Copied to
clipboard

G = C7⋊Q16⋊C4  order 448 = 26·7

1st semidirect product of C7⋊Q16 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C7⋊Q16⋊1C4, Q8.1(C4×D7), C14.35(C4×D4), C56⋊C4.6C2, C4⋊C4.143D14, C7⋊2(Q16⋊C4), (C2×C8).172D14, Q8⋊C4.7D7, (C2×Q8).97D14, (Q8×Dic7).1C2, C22.74(D4×D7), C28.10(C22×C4), Dic14.2(C2×C4), C4.Dic14.1C2, C28.155(C4○D4), C4.52(D4⋊2D7), (C2×C56).190C22, (C2×C28).229C23, C2.1(Q16⋊D7), Dic7⋊3Q8.1C2, (C2×Dic7).146D4, C28.44D4.6C2, C2.3(SD16⋊D7), C4⋊Dic7.79C22, (Q8×C14).12C22, C14.52(C8.C22), (C4×Dic7).15C22, C2.19(Dic7⋊4D4), (C2×Dic14).62C22, C7⋊C8.1(C2×C4), C4.10(C2×C4×D7), (C7×Q8).1(C2×C4), (C2×C7⋊C8).26C22, (C2×C7⋊Q16).1C2, (C2×C14).242(C2×D4), (C7×C4⋊C4).30C22, (C7×Q8⋊C4).7C2, (C2×C4).336(C22×D7), SmallGroup(448,323)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C28 — C7⋊Q16⋊C4
Chief series
C1 — C7 — C14 — C28 — C2×C28 — C4×Dic7 — Dic7⋊3Q8 — C7⋊Q16⋊C4
Lower central
C7 — C14 — C28 — C7⋊Q16⋊C4
Upper central
C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for C7⋊Q16⋊C4
 G = < a,b,c,d | a7=b8=d4=1, c2=b4, bab-1=a-1, ac=ca, ad=da, cbc-1=b-1, dbd-1=b3, dcd-1=b6c >

Subgroups: 436 in 108 conjugacy classes, 49 normal (37 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C2×Q8, Dic7, C28, C28, C2×C14, C8⋊C4, Q8⋊C4, Q8⋊C4, C4.Q8, C4×Q8, C2×Q16, C7⋊C8, C56, Dic14, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, Q16⋊C4, C2×C7⋊C8, C4×Dic7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C7⋊Q16, C7×C4⋊C4, C2×C56, C2×Dic14, Q8×C14, C4.Dic14, C56⋊C4, C28.44D4, C7×Q8⋊C4, Dic7⋊3Q8, C2×C7⋊Q16, Q8×Dic7, C7⋊Q16⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C8.C22, C4×D7, C22×D7, Q16⋊C4, C2×C4×D7, D4×D7, D4⋊2D7, Dic7⋊4D4, SD16⋊D7, Q16⋊D7, C7⋊Q16⋊C4

Smallest permutation representation of C7⋊Q16⋊C4
►Regular action on 448 points
Generators in S448
(1 192 197 32 35 53 232)(2 225 54 36 25 198 185)(3 186 199 26 37 55 226)(4 227 56 38 27 200 187)(5 188 193 28 39 49 228)(6 229 50 40 29 194 189)(7 190 195 30 33 51 230)(8 231 52 34 31 196 191)(9 405 170 267 24 412 246)(10 247 413 17 268 171 406)(11 407 172 269 18 414 248)(12 241 415 19 270 173 408)(13 401 174 271 20 416 242)(14 243 409 21 272 175 402)(15 403 176 265 22 410 244)(16 245 411 23 266 169 404)(41 437 206 209 391 253 179)(42 180 254 392 210 207 438)(43 439 208 211 385 255 181)(44 182 256 386 212 201 440)(45 433 202 213 387 249 183)(46 184 250 388 214 203 434)(47 435 204 215 389 251 177)(48 178 252 390 216 205 436)(57 325 370 95 300 311 68)(58 69 312 301 96 371 326)(59 327 372 89 302 305 70)(60 71 306 303 90 373 328)(61 321 374 91 304 307 72)(62 65 308 297 92 375 322)(63 323 376 93 298 309 66)(64 67 310 299 94 369 324)(73 384 277 107 355 141 284)(74 285 142 356 108 278 377)(75 378 279 109 357 143 286)(76 287 144 358 110 280 379)(77 380 273 111 359 137 288)(78 281 138 360 112 274 381)(79 382 275 105 353 139 282)(80 283 140 354 106 276 383)(81 365 129 314 99 166 296)(82 289 167 100 315 130 366)(83 367 131 316 101 168 290)(84 291 161 102 317 132 368)(85 361 133 318 103 162 292)(86 293 163 104 319 134 362)(87 363 135 320 97 164 294)(88 295 165 98 313 136 364)(113 331 158 147 349 338 124)(114 125 339 350 148 159 332)(115 333 160 149 351 340 126)(116 127 341 352 150 153 334)(117 335 154 151 345 342 128)(118 121 343 346 152 155 336)(119 329 156 145 347 344 122)(120 123 337 348 146 157 330)(217 257 422 425 233 447 395)(218 396 448 234 426 423 258)(219 259 424 427 235 441 397)(220 398 442 236 428 417 260)(221 261 418 429 237 443 399)(222 400 444 238 430 419 262)(223 263 420 431 239 445 393)(224 394 446 240 432 421 264)

