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G = Q16×C25order 400 = 24·52

Direct product of C25 and Q16

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: Q16×C25, C8.C50, Q8.C50, C200.3C2, C40.4C10, C50.16D4, C100.19C22, C5.(C5×Q16), (C5×Q16).C5, C4.3(C2×C50), C2.5(D4×C25), C10.16(C5×D4), (Q8×C25).2C2, (C5×Q8).3C10, C20.19(C2×C10), SmallGroup(400,27)

Series: Derived Chief Lower central Upper central

C1C4 — Q16×C25
C1C2C10C20C100Q8×C25 — Q16×C25
C1C2C4 — Q16×C25
C1C50C100 — Q16×C25

Generators and relations for Q16×C25
 G = < a,b,c | a25=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

2C4
2C4
2C20
2C20
2C100
2C100

Smallest permutation representation of Q16×C25
Regular action on 400 points
Generators in S400
(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)
(1 334 391 81 204 108 47 294)(2 335 392 82 205 109 48 295)(3 336 393 83 206 110 49 296)(4 337 394 84 207 111 50 297)(5 338 395 85 208 112 26 298)(6 339 396 86 209 113 27 299)(7 340 397 87 210 114 28 300)(8 341 398 88 211 115 29 276)(9 342 399 89 212 116 30 277)(10 343 400 90 213 117 31 278)(11 344 376 91 214 118 32 279)(12 345 377 92 215 119 33 280)(13 346 378 93 216 120 34 281)(14 347 379 94 217 121 35 282)(15 348 380 95 218 122 36 283)(16 349 381 96 219 123 37 284)(17 350 382 97 220 124 38 285)(18 326 383 98 221 125 39 286)(19 327 384 99 222 101 40 287)(20 328 385 100 223 102 41 288)(21 329 386 76 224 103 42 289)(22 330 387 77 225 104 43 290)(23 331 388 78 201 105 44 291)(24 332 389 79 202 106 45 292)(25 333 390 80 203 107 46 293)(51 169 325 247 356 130 182 252)(52 170 301 248 357 131 183 253)(53 171 302 249 358 132 184 254)(54 172 303 250 359 133 185 255)(55 173 304 226 360 134 186 256)(56 174 305 227 361 135 187 257)(57 175 306 228 362 136 188 258)(58 151 307 229 363 137 189 259)(59 152 308 230 364 138 190 260)(60 153 309 231 365 139 191 261)(61 154 310 232 366 140 192 262)(62 155 311 233 367 141 193 263)(63 156 312 234 368 142 194 264)(64 157 313 235 369 143 195 265)(65 158 314 236 370 144 196 266)(66 159 315 237 371 145 197 267)(67 160 316 238 372 146 198 268)(68 161 317 239 373 147 199 269)(69 162 318 240 374 148 200 270)(70 163 319 241 375 149 176 271)(71 164 320 242 351 150 177 272)(72 165 321 243 352 126 178 273)(73 166 322 244 353 127 179 274)(74 167 323 245 354 128 180 275)(75 168 324 246 355 129 181 251)
(1 367 204 62)(2 368 205 63)(3 369 206 64)(4 370 207 65)(5 371 208 66)(6 372 209 67)(7 373 210 68)(8 374 211 69)(9 375 212 70)(10 351 213 71)(11 352 214 72)(12 353 215 73)(13 354 216 74)(14 355 217 75)(15 356 218 51)(16 357 219 52)(17 358 220 53)(18 359 221 54)(19 360 222 55)(20 361 223 56)(21 362 224 57)(22 363 225 58)(23 364 201 59)(24 365 202 60)(25 366 203 61)(26 197 395 315)(27 198 396 316)(28 199 397 317)(29 200 398 318)(30 176 399 319)(31 177 400 320)(32 178 376 321)(33 179 377 322)(34 180 378 323)(35 181 379 324)(36 182 380 325)(37 183 381 301)(38 184 382 302)(39 185 383 303)(40 186 384 304)(41 187 385 305)(42 188 386 306)(43 189 387 307)(44 190 388 308)(45 191 389 309)(46 192 390 310)(47 193 391 311)(48 194 392 312)(49 195 393 313)(50 196 394 314)(76 175 289 136)(77 151 290 137)(78 152 291 138)(79 153 292 139)(80 154 293 140)(81 155 294 141)(82 156 295 142)(83 157 296 143)(84 158 297 144)(85 159 298 145)(86 160 299 146)(87 161 300 147)(88 162 276 148)(89 163 277 149)(90 164 278 150)(91 165 279 126)(92 166 280 127)(93 167 281 128)(94 168 282 129)(95 169 283 130)(96 170 284 131)(97 171 285 132)(98 172 286 133)(99 173 287 134)(100 174 288 135)(101 256 327 226)(102 257 328 227)(103 258 329 228)(104 259 330 229)(105 260 331 230)(106 261 332 231)(107 262 333 232)(108 263 334 233)(109 264 335 234)(110 265 336 235)(111 266 337 236)(112 267 338 237)(113 268 339 238)(114 269 340 239)(115 270 341 240)(116 271 342 241)(117 272 343 242)(118 273 344 243)(119 274 345 244)(120 275 346 245)(121 251 347 246)(122 252 348 247)(123 253 349 248)(124 254 350 249)(125 255 326 250)

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

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

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

175 conjugacy classes

class 1  2 4A4B4C5A5B5C5D8A8B10A10B10C10D20A20B20C20D20E···20L25A···25T40A···40H50A···50T100A···100T100U···100BH200A···200AN
order12444555588101010102020202020···2025···2540···4050···50100···100100···100200···200
size11244111122111122224···41···12···21···12···24···42···2

175 irreducible representations

dim111111111222222
type++++-
imageC1C2C2C5C10C10C25C50C50D4Q16C5×D4C5×Q16D4×C25Q16×C25
kernelQ16×C25C200Q8×C25C5×Q16C40C5×Q8Q16C8Q8C50C25C10C5C2C1
# reps11244820204012482040

Matrix representation of Q16×C25 in GL2(𝔽401) generated by

1250
0125
,
053
17453
,
334263
26567
G:=sub<GL(2,GF(401))| [125,0,0,125],[0,174,53,53],[334,265,263,67] >;

Q16×C25 in GAP, Magma, Sage, TeX

Q_{16}\times C_{25}
% in TeX

G:=Group("Q16xC25");
// GroupNames label

G:=SmallGroup(400,27);
// by ID

G=gap.SmallGroup(400,27);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-5,-2,1200,265,1207,194,5283,2649,261]);
// Polycyclic

G:=Group<a,b,c|a^25=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Q16×C25 in TeX

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