metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C25⋊2Q16, Q8.D25, C4.3D50, C50.9D4, C20.3D10, C100.3C22, Dic50.2C2, C5.(C5⋊Q16), (C5×Q8).1D5, C25⋊2C8.1C2, (Q8×C25).1C2, C2.6(C25⋊D4), C10.16(C5⋊D4), SmallGroup(400,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C25⋊Q16
G = < a,b,c | a25=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C25⋊Q16
►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)
(1 356 60 347 35 393 77 307)(2 355 61 346 36 392 78 306)(3 354 62 345 37 391 79 305)(4 353 63 344 38 390 80 304)(5 352 64 343 39 389 81 303)(6 351 65 342 40 388 82 302)(7 375 66 341 41 387 83 301)(8 374 67 340 42 386 84 325)(9 373 68 339 43 385 85 324)(10 372 69 338 44 384 86 323)(11 371 70 337 45 383 87 322)(12 370 71 336 46 382 88 321)(13 369 72 335 47 381 89 320)(14 368 73 334 48 380 90 319)(15 367 74 333 49 379 91 318)(16 366 75 332 50 378 92 317)(17 365 51 331 26 377 93 316)(18 364 52 330 27 376 94 315)(19 363 53 329 28 400 95 314)(20 362 54 328 29 399 96 313)(21 361 55 327 30 398 97 312)(22 360 56 326 31 397 98 311)(23 359 57 350 32 396 99 310)(24 358 58 349 33 395 100 309)(25 357 59 348 34 394 76 308)(101 265 188 205 130 288 172 235)(102 264 189 204 131 287 173 234)(103 263 190 203 132 286 174 233)(104 262 191 202 133 285 175 232)(105 261 192 201 134 284 151 231)(106 260 193 225 135 283 152 230)(107 259 194 224 136 282 153 229)(108 258 195 223 137 281 154 228)(109 257 196 222 138 280 155 227)(110 256 197 221 139 279 156 226)(111 255 198 220 140 278 157 250)(112 254 199 219 141 277 158 249)(113 253 200 218 142 276 159 248)(114 252 176 217 143 300 160 247)(115 251 177 216 144 299 161 246)(116 275 178 215 145 298 162 245)(117 274 179 214 146 297 163 244)(118 273 180 213 147 296 164 243)(119 272 181 212 148 295 165 242)(120 271 182 211 149 294 166 241)(121 270 183 210 150 293 167 240)(122 269 184 209 126 292 168 239)(123 268 185 208 127 291 169 238)(124 267 186 207 128 290 170 237)(125 266 187 206 129 289 171 236)
(1 139 35 110)(2 140 36 111)(3 141 37 112)(4 142 38 113)(5 143 39 114)(6 144 40 115)(7 145 41 116)(8 146 42 117)(9 147 43 118)(10 148 44 119)(11 149 45 120)(12 150 46 121)(13 126 47 122)(14 127 48 123)(15 128 49 124)(16 129 50 125)(17 130 26 101)(18 131 27 102)(19 132 28 103)(20 133 29 104)(21 134 30 105)(22 135 31 106)(23 136 32 107)(24 137 33 108)(25 138 34 109)(51 188 93 172)(52 189 94 173)(53 190 95 174)(54 191 96 175)(55 192 97 151)(56 193 98 152)(57 194 99 153)(58 195 100 154)(59 196 76 155)(60 197 77 156)(61 198 78 157)(62 199 79 158)(63 200 80 159)(64 176 81 160)(65 177 82 161)(66 178 83 162)(67 179 84 163)(68 180 85 164)(69 181 86 165)(70 182 87 166)(71 183 88 167)(72 184 89 168)(73 185 90 169)(74 186 91 170)(75 187 92 171)(201 398 231 361)(202 399 232 362)(203 400 233 363)(204 376 234 364)(205 377 235 365)(206 378 236 366)(207 379 237 367)(208 380 238 368)(209 381 239 369)(210 382 240 370)(211 383 241 371)(212 384 242 372)(213 385 243 373)(214 386 244 374)(215 387 245 375)(216 388 246 351)(217 389 247 352)(218 390 248 353)(219 391 249 354)(220 392 250 355)(221 393 226 356)(222 394 227 357)(223 395 228 358)(224 396 229 359)(225 397 230 360)(251 302 299 342)(252 303 300 343)(253 304 276 344)(254 305 277 345)(255 306 278 346)(256 307 279 347)(257 308 280 348)(258 309 281 349)(259 310 282 350)(260 311 283 326)(261 312 284 327)(262 313 285 328)(263 314 286 329)(264 315 287 330)(265 316 288 331)(266 317 289 332)(267 318 290 333)(268 319 291 334)(269 320 292 335)(270 321 293 336)(271 322 294 337)(272 323 295 338)(273 324 296 339)(274 325 297 340)(275 301 298 341)
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), (1,356,60,347,35,393,77,307)(2,355,61,346,36,392,78,306)(3,354,62,345,37,391,79,305)(4,353,63,344,38,390,80,304)(5,352,64,343,39,389,81,303)(6,351,65,342,40,388,82,302)(7,375,66,341,41,387,83,301)(8,374,67,340,42,386,84,325)(9,373,68,339,43,385,85,324)(10,372,69,338,44,384,86,323)(11,371,70,337,45,383,87,322)(12,370,71,336,46,382,88,321)(13,369,72,335,47,381,89,320)(14,368,73,334,48,380,90,319)(15,367,74,333,49,379,91,318)(16,366,75,332,50,378,92,317)(17,365,51,331,26,377,93,316)(18,364,52,330,27,376,94,315)(19,363,53,329,28,400,95,314)(20,362,54,328,29,399,96,313)(21,361,55,327,30,398,97,312)(22,360,56,326,31,397,98,311)(23,359,57,350,32,396,99,310)(24,358,58,349,33,395,100,309)(25,357,59,348,34,394,76,308)(101,265,188,205,130,288,172,235)(102,264,189,204,131,287,173,234)(103,263,190,203,132,286,174,233)(104,262,191,202,133,285,175,232)(105,261,192,201,134,284,151,231)(106,260,193,225,135,283,152,230)(107,259,194,224,136,282,153,229)(108,258,195,223,137,281,154,228)(109,257,196,222,138,280,155,227)(110,256,197,221,139,279,156,226)(111,255,198,220,140,278,157,250)(112,254,199,219,141,277,158,249)(113,253,200,218,142,276,159,248)(114,252,176,217,143,300,160,247)(115,251,177,216,144,299,161,246)(116,275,178,215,145,298,162,245)(117,274,179,214,146,297,163,244)(118,273,180,213,147,296,164,243)(119,272,181,212,148,295,165,242)(120,271,182,211,149,294,166,241)(121,270,183,210,150,293,167,240)(122,269,184,209,126,292,168,239)(123,268,185,208,127,291,169,238)(124,267,186,207,128,290,170,237)(125,266,187,206,129,289,171,236), (1,139,35,110)(2,140,36,111)(3,141,37,112)(4,142,38,113)(5,143,39,114)(6,144,40,115)(7,145,41,116)(8,146,42,117)(9,147,43,118)(10,148,44,119)(11,149,45,120)(12,150,46,121)(13,126,47,122)(14,127,48,123)(15,128,49,124)(16,129,50,125)(17,130,26,101)(18,131,27,102)(19,132,28,103)(20,133,29,104)(21,134,30,105)(22,135,31,106)(23,136,32,107)(24,137,33,108)(25,138,34,109)(51,188,93,172)(52,189,94,173)(53,190,95,174)(54,191,96,175)(55,192,97,151)(56,193,98,152)(57,194,99,153)(58,195,100,154)(59,196,76,155)(60,197,77,156)(61,198,78,157)(62,199,79,158)(63,200,80,159)(64,176,81,160)(65,177,82,161)(66,178,83,162)(67,179,84,163)(68,180,85,164)(69,181,86,165)(70,182,87,166)(71,183,88,167)(72,184,89,168)(73,185,90,169)(74,186,91,170)(75,187,92,171)(201,398,231,361)(202,399,232,362)(203,400,233,363)(204,376,234,364)(205,377,235,365)(206,378,236,366)(207,379,237,367)(208,380,238,368)(209,381,239,369)(210,382,240,370)(211,383,241,371)(212,384,242,372)(213,385,243,373)(214,386,244,374)(215,387,245,375)(216,388,246,351)(217,389,247,352)(218,390,248,353)(219,391,249,354)(220,392,250,355)(221,393,226,356)(222,394,227,357)(223,395,228,358)(224,396,229,359)(225,397,230,360)(251,302,299,342)(252,303,300,343)(253,304,276,344)(254,305,277,345)(255,306,278,346)(256,307,279,347)(257,308,280,348)(258,309,281,349)(259,310,282,350)(260,311,283,326)(261,312,284,327)(262,313,285,328)(263,314,286,329)(264,315,287,330)(265,316,288,331)(266,317,289,332)(267,318,290,333)(268,319,291,334)(269,320,292,335)(270,321,293,336)(271,322,294,337)(272,323,295,338)(273,324,296,339)(274,325,297,340)(275,301,298,341)>;
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), (1,356,60,347,35,393,77,307)(2,355,61,346,36,392,78,306)(3,354,62,345,37,391,79,305)(4,353,63,344,38,390,80,304)(5,352,64,343,39,389,81,303)(6,351,65,342,40,388,82,302)(7,375,66,341,41,387,83,301)(8,374,67,340,42,386,84,325)(9,373,68,339,43,385,85,324)(10,372,69,338,44,384,86,323)(11,371,70,337,45,383,87,322)(12,370,71,336,46,382,88,321)(13,369,72,335,47,381,89,320)(14,368,73,334,48,380,90,319)(15,367,74,333,49,379,91,318)(16,366,75,332,50,378,92,317)(17,365,51,331,26,377,93,316)(18,364,52,330,27,376,94,315)(19,363,53,329,28,400,95,314)(20,362,54,328,29,399,96,313)(21,361,55,327,30,398,97,312)(22,360,56,326,31,397,98,311)(23,359,57,350,32,396,99,310)(24,358,58,349,33,395,100,309)(25,357,59,348,34,394,76,308)(101,265,188,205,130,288,172,235)(102,264,189,204,131,287,173,234)(103,263,190,203,132,286,174,233)(104,262,191,202,133,285,175,232)(105,261,192,201,134,284,151,231)(106,260,193,225,135,283,152,230)(107,259,194,224,136,282,153,229)(108,258,195,223,137,281,154,228)(109,257,196,222,138,280,155,227)(110,256,197,221,139,279,156,226)(111,255,198,220,140,278,157,250)(112,254,199,219,141,277,158,249)(113,253,200,218,142,276,159,248)(114,252,176,217,143,300,160,247)(115,251,177,216,144,299,161,246)(116,275,178,215,145,298,162,245)(117,274,179,214,146,297,163,244)(118,273,180,213,147,296,164,243)(119,272,181,212,148,295,165,242)(120,271,182,211,149,294,166,241)(121,270,183,210,150,293,167,240)(122,269,184,209,126,292,168,239)(123,268,185,208,127,291,169,238)(124,267,186,207,128,290,170,237)(125,266,187,206,129,289,171,236), (1,139,35,110)(2,140,36,111)(3,141,37,112)(4,142,38,113)(5,143,39,114)(6,144,40,115)(7,145,41,116)(8,146,42,117)(9,147,43,118)(10,148,44,119)(11,149,45,120)(12,150,46,121)(13,126,47,122)(14,127,48,123)(15,128,49,124)(16,129,50,125)(17,130,26,101)(18,131,27,102)(19,132,28,103)(20,133,29,104)(21,134,30,105)(22,135,31,106)(23,136,32,107)(24,137,33,108)(25,138,34,109)(51,188,93,172)(52,189,94,173)(53,190,95,174)(54,191,96,175)(55,192,97,151)(56,193,98,152)(57,194,99,153)(58,195,100,154)(59,196,76,155)(60,197,77,156)(61,198,78,157)(62,199,79,158)(63,200,80,159)(64,176,81,160)(65,177,82,161)(66,178,83,162)(67,179,84,163)(68,180,85,164)(69,181,86,165)(70,182,87,166)(71,183,88,167)(72,184,89,168)(73,185,90,169)(74,186,91,170)(75,187,92,171)(201,398,231,361)(202,399,232,362)(203,400,233,363)(204,376,234,364)(205,377,235,365)(206,378,236,366)(207,379,237,367)(208,380,238,368)(209,381,239,369)(210,382,240,370)(211,383,241,371)(212,384,242,372)(213,385,243,373)(214,386,244,374)(215,387,245,375)(216,388,246,351)(217,389,247,352)(218,390,248,353)(219,391,249,354)(220,392,250,355)(221,393,226,356)(222,394,227,357)(223,395,228,358)(224,396,229,359)(225,397,230,360)(251,302,299,342)(252,303,300,343)(253,304,276,344)(254,305,277,345)(255,306,278,346)(256,307,279,347)(257,308,280,348)(258,309,281,349)(259,310,282,350)(260,311,283,326)(261,312,284,327)(262,313,285,328)(263,314,286,329)(264,315,287,330)(265,316,288,331)(266,317,289,332)(267,318,290,333)(268,319,291,334)(269,320,292,335)(270,321,293,336)(271,322,294,337)(272,323,295,338)(273,324,296,339)(274,325,297,340)(275,301,298,341) );
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)], [(1,356,60,347,35,393,77,307),(2,355,61,346,36,392,78,306),(3,354,62,345,37,391,79,305),(4,353,63,344,38,390,80,304),(5,352,64,343,39,389,81,303),(6,351,65,342,40,388,82,302),(7,375,66,341,41,387,83,301),(8,374,67,340,42,386,84,325),(9,373,68,339,43,385,85,324),(10,372,69,338,44,384,86,323),(11,371,70,337,45,383,87,322),(12,370,71,336,46,382,88,321),(13,369,72,335,47,381,89,320),(14,368,73,334,48,380,90,319),(15,367,74,333,49,379,91,318),(16,366,75,332,50,378,92,317),(17,365,51,331,26,377,93,316),(18,364,52,330,27,376,94,315),(19,363,53,329,28,400,95,314),(20,362,54,328,29,399,96,313),(21,361,55,327,30,398,97,312),(22,360,56,326,31,397,98,311),(23,359,57,350,32,396,99,310),(24,358,58,349,33,395,100,309),(25,357,59,348,34,394,76,308),(101,265,188,205,130,288,172,235),(102,264,189,204,131,287,173,234),(103,263,190,203,132,286,174,233),(104,262,191,202,133,285,175,232),(105,261,192,201,134,284,151,231),(106,260,193,225,135,283,152,230),(107,259,194,224,136,282,153,229),(108,258,195,223,137,281,154,228),(109,257,196,222,138,280,155,227),(110,256,197,221,139,279,156,226),(111,255,198,220,140,278,157,250),(112,254,199,219,141,277,158,249),(113,253,200,218,142,276,159,248),(114,252,176,217,143,300,160,247),(115,251,177,216,144,299,161,246),(116,275,178,215,145,298,162,245),(117,274,179,214,146,297,163,244),(118,273,180,213,147,296,164,243),(119,272,181,212,148,295,165,242),(120,271,182,211,149,294,166,241),(121,270,183,210,150,293,167,240),(122,269,184,209,126,292,168,239),(123,268,185,208,127,291,169,238),(124,267,186,207,128,290,170,237),(125,266,187,206,129,289,171,236)], [(1,139,35,110),(2,140,36,111),(3,141,37,112),(4,142,38,113),(5,143,39,114),(6,144,40,115),(7,145,41,116),(8,146,42,117),(9,147,43,118),(10,148,44,119),(11,149,45,120),(12,150,46,121),(13,126,47,122),(14,127,48,123),(15,128,49,124),(16,129,50,125),(17,130,26,101),(18,131,27,102),(19,132,28,103),(20,133,29,104),(21,134,30,105),(22,135,31,106),(23,136,32,107),(24,137,33,108),(25,138,34,109),(51,188,93,172),(52,189,94,173),(53,190,95,174),(54,191,96,175),(55,192,97,151),(56,193,98,152),(57,194,99,153),(58,195,100,154),(59,196,76,155),(60,197,77,156),(61,198,78,157),(62,199,79,158),(63,200,80,159),(64,176,81,160),(65,177,82,161),(66,178,83,162),(67,179,84,163),(68,180,85,164),(69,181,86,165),(70,182,87,166),(71,183,88,167),(72,184,89,168),(73,185,90,169),(74,186,91,170),(75,187,92,171),(201,398,231,361),(202,399,232,362),(203,400,233,363),(204,376,234,364),(205,377,235,365),(206,378,236,366),(207,379,237,367),(208,380,238,368),(209,381,239,369),(210,382,240,370),(211,383,241,371),(212,384,242,372),(213,385,243,373),(214,386,244,374),(215,387,245,375),(216,388,246,351),(217,389,247,352),(218,390,248,353),(219,391,249,354),(220,392,250,355),(221,393,226,356),(222,394,227,357),(223,395,228,358),(224,396,229,359),(225,397,230,360),(251,302,299,342),(252,303,300,343),(253,304,276,344),(254,305,277,345),(255,306,278,346),(256,307,279,347),(257,308,280,348),(258,309,281,349),(259,310,282,350),(260,311,283,326),(261,312,284,327),(262,313,285,328),(263,314,286,329),(264,315,287,330),(265,316,288,331),(266,317,289,332),(267,318,290,333),(268,319,291,334),(269,320,292,335),(270,321,293,336),(271,322,294,337),(272,323,295,338),(273,324,296,339),(274,325,297,340),(275,301,298,341)]])
67 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 10A | 10B | 20A | ··· | 20F | 25A | ··· | 25J | 50A | ··· | 50J | 100A | ··· | 100AD |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 20 | ··· | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 2 | 4 | 100 | 2 | 2 | 50 | 50 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | D4 | D5 | Q16 | D10 | C5⋊D4 | D25 | D50 | C25⋊D4 | C5⋊Q16 | C25⋊Q16 |
| kernel | C25⋊Q16 | C25⋊2C8 | Dic50 | Q8×C25 | C50 | C5×Q8 | C25 | C20 | C10 | Q8 | C4 | C2 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 10 | 10 | 20 | 2 | 10 |
Matrix representation of C25⋊Q16 ►in GL4(𝔽401) generated by
| 253 | 88 | 0 | 0 |
| 56 | 197 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 161 | 142 | 0 | 0 |
| 354 | 240 | 0 | 0 |
| 0 | 0 | 0 | 348 |
| 0 | 0 | 227 | 348 |
| 295 | 297 | 0 | 0 |
| 189 | 106 | 0 | 0 |
| 0 | 0 | 76 | 199 |
| 0 | 0 | 376 | 325 |
G:=sub<GL(4,GF(401))| [253,56,0,0,88,197,0,0,0,0,1,0,0,0,0,1],[161,354,0,0,142,240,0,0,0,0,0,227,0,0,348,348],[295,189,0,0,297,106,0,0,0,0,76,376,0,0,199,325] >;
C25⋊Q16 in GAP, Magma, Sage, TeX
C_{25}\rtimes Q_{16} % in TeX
G:=Group("C25:Q16"); // GroupNames label
G:=SmallGroup(400,17);
// by ID
G=gap.SmallGroup(400,17);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,73,55,218,116,50,4324,628,11525]);
// Polycyclic
G:=Group<a,b,c|a^25=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export