(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)

(1 389 5 385)(2 388 6 392)(3 387 7 391)(4 386 8 390)(9 441 13 445)(10 448 14 444)(11 447 15 443)(12 446 16 442)(17 423 21 419)(18 422 22 418)(19 421 23 417)(20 420 24 424)(25 46 29 42)(26 45 30 41)(27 44 31 48)(28 43 32 47)(33 437 37 433)(34 436 38 440)(35 435 39 439)(36 434 40 438)(49 208 53 204)(50 207 54 203)(51 206 55 202)(52 205 56 201)(57 380 61 384)(58 379 62 383)(59 378 63 382)(60 377 64 381)(65 80 69 76)(66 79 70 75)(67 78 71 74)(68 77 72 73)(81 347 85 351)(82 346 86 350)(83 345 87 349)(84 352 88 348)(89 357 93 353)(90 356 94 360)(91 355 95 359)(92 354 96 358)(97 331 101 335)(98 330 102 334)(99 329 103 333)(100 336 104 332)(105 372 109 376)(106 371 110 375)(107 370 111 374)(108 369 112 373)(113 316 117 320)(114 315 118 319)(115 314 119 318)(116 313 120 317)(121 134 125 130)(122 133 126 129)(123 132 127 136)(124 131 128 135)(137 304 141 300)(138 303 142 299)(139 302 143 298)(140 301 144 297)(145 292 149 296)(146 291 150 295)(147 290 151 294)(148 289 152 293)(153 165 157 161)(154 164 158 168)(155 163 159 167)(156 162 160 166)(169 220 173 224)(170 219 174 223)(171 218 175 222)(172 217 176 221)(177 193 181 197)(178 200 182 196)(179 199 183 195)(180 198 184 194)(185 250 189 254)(186 249 190 253)(187 256 191 252)(188 255 192 251)(209 226 213 230)(210 225 214 229)(211 232 215 228)(212 231 216 227)(233 244 237 248)(234 243 238 247)(235 242 239 246)(236 241 240 245)(257 265 261 269)(258 272 262 268)(259 271 263 267)(260 270 264 266)(273 321 277 325)(274 328 278 324)(275 327 279 323)(276 326 280 322)(281 306 285 310)(282 305 286 309)(283 312 287 308)(284 311 288 307)(337 368 341 364)(338 367 342 363)(339 366 343 362)(340 365 344 361)(393 405 397 401)(394 404 398 408)(395 403 399 407)(396 402 400 406)(409 430 413 426)(410 429 414 425)(411 428 415 432)(412 427 416 431)

(1 103 239 276)(2 98 240 279)(3 101 233 274)(4 104 234 277)(5 99 235 280)(6 102 236 275)(7 97 237 278)(8 100 238 273)(9 60 251 154)(10 63 252 157)(11 58 253 160)(12 61 254 155)(13 64 255 158)(14 59 256 153)(15 62 249 156)(16 57 250 159)(17 93 205 123)(18 96 206 126)(19 91 207 121)(20 94 208 124)(21 89 201 127)(22 92 202 122)(23 95 203 125)(24 90 204 128)(25 88 224 286)(26 83 217 281)(27 86 218 284)(28 81 219 287)(29 84 220 282)(30 87 221 285)(31 82 222 288)(32 85 223 283)(33 363 261 142)(34 366 262 137)(35 361 263 140)(36 364 264 143)(37 367 257 138)(38 362 258 141)(39 365 259 144)(40 368 260 139)(41 351 172 312)(42 346 173 307)(43 349 174 310)(44 352 175 305)(45 347 176 308)(46 350 169 311)(47 345 170 306)(48 348 171 309)(49 129 424 358)(50 132 417 353)(51 135 418 356)(52 130 419 359)(53 133 420 354)(54 136 421 357)(55 131 422 360)(56 134 423 355)(65 183 145 403)(66 178 146 406)(67 181 147 401)(68 184 148 404)(69 179 149 407)(70 182 150 402)(71 177 151 405)(72 180 152 408)(73 200 293 396)(74 195 294 399)(75 198 295 394)(76 193 296 397)(77 196 289 400)(78 199 290 395)(79 194 291 398)(80 197 292 393)(105 229 317 428)(106 232 318 431)(107 227 319 426)(108 230 320 429)(109 225 313 432)(110 228 314 427)(111 231 315 430)(112 226 316 425)(113 416 369 211)(114 411 370 214)(115 414 371 209)(116 409 372 212)(117 412 373 215)(118 415 374 210)(119 410 375 213)(120 413 376 216)(161 442 382 189)(162 445 383 192)(163 448 384 187)(164 443 377 190)(165 446 378 185)(166 441 379 188)(167 444 380 191)(168 447 381 186)(241 321 392 336)(242 324 385 331)(243 327 386 334)(244 322 387 329)(245 325 388 332)(246 328 389 335)(247 323 390 330)(248 326 391 333)(265 297 433 344)(266 300 434 339)(267 303 435 342)(268 298 436 337)(269 301 437 340)(270 304 438 343)(271 299 439 338)(272 302 440 341)
 
