,200,175,394,405),(110,406,395,176,193,311,414),(111,415,312,194,169,396,407),(112,408,397,170,195,305,416),(121,370,388,448,130,260,265),(122,266,261,131,441,389,371),(123,372,390,442,132,262,267),(124,268,263,133,443,391,373),(125,374,392,444,134,264,269),(126,270,257,135,445,385,375),(127,376,386,446,136,258,271),(128,272,259,129,447,387,369)], [(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)], [(1,333,5,329),(2,332,6,336),(3,331,7,335),(4,330,8,334),(9,325,13,321),(10,324,14,328),(11,323,15,327),(12,322,16,326),(17,317,21,313),(18,316,22,320),(19,315,23,319),(20,314,24,318),(25,256,29,252),(26,255,30,251),(27,254,31,250),(28,253,32,249),(33,152,37,148),(34,151,38,147),(35,150,39,146),(36,149,40,145),(41,165,45,161),(42,164,46,168),(43,163,47,167),(44,162,48,166),(49,157,53,153),(50,156,54,160),(51,155,55,159),(52,154,56,158),(57,201,61,205),(58,208,62,204),(59,207,63,203),(60,206,64,202),(65,292,69,296),(66,291,70,295),(67,290,71,294),(68,289,72,293),(73,221,77,217),(74,220,78,224),(75,219,79,223),(76,218,80,222),(81,304,85,300),(82,303,86,299),(83,302,87,298),(84,301,88,297),(89,113,93,117),(90,120,94,116),(91,119,95,115),(92,118,96,114),(97,141,101,137),(98,140,102,144),(99,139,103,143),(100,138,104,142),(105,259,109,263),(106,258,110,262),(107,257,111,261),(108,264,112,260),(121,393,125,397),(122,400,126,396),(123,399,127,395),(124,398,128,394),(129,413,133,409),(130,412,134,416),(131,411,135,415),(132,410,136,414),(169,371,173,375),(170,370,174,374),(171,369,175,373),(172,376,176,372),(177,363,181,367),(178,362,182,366),(179,361,183,365),(180,368,184,364),(185,384,189,380),(186,383,190,379),(187,382,191,378),(188,381,192,377),(193,390,197,386),(194,389,198,385),(195,388,199,392),(196,387,200,391),(209,439,213,435),(210,438,214,434),(211,437,215,433),(212,436,216,440),(225,349,229,345),(226,348,230,352),(227,347,231,351),(228,346,232,350),(233,341,237,337),(234,340,238,344),(235,339,239,343),(236,338,240,342),(241,360,245,356),(242,359,246,355),(243,358,247,354),(244,357,248,353),(265,404,269,408),(266,403,270,407),(267,402,271,406),(268,401,272,405),(273,424,277,420),(274,423,278,419),(275,422,279,418),(276,421,280,417),(281,430,285,426),(282,429,286,425),(283,428,287,432),(284,427,288,431),(305,448,309,444),(306,447,310,443),(307,446,311,442),(308,445,312,441)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 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 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | Q16 | C4○D4 | D14 | D14 | D14 | C4○D8 | C7⋊D4 | C4×D7 | C4○D28 | C7⋊Q16 | D4.8D14 |
| kernel | C4×C7⋊Q16 | C4×C7⋊C8 | C28.Q8 | C14.Q16 | Q8⋊Dic7 | C4×Dic14 | C2×C7⋊Q16 | Q8×C28 | C7⋊Q16 | C2×C28 | C4×Q8 | C28 | C28 | C42 | C4⋊C4 | C2×Q8 | C14 | C2×C4 | Q8 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 3 | 4 | 2 | 3 | 3 | 3 | 4 | 12 | 12 | 12 | 6 | 6 |
Matrix representation of C4×C7⋊Q16 ►in GL5(𝔽113)
| 15 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 10 | 112 |
| 0 | 0 | 0 | 11 | 112 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 62 | 0 | 0 |
| 0 | 82 | 62 | 0 | 0 |
| 0 | 0 | 0 | 24 | 9 |
| 0 | 0 | 0 | 24 | 89 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 89 | 14 | 0 | 0 |
| 0 | 96 | 24 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(5,GF(113))| [15,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,112],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,10,11,0,0,0,112,112],[112,0,0,0,0,0,0,82,0,0,0,62,62,0,0,0,0,0,24,24,0,0,0,9,89],[1,0,0,0,0,0,89,96,0,0,0,14,24,0,0,0,0,0,112,0,0,0,0,0,112] >;
C4×C7⋊Q16 in GAP, Magma, Sage, TeX
C_4\times C_7\rtimes Q_{16} % in TeX
G:=Group("C4xC7:Q16"); // GroupNames label
G:=SmallGroup(448,563);
// by ID
G=gap.SmallGroup(448,563);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,232,58,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^7=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations