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## G = Q16×C52order 400 = 24·52

### Direct product of C52 and Q16

direct product, metacyclic, nilpotent (class 3), monomial

Aliases: Q16×C52, C40.5C10, C4.3C102, C8.(C5×C10), Q8.(C5×C10), (C5×C40).5C2, C10.22(C5×D4), (C5×C10).43D4, (C5×Q8).4C10, C2.5(D4×C52), C20.25(C2×C10), (Q8×C52).3C2, (C5×C20).52C22, SmallGroup(400,115)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — Q16×C52
 Chief series C1 — C2 — C4 — C20 — C5×C20 — Q8×C52 — Q16×C52
 Lower central C1 — C2 — C4 — Q16×C52
 Upper central C1 — C5×C10 — C5×C20 — Q16×C52

Generators and relations for Q16×C52
G = < a,b,c,d | a5=b5=c8=1, d2=c4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 88 in 72 conjugacy classes, 56 normal (10 characteristic)
C1, C2, C4, C4, C5, C8, Q8, C10, Q16, C20, C20, C52, C40, C5×Q8, C5×C10, C5×Q16, C5×C20, C5×C20, C5×C40, Q8×C52, Q16×C52
Quotients: C1, C2, C22, C5, D4, C10, Q16, C2×C10, C52, C5×D4, C5×C10, C5×Q16, C102, D4×C52, Q16×C52

Smallest permutation representation of Q16×C52
Regular action on 400 points
Generators in S400
(1 298 387 87 363)(2 299 388 88 364)(3 300 389 81 365)(4 301 390 82 366)(5 302 391 83 367)(6 303 392 84 368)(7 304 385 85 361)(8 297 386 86 362)(9 288 341 33 317)(10 281 342 34 318)(11 282 343 35 319)(12 283 344 36 320)(13 284 337 37 313)(14 285 338 38 314)(15 286 339 39 315)(16 287 340 40 316)(17 256 349 41 325)(18 249 350 42 326)(19 250 351 43 327)(20 251 352 44 328)(21 252 345 45 321)(22 253 346 46 322)(23 254 347 47 323)(24 255 348 48 324)(25 309 195 280 171)(26 310 196 273 172)(27 311 197 274 173)(28 312 198 275 174)(29 305 199 276 175)(30 306 200 277 176)(31 307 193 278 169)(32 308 194 279 170)(49 333 179 264 155)(50 334 180 257 156)(51 335 181 258 157)(52 336 182 259 158)(53 329 183 260 159)(54 330 184 261 160)(55 331 177 262 153)(56 332 178 263 154)(57 148 123 208 69)(58 149 124 201 70)(59 150 125 202 71)(60 151 126 203 72)(61 152 127 204 65)(62 145 128 205 66)(63 146 121 206 67)(64 147 122 207 68)(73 357 236 135 381)(74 358 237 136 382)(75 359 238 129 383)(76 360 239 130 384)(77 353 240 131 377)(78 354 233 132 378)(79 355 234 133 379)(80 356 235 134 380)(89 373 212 120 397)(90 374 213 113 398)(91 375 214 114 399)(92 376 215 115 400)(93 369 216 116 393)(94 370 209 117 394)(95 371 210 118 395)(96 372 211 119 396)(97 143 229 221 106)(98 144 230 222 107)(99 137 231 223 108)(100 138 232 224 109)(101 139 225 217 110)(102 140 226 218 111)(103 141 227 219 112)(104 142 228 220 105)(161 246 294 185 270)(162 247 295 186 271)(163 248 296 187 272)(164 241 289 188 265)(165 242 290 189 266)(166 243 291 190 267)(167 244 292 191 268)(168 245 293 192 269)
(1 71 118 355 112)(2 72 119 356 105)(3 65 120 357 106)(4 66 113 358 107)(5 67 114 359 108)(6 68 115 360 109)(7 69 116 353 110)(8 70 117 354 111)(9 248 256 309 155)(10 241 249 310 156)(11 242 250 311 157)(12 243 251 312 158)(13 244 252 305 159)(14 245 253 306 160)(15 246 254 307 153)(16 247 255 308 154)(17 25 264 317 163)(18 26 257 318 164)(19 27 258 319 165)(20 28 259 320 166)(21 29 260 313 167)(22 30 261 314 168)(23 31 262 315 161)(24 32 263 316 162)(33 272 325 171 179)(34 265 326 172 180)(35 266 327 173 181)(36 267 328 174 182)(37 268 321 175 183)(38 269 322 176 184)(39 270 323 169 177)(40 271 324 170 178)(41 280 333 341 187)(42 273 334 342 188)(43 274 335 343 189)(44 275 336 344 190)(45 276 329 337 191)(46 277 330 338 192)(47 278 331 339 185)(48 279 332 340 186)(49 288 296 349 195)(50 281 289 350 196)(51 282 290 351 197)(52 283 291 352 198)(53 284 292 345 199)(54 285 293 346 200)(55 286 294 347 193)(56 287 295 348 194)(57 393 240 101 304)(58 394 233 102 297)(59 395 234 103 298)(60 396 235 104 299)(61 397 236 97 300)(62 398 237 98 301)(63 399 238 99 302)(64 400 239 100 303)(73 221 365 204 212)(74 222 366 205 213)(75 223 367 206 214)(76 224 368 207 215)(77 217 361 208 216)(78 218 362 201 209)(79 219 363 202 210)(80 220 364 203 211)(81 127 373 381 229)(82 128 374 382 230)(83 121 375 383 231)(84 122 376 384 232)(85 123 369 377 225)(86 124 370 378 226)(87 125 371 379 227)(88 126 372 380 228)(89 135 143 389 152)(90 136 144 390 145)(91 129 137 391 146)(92 130 138 392 147)(93 131 139 385 148)(94 132 140 386 149)(95 133 141 387 150)(96 134 142 388 151)
(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)
(1 242 5 246)(2 241 6 245)(3 248 7 244)(4 247 8 243)(9 110 13 106)(10 109 14 105)(11 108 15 112)(12 107 16 111)(17 208 21 204)(18 207 22 203)(19 206 23 202)(20 205 24 201)(25 216 29 212)(26 215 30 211)(27 214 31 210)(28 213 32 209)(33 225 37 229)(34 232 38 228)(35 231 39 227)(36 230 40 226)(41 148 45 152)(42 147 46 151)(43 146 47 150)(44 145 48 149)(49 240 53 236)(50 239 54 235)(51 238 55 234)(52 237 56 233)(57 345 61 349)(58 352 62 348)(59 351 63 347)(60 350 64 346)(65 256 69 252)(66 255 70 251)(67 254 71 250)(68 253 72 249)(73 264 77 260)(74 263 78 259)(75 262 79 258)(76 261 80 257)(81 272 85 268)(82 271 86 267)(83 270 87 266)(84 269 88 265)(89 280 93 276)(90 279 94 275)(91 278 95 274)(92 277 96 273)(97 288 101 284)(98 287 102 283)(99 286 103 282)(100 285 104 281)(113 308 117 312)(114 307 118 311)(115 306 119 310)(116 305 120 309)(121 323 125 327)(122 322 126 326)(123 321 127 325)(124 328 128 324)(129 331 133 335)(130 330 134 334)(131 329 135 333)(132 336 136 332)(137 339 141 343)(138 338 142 342)(139 337 143 341)(140 344 144 340)(153 355 157 359)(154 354 158 358)(155 353 159 357)(156 360 160 356)(161 363 165 367)(162 362 166 366)(163 361 167 365)(164 368 168 364)(169 371 173 375)(170 370 174 374)(171 369 175 373)(172 376 176 372)(177 379 181 383)(178 378 182 382)(179 377 183 381)(180 384 184 380)(185 387 189 391)(186 386 190 390)(187 385 191 389)(188 392 192 388)(193 395 197 399)(194 394 198 398)(195 393 199 397)(196 400 200 396)(217 313 221 317)(218 320 222 316)(219 319 223 315)(220 318 224 314)(289 303 293 299)(290 302 294 298)(291 301 295 297)(292 300 296 304)

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

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

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

175 conjugacy classes

 class 1 2 4A 4B 4C 5A ··· 5X 8A 8B 10A ··· 10X 20A ··· 20X 20Y ··· 20BT 40A ··· 40AV order 1 2 4 4 4 5 ··· 5 8 8 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 2 4 4 1 ··· 1 2 2 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

175 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + - image C1 C2 C2 C5 C10 C10 D4 Q16 C5×D4 C5×Q16 kernel Q16×C52 C5×C40 Q8×C52 C5×Q16 C40 C5×Q8 C5×C10 C52 C10 C5 # reps 1 1 2 24 24 48 1 2 24 48

Matrix representation of Q16×C52 in GL4(𝔽41) generated by

 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 37 0 0 0 0 37
,
 20 14 0 0 27 21 0 0 0 0 17 17 0 0 12 0
,
 0 1 0 0 1 0 0 0 0 0 20 13 0 0 7 21
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,37,0,0,0,0,37],[20,27,0,0,14,21,0,0,0,0,17,12,0,0,17,0],[0,1,0,0,1,0,0,0,0,0,20,7,0,0,13,21] >;

Q16×C52 in GAP, Magma, Sage, TeX

Q_{16}\times C_5^2
% in TeX

G:=Group("Q16xC5^2");
// GroupNames label

G:=SmallGroup(400,115);
// by ID

G=gap.SmallGroup(400,115);
# by ID

G:=PCGroup([6,-2,-2,-5,-5,-2,-2,1200,1225,1207,9004,4510,88]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