G:=sub<Sym(448)| (1,192,197,32,35,53,232)(2,225,54,36,25,198,185)(3,186,199,26,37,55,226)(4,227,56,38,27,200,187)(5,188,193,28,39,49,228)(6,229,50,40,29,194,189)(7,190,195,30,33,51,230)(8,231,52,34,31,196,191)(9,405,170,267,24,412,246)(10,247,413,17,268,171,406)(11,407,172,269,18,414,248)(12,241,415,19,270,173,408)(13,401,174,271,20,416,242)(14,243,409,21,272,175,402)(15,403,176,265,22,410,244)(16,245,411,23,266,169,404)(41,437,206,209,391,253,179)(42,180,254,392,210,207,438)(43,439,208,211,385,255,181)(44,182,256,386,212,201,440)(45,433,202,213,387,249,183)(46,184,250,388,214,203,434)(47,435,204,215,389,251,177)(48,178,252,390,216,205,436)(57,325,370,95,300,311,68)(58,69,312,301,96,371,326)(59,327,372,89,302,305,70)(60,71,306,303,90,373,328)(61,321,374,91,304,307,72)(62,65,308,297,92,375,322)(63,323,376,93,298,309,66)(64,67,310,299,94,369,324)(73,384,277,107,355,141,284)(74,285,142,356,108,278,377)(75,378,279,109,357,143,286)(76,287,144,358,110,280,379)(77,380,273,111,359,137,288)(78,281,138,360,112,274,381)(79,382,275,105,353,139,282)(80,283,140,354,106,276,383)(81,365,129,314,99,166,296)(82,289,167,100,315,130,366)(83,367,131,316,101,168,290)(84,291,161,102,317,132,368)(85,361,133,318,103,162,292)(86,293,163,104,319,134,362)(87,363,135,320,97,164,294)(88,295,165,98,313,136,364)(113,331,158,147,349,338,124)(114,125,339,350,148,159,332)(115,333,160,149,351,340,126)(116,127,341,352,150,153,334)(117,335,154,151,345,342,128)(118,121,343,346,152,155,336)(119,329,156,145,347,344,122)(120,123,337,348,146,157,330)(217,257,422,425,233,447,395)(218,396,448,234,426,423,258)(219,259,424,427,235,441,397)(220,398,442,236,428,417,260)(221,261,418,429,237,443,399)(222,400,444,238,430,419,262)(223,263,420,431,239,445,393)(224,394,446,240,432,421,264), (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), (1,389,5,385)(2,388,6,392)(3,387,7,391)(4,386,8,390)(9,441,13,445)(10,448,14,444)(11,447,15,443)(12,446,16,442)(17,423,21,419)(18,422,22,418)(19,421,23,417)(20,420,24,424)(25,46,29,42)(26,45,30,41)(27,44,31,48)(28,43,32,47)(33,437,37,433)(34,436,38,440)(35,435,39,439)(36,434,40,438)(49,208,53,204)(50,207,54,203)(51,206,55,202)(52,205,56,201)(57,380,61,384)(58,379,62,383)(59,378,63,382)(60,377,64,381)(65,80,69,76)(66,79,70,75)(67,78,71,74)(68,77,72,73)(81,347,85,351)(82,346,86,350)(83,345,87,349)(84,352,88,348)(89,357,93,353)(90,356,94,360)(91,355,95,359)(92,354,96,358)(97,331,101,335)(98,330,102,334)(99,329,103,333)(100,336,104,332)(105,372,109,376)(106,371,110,375)(107,370,111,374)(108,369,112,373)(113,316,117,320)(114,315,118,319)(115,314,119,318)(116,313,120,317)(121,134,125,130)(122,133,126,129)(123,132,127,136)(124,131,128,135)(137,304,141,300)(138,303,142,299)(139,302,143,298)(140,301,144,297)(145,292,149,296)(146,291,150,295)(147,290,151,294)(148,289,152,293)(153,165,157,161)(154,164,158,168)(155,163,159,167)(156,162,160,166)(169,220,173,224)(170,219,174,223)(171,218,175,222)(172,217,176,221)(177,193,181,197)(178,200,182,196)(179,199,183,195)(180,198,184,194)(185,250,189,254)(186,249,190,253)(187,256,191,252)(188,255,192,251)(209,226,213,230)(210,225,214,229)(211,232,215,228)(212,231,216,227)(233,244,237,248)(234,243,238,247)(235,242,239,246)(236,241,240,245)(257,265,261,269)(258,272,262,268)(259,271,263,267)(260,270,264,266)(273,321,277,325)(274,328,278,324)(275,327,279,323)(276,326,280,322)(281,306,285,310)(282,305,286,309)(283,312,287,308)(284,311,288,307)(337,368,341,364)(338,367,342,363)(339,366,343,362)(340,365,344,361)(393,405,397,401)(394,404,398,408)(395,403,399,407)(396,402,400,406)(409,430,413,426)(410,429,414,425)(411,428,415,432)(412,427,416,431), (1,103,239,276)(2,98,240,279)(3,101,233,274)(4,104,234,277)(5,99,235,280)(6,102,236,275)(7,97,237,278)(8,100,238,273)(9,60,251,154)(10,63,252,157)(11,58,253,160)(12,61,254,155)(13,64,255,158)(14,59,256,153)(15,62,249,156)(16,57,250,159)(17,93,205,123)(18,96,206,126)(19,91,207,121)(20,94,208,124)(21,89,201,127)(22,92,202,122)(23,95,203,125)(24,90,204,128)(25,88,224,286)(26,83,217,281)(27,86,218,284)(28,81,219,287)(29,84,220,282)(30,87,221,285)(31,82,222,288)(32,85,223,283)(33,363,261,142)(34,366,262,137)(35,361,263,140)(36,364,264,143)(37,367,257,138)(38,362,258,141)(39,365,259,144)(40,368,260,139)(41,351,172,312)(42,346,173,307)(43,349,174,310)(44,352,175,305)(45,347,176,308)(46,350,169,311)(47,345,170,306)(48,348,171,309)(49,129,424,358)(50,132,417,353)(51,135,418,356)(52,130,419,359)(53,133,420,354)(54,136,421,357)(55,131,422,360)(56,134,423,355)(65,183,145,403)(66,178,146,406)(67,181,147,401)(68,184,148,404)(69,179,149,407)(70,182,150,402)(71,177,151,405)(72,180,152,408)(73,200,293,396)(74,195,294,399)(75,198,295,394)(76,193,296,397)(77,196,289,400)(78,199,290,395)(79,194,291,398)(80,197,292,393)(105,229,317,428)(106,232,318,431)(107,227,319,426)(108,230,320,429)(109,225,313,432)(110,228,314,427)(111,231,315,430)(112,226,316,425)(113,416,369,211)(114,411,370,214)(115,414,371,209)(116,409,372,212)(117,412,373,215)(118,415,374,210)(119,410,375,213)(120,413,376,216)(161,442,382,189)(162,445,383,192)(163,448,384,187)(164,443,377,190)(165,446,378,185)(166,441,379,188)(167,444,380,191)(168,447,381,186)(241,321,392,336)(242,324,385,331)(243,327,386,334)(244,322,387,329)(245,325,388,332)(246,328,389,335)(247,323,390,330)(248,326,391,333)(265,297,433,344)(266,300,434,339)(267,303,435,342)(268,298,436,337)(269,301,437,340)(270,304,438,343)(271,299,439,338)(272,302,440,341)>;
 
G:=Group( (1,192,197,32,35,53,232)(2,225,54,36,25,198,185)(3,186,199,26,37,55,226)(4,227,56,38,27,200,187)(5,188,193,28,39,49,228)(6,229,50,40,29,194,189)(7,190,195,30,33,51,230)(8,231,52,34,31,196,191)(9,405,170,267,24,412,246)(10,247,413,17,268,171,406)(11,407,172,269,18,414,248)(12,241,415,19,270,173,408)(13,401,174,271,20,416,242)(14,243,409,21,272,175,402)(15,403,176,265,22,410,244)(16,245,411,23,266,169,404)(41,437,206,209,391,253,179)(42,180,254,392,210,207,438)(43,439,208,211,385,255,181)(44,182,256,386,212,201,440)(45,433,202,213,387,249,183)(46,184,250,388,214,203,434)(47,435,204,215,389,251,177)(48,178,252,390,216,205,436)(57,325,370,95,300,311,68)(58,69,312,301,96,371,326)(59,327,372,89,302,305,70)(60,71,306,303,90,373,328)(61,321,374,91,304,307,72)(62,65,308,297,92,375,322)(63,323,376,93,298,309,66)(64,67,310,299,94,369,324)(73,384,277,107,355,141,284)(74,285,142,356,108,278,377)(75,378,279,109,357,143,286)(76,287,144,358,110,280,379)(77,380,273,111,359,137,288)(78,281,138,360,112,274,381)(79,382,275,105,353,139,282)(80,283,140,354,106,276,383)(81,365,129,314,99,166,296)(82,289,167,100,315,130,366)(83,367,131,316,101,168,290)(84,291,161,102,317,132,368)(85,361,133,318,103,162,292)(86,293,163,104,319,134,362)(87,363,135,320,97,164,294)(88,295,165,98,313,136,364)(113,331,158,147,349,338,124)(114,125,339,350,148,159,332)(115,333,160,149,351,340,126)(116,127,341,352,150,153,334)(117,335,154,151,345,342,128)(118,121,343,346,152,155,336)(119,329,156,145,347,344,122)(120,123,337,348,146,157,330)(217,257,422,425,233,447,395)(218,396,448,234,426,423,258)(219,259,424,427,235,441,397)(220,398,442,236,428,417,260)(221,261,418,429,237,443,399)(222,400,444,238,430,419,262)(223,263,420,431,239,445,393)(224,394,446,240,432,421,264), (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), (1,389,5,385)(2,388,6,392)(3,387,7,391)(4,386,8,390)(9,441,13,445)(10,448,14,444)(11,447,15,443)(12,446,16,442)(17,423,21,419)(18,422,22,418)(19,421,23,417)(20,420,24,424)(25,46,29,42)(26,45,30,41)(27,44,31,48)(28,43,32,47)(33,437,37,433)(34,436,38,440)(35,435,39,439)(36,434,40,438)(49,208,53,204)(50,207,54,203)(51,206,55,202)(52,205,56,201)(57,380,61,384)(58,379,62,383)(59,378,63,382)(60,377,64,381)(65,80,69,76)(66,79,70,75)(67,78,71,74)(68,77,72,73)(81,347,85,351)(82,346,86,350)(83,345,87,349)(84,352,88,348)(89,357,93,353)(90,356,94,360)(91,355,95,359)(92,354,96,358)(97,331,101,335)(98,330,102,334)(99,329,103,333)(100,336,104,332)(105,372,109,376)(106,371,110,375)(107,370,111,374)(108,369,112,373)(113,316,117,320)(114,315,118,319)(115,314,119,318)(116,313,120,317)(121,134,125,130)(122,133,126,129)(123,132,127,136)(124,131,128,135)(137,304,141,300)(138,303,142,299)(139,302,143,298)(140,301,144,297)(145,292,149,296)(146,291,150,295)(147,290,151,294)(148,289,152,293)(153,165,157,161)(154,164,158,168)(155,163,159,167)(156,162,160,166)(169,220,173,224)(170,219,174,223)(171,218,175,222)(172,217,176,221)(177,193,181,197)(178,200,182,196)(179,199,183,195)(180,198,184,194)(185,250,189,254)(186,249,190,253)(187,256,191,252)(188,255,192,251)(209,226,213,230)(210,225,214,229)(211,232,215,228)(212,231,216,227)(233,244,237,248)(234,243,238,247)(235,242,239,246)(236,241,240,245)(257,265,261,269)(258,272,262,268)(259,271,263,267)(260,270,264,266)(273,321,277,325)(274,328,278,324)(275,327,279,323)(276,326,280,322)(281,306,285,310)(282,305,286,309)(283,312,287,308)(284,311,288,307)(337,368,341,364)(338,367,342,363)(339,366,343,362)(340,365,344,361)(393,405,397,401)(394,404,398,408)(395,403,399,407)(396,402,400,406)(409,430,413,426)(410,429,414,425)(411,428,415,432)(412,427,416,431), (1,103,239,276)(2,98,240,279)(3,101,233,274)(4,104,234,277)(5,99,235,280)(6,102,236,275)(7,97,237,278)(8,100,238,273)(9,60,251,154)(10,63,252,157)(11,58,253,160)(12,61,254,155)(13,64,255,158)(14,59,256,153)(15,62,249,156)(16,57,250,159)(17,93,205,123)(18,96,206,126)(19,91,207,121)(20,94,208,124)(21,89,201,127)(22,92,202,122)(23,95,203,125)(24,90,204,128)(25,88,224,286)(26,83,217,281)(27,86,218,284)(28,81,219,287)(29,84,220,282)(30,87,221,285)(31,82,222,288)(32,85,223,283)(33,363,261,142)(34,366,262,137)(35,361,263,140)(36,364,264,143)(37,367,257,138)(38,362,258,141)(39,365,259,144)(40,368,260,139)(41,351,172,312)(42,346,173,307)(43,349,174,310)(44,352,175,305)(45,347,176,308)(46,350,169,311)(47,345,170,306)(48,348,171,309)(49,129,424,358)(50,132,417,353)(51,135,418,356)(52,130,419,359)(53,133,420,354)(54,136,421,357)(55,131,422,360)(56,134,423,355)(65,183,145,403)(66,178,146,406)(67,181,147,401)(68,184,148,404)(69,179,149,407)(70,182,150,402)(71,177,151,405)(72,180,152,408)(73,200,293,396)(74,195,294,399)(75,198,295,394)(76,193,296,397)(77,196,289,400)(78,199,290,395)(79,194,291,398)(80,197,292,393)(105,229,317,428)(106,232,318,431)(107,227,319,426)(108,230,320,429)(109,225,313,432)(110,228,314,427)(111,231,315,430)(112,226,316,425)(113,416,369,211)(114,411,370,214)(115,414,371,209)(116,409,372,212)(117,412,373,215)(118,415,374,210)(119,410,375,213)(120,413,376,216)(161,442,382,189)(162,445,383,192)(163,448,384,187)(164,443,377,190)(165,446,378,185)(166,441,379,188)(167,444,380,191)(168,447,381,186)(241,321,392,336)(242,324,385,331)(243,327,386,334)(244,322,387,329)(245,325,388,332)(246,328,389,335)(247,323,390,330)(248,326,391,333)(265,297,433,344)(266,300,434,339)(267,303,435,342)(268,298,436,337)(269,301,437,340)(270,304,438,343)(271,299,439,338)(272,302,440,341) );
 
G=PermutationGroup([[(1,192,197,32,35,53,232),(2,225,54,36,25,198,185),(3,186,199,26,37,55,226),(4,227,56,38,27,200,187),(5,188,193,28,39,49,228),(6,229,50,40,29,194,189),(7,190,195,30,33,51,230),(8,231,52,34,31,196,191),(9,405,170,267,24,412,246),(10,247,413,17,268,171,406),(11,407,172,269,18,414,248),(12,241,415,19,270,173,408),(13,401,174,271,20,416,242),(14,243,409,21,272,175,402),(15,403,176,265,22,410,244),(16,245,411,23,266,169,404),(41,437,206,209,391,253,179),(42,180,254,392,210,207,438),(43,439,208,211,385,255,181),(44,182,256,386,212,201,440),(45,433,202,213,387,249,183),(46,184,250,388,214,203,434),(47,435,204,215,389,251,177),(48,178,252,390,216,205,436),(57,325,370,95,300,311,68),(58,69,312,301,96,371,326),(59,327,372,89,302,305,70),(60,71,306,303,90,373,328),(61,321,374,91,304,307,72),(62,65,308,297,92,375,322),(63,323,376,93,298,309,66),(64,67,310,299,94,369,324),(73,384,277,107,355,141,284),(74,285,142,356,108,278,377),(75,378,279,109,357,143,286),(76,287,144,358,110,280,379),(77,380,273,111,359,137,288),(78,281,138,360,112,274,381),(79,382,275,105,353,139,282),(80,283,140,354,106,276,383),(81,365,129,314,99,166,296),(82,289,167,100,315,130,366),(83,367,131,316,101,168,290),(84,291,161,102,317,132,368),(85,361,133,318,103,162,292),(86,293,163,104,319,134,362),(87,363,135,320,97,164,294),(88,295,165,98,313,136,364),(113,331,158,147,349,338,124),(114,125,339,350,148,159,332),(115,333,160,149,351,340,126),(116,127,341,352,150,153,334),(117,335,154,151,345,342,128),(118,121,343,346,152,155,336),(119,329,156,145,347,344,122),(120,123,337,348,146,157,330),(217,257,422,425,233,447,395),(218,396,448,234,426,423,258),(219,259,424,427,235,441,397),(220,398,442,236,428,417,260),(221,261,418,429,237,443,399),(222,400,444,238,430,419,262),(223,263,420,431,239,445,393),(224,394,446,240,432,421,264)], [(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)], [(1,389,5,385),(2,388,6,392),(3,387,7,391),(4,386,8,390),(9,441,13,445),(10,448,14,444),(11,447,15,443),(12,446,16,442),(17,423,21,419),(18,422,22,418),(19,421,23,417),(20,420,24,424),(25,46,29,42),(26,45,30,41),(27,44,31,48),(28,43,32,47),(33,437,37,433),(34,436,38,440),(35,435,39,439),(36,434,40,438),(49,208,53,204),(50,207,54,203),(51,206,55,202),(52,205,56,201),(57,380,61,384),(58,379,62,383),(59,378,63,382),(60,377,64,381),(65,80,69,76),(66,79,70,75),(67,78,71,74),(68,77,72,73),(81,347,85,351),(82,346,86,350),(83,345,87,349),(84,352,88,348),(89,357,93,353),(90,356,94,360),(91,355,95,359),(92,354,96,358),(97,331,101,335),(98,330,102,334),(99,329,103,333),(100,336,104,332),(105,372,109,376),(106,371,110,375),(107,370,111,374),(108,369,112,373),(113,316,117,320),(114,315,118,319),(115,314,119,318),(116,313,120,317),(121,134,125,130),(122,133,126,129),(123,132,127,136),(124,131,128,135),(137,304,141,300),(138,303,142,299),(139,302,143,298),(140,301,144,297),(145,292,149,296),(146,291,150,295),(147,290,151,294),(148,289,152,293),(153,165,157,161),(154,164,158,168),(155,163,159,167),(156,162,160,166),(169,220,173,224),(170,219,174,223),(171,218,175,222),(172,217,176,221),(177,193,181,197),(178,200,182,196),(179,199,183,195),(180,198,184,194),(185,250,189,254),(186,249,190,253),(187,256,191,252),(188,255,192,251),(209,226,213,230),(210,225,214,229),(211,232,215,228),(212,231,216,227),(233,244,237,248),(234,243,238,247),(235,242,239,246),(236,241,240,245),(257,265,261,269),(258,272,262,268),(259,271,263,267),(260,270,264,266),(273,321,277,325),(274,328,278,324),(275,327,279,323),(276,326,280,322),(281,306,285,310),(282,305,286,309),(283,312,287,308),(284,311,288,307),(337,368,341,364),(338,367,342,363),(339,366,343,362),(340,365,344,361),(393,405,397,401),(394,404,398,408),(395,403,399,407),(396,402,400,406),(409,430,413,426),(410,429,414,425),(411,428,415,432),(412,427,416,431)], [(1,103,239,276),(2,98,240,279),(3,101,233,274),(4,104,234,277),(5,99,235,280),(6,102,236,275),(7,97,237,278),(8,100,238,273),(9,60,251,154),(10,63,252,157),(11,58,253,160),(12,61,254,155),(13,64,255,158),(14,59,256,153),(15,62,249,156),(16,57,250,159),(17,93,205,123),(18,96,206,126),(19,91,207,121),(20,94,208,124),(21,89,201,127),(22,92,202,122),(23,95,203,125),(24,90,204,128),(25,88,224,286),(26,83,217,281),(27,86,218,284),(28,81,219,287),(29,84,220,282),(30,87,221,285),(31,82,222,288),(32,85,223,283),(33,363,261,142),(34,366,262,137),(35,361,263,140),(36,364,264,143),(37,367,257,138),(38,362,258,141),(39,365,259,144),(40,368,260,139),(41,351,172,312),(42,346,173,307),(43,349,174,310),(44,352,175,305),(45,347,176,308),(46,350,169,311),(47,345,170,306),(48,348,171,309),(49,129,424,358),(50,132,417,353),(51,135,418,356),(52,130,419,359),(53,133,420,354),(54,136,421,357),(55,131,422,360),(56,134,423,355),(65,183,145,403),(66,178,146,406),(67,181,147,401),(68,184,148,404),(69,179,149,407),(70,182,150,402),(71,177,151,405),(72,180,152,408),(73,200,293,396),(74,195,294,399),(75,198,295,394),(76,193,296,397),(77,196,289,400),(78,199,290,395),(79,194,291,398),(80,197,292,393),(105,229,317,428),(106,232,318,431),(107,227,319,426),(108,230,320,429),(109,225,313,432),(110,228,314,427),(111,231,315,430),(112,226,316,425),(113,416,369,211),(114,411,370,214),(115,414,371,209),(116,409,372,212),(117,412,373,215),(118,415,374,210),(119,410,375,213),(120,413,376,216),(161,442,382,189),(162,445,383,192),(163,448,384,187),(164,443,377,190),(165,446,378,185),(166,441,379,188),(167,444,380,191),(168,447,381,186),(241,321,392,336),(242,324,385,331),(243,327,386,334),(244,322,387,329),(245,325,388,332),(246,328,389,335),(247,323,390,330),(248,326,391,333),(265,297,433,344),(266,300,434,339),(267,303,435,342),(268,298,436,337),(269,301,437,340),(270,304,438,343),(271,299,439,338),(272,302,440,341)]])
 

64 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N7A7B7C8A8B8C8D14A···14I28A···28F28G···28R56A···56L
order122244444444444444777888814···1428···2828···2856···56
size111122444414141414282828282224428282···24···48···84···4

64 irreducible representations

dim111111111222222244444
type+++++++++++++--+-
imageC1C2C2C2C2C2C2C2C4D4D7C4○D4D14D14D14C4×D7C8.C22D4⋊2D7D4×D7SD16⋊D7Q16⋊D7
kernelC7⋊Q16⋊C4C4.Dic14C56⋊C4C28.44D4C7×Q8⋊C4Dic7⋊3Q8C2×C7⋊Q16Q8×Dic7C7⋊Q16C2×Dic7Q8⋊C4C28C4⋊C4C2×C8C2×Q8Q8C14C4C22C2C2
# reps1111111182323331223366

Matrix representation of C7⋊Q16⋊C4 ►in GL8(𝔽113)

10112000000
11112000000
1091120000
10112100000
00008911200
000010010300
000032079112
0000238110
,
591362810000
828623190000
4525100440000
8112102940000
000093297673
000059575789
000051553598
00004169741
,
0011210000
104111190000
0011200000
1011200000
000053571164
000047604657
000065424156
00009747072
,
29647190000
36128110000
3112591070000
3810653240000
000097845992
0000265746111
000022227429
0000225951111

G:=sub<GL(8,GF(113))| [10,11,1,10,0,0,0,0,112,112,0,112,0,0,0,0,0,0,9,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,89,100,32,23,0,0,0,0,112,103,0,81,0,0,0,0,0,0,79,1,0,0,0,0,0,0,112,0],[59,82,45,81,0,0,0,0,13,86,25,12,0,0,0,0,62,23,100,102,0,0,0,0,81,19,44,94,0,0,0,0,0,0,0,0,93,59,51,4,0,0,0,0,29,57,55,16,0,0,0,0,76,57,35,97,0,0,0,0,73,89,98,41],[0,104,0,1,0,0,0,0,0,1,0,0,0,0,0,0,112,111,112,112,0,0,0,0,1,9,0,0,0,0,0,0,0,0,0,0,53,47,65,9,0,0,0,0,57,60,42,74,0,0,0,0,11,46,41,70,0,0,0,0,64,57,56,72],[29,36,3,38,0,0,0,0,6,1,112,106,0,0,0,0,47,28,59,53,0,0,0,0,19,11,107,24,0,0,0,0,0,0,0,0,97,26,22,22,0,0,0,0,84,57,22,59,0,0,0,0,59,46,74,51,0,0,0,0,92,111,29,111] >;
 

C7⋊Q16⋊C4 in GAP, Magma, Sage, TeX 

C_7\rtimes Q_{16}\rtimes C_4
 
% in TeX
 
G:=Group("C7:Q16:C4");
 
// GroupNames label
 
G:=SmallGroup(448,323);
 
// by ID
 
G=gap.SmallGroup(448,323);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,344,758,135,184,570,297,136,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^7=b^8=d^4=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^6*c>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