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
Generators and relations for Q16×C25
G = < a,b,c | a25=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
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 94 178 143 368 57 344 247)(2 95 179 144 369 58 345 248)(3 96 180 145 370 59 346 249)(4 97 181 146 371 60 347 250)(5 98 182 147 372 61 348 226)(6 99 183 148 373 62 349 227)(7 100 184 149 374 63 350 228)(8 76 185 150 375 64 326 229)(9 77 186 126 351 65 327 230)(10 78 187 127 352 66 328 231)(11 79 188 128 353 67 329 232)(12 80 189 129 354 68 330 233)(13 81 190 130 355 69 331 234)(14 82 191 131 356 70 332 235)(15 83 192 132 357 71 333 236)(16 84 193 133 358 72 334 237)(17 85 194 134 359 73 335 238)(18 86 195 135 360 74 336 239)(19 87 196 136 361 75 337 240)(20 88 197 137 362 51 338 241)(21 89 198 138 363 52 339 242)(22 90 199 139 364 53 340 243)(23 91 200 140 365 54 341 244)(24 92 176 141 366 55 342 245)(25 93 177 142 367 56 343 246)(26 212 307 285 382 170 101 270)(27 213 308 286 383 171 102 271)(28 214 309 287 384 172 103 272)(29 215 310 288 385 173 104 273)(30 216 311 289 386 174 105 274)(31 217 312 290 387 175 106 275)(32 218 313 291 388 151 107 251)(33 219 314 292 389 152 108 252)(34 220 315 293 390 153 109 253)(35 221 316 294 391 154 110 254)(36 222 317 295 392 155 111 255)(37 223 318 296 393 156 112 256)(38 224 319 297 394 157 113 257)(39 225 320 298 395 158 114 258)(40 201 321 299 396 159 115 259)(41 202 322 300 397 160 116 260)(42 203 323 276 398 161 117 261)(43 204 324 277 399 162 118 262)(44 205 325 278 400 163 119 263)(45 206 301 279 376 164 120 264)(46 207 302 280 377 165 121 265)(47 208 303 281 378 166 122 266)(48 209 304 282 379 167 123 267)(49 210 305 283 380 168 124 268)(50 211 306 284 381 169 125 269)
(1 29 368 385)(2 30 369 386)(3 31 370 387)(4 32 371 388)(5 33 372 389)(6 34 373 390)(7 35 374 391)(8 36 375 392)(9 37 351 393)(10 38 352 394)(11 39 353 395)(12 40 354 396)(13 41 355 397)(14 42 356 398)(15 43 357 399)(16 44 358 400)(17 45 359 376)(18 46 360 377)(19 47 361 378)(20 48 362 379)(21 49 363 380)(22 50 364 381)(23 26 365 382)(24 27 366 383)(25 28 367 384)(51 282 88 267)(52 283 89 268)(53 284 90 269)(54 285 91 270)(55 286 92 271)(56 287 93 272)(57 288 94 273)(58 289 95 274)(59 290 96 275)(60 291 97 251)(61 292 98 252)(62 293 99 253)(63 294 100 254)(64 295 76 255)(65 296 77 256)(66 297 78 257)(67 298 79 258)(68 299 80 259)(69 300 81 260)(70 276 82 261)(71 277 83 262)(72 278 84 263)(73 279 85 264)(74 280 86 265)(75 281 87 266)(101 341 307 200)(102 342 308 176)(103 343 309 177)(104 344 310 178)(105 345 311 179)(106 346 312 180)(107 347 313 181)(108 348 314 182)(109 349 315 183)(110 350 316 184)(111 326 317 185)(112 327 318 186)(113 328 319 187)(114 329 320 188)(115 330 321 189)(116 331 322 190)(117 332 323 191)(118 333 324 192)(119 334 325 193)(120 335 301 194)(121 336 302 195)(122 337 303 196)(123 338 304 197)(124 339 305 198)(125 340 306 199)(126 156 230 223)(127 157 231 224)(128 158 232 225)(129 159 233 201)(130 160 234 202)(131 161 235 203)(132 162 236 204)(133 163 237 205)(134 164 238 206)(135 165 239 207)(136 166 240 208)(137 167 241 209)(138 168 242 210)(139 169 243 211)(140 170 244 212)(141 171 245 213)(142 172 246 214)(143 173 247 215)(144 174 248 216)(145 175 249 217)(146 151 250 218)(147 152 226 219)(148 153 227 220)(149 154 228 221)(150 155 229 222)
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), (1,94,178,143,368,57,344,247)(2,95,179,144,369,58,345,248)(3,96,180,145,370,59,346,249)(4,97,181,146,371,60,347,250)(5,98,182,147,372,61,348,226)(6,99,183,148,373,62,349,227)(7,100,184,149,374,63,350,228)(8,76,185,150,375,64,326,229)(9,77,186,126,351,65,327,230)(10,78,187,127,352,66,328,231)(11,79,188,128,353,67,329,232)(12,80,189,129,354,68,330,233)(13,81,190,130,355,69,331,234)(14,82,191,131,356,70,332,235)(15,83,192,132,357,71,333,236)(16,84,193,133,358,72,334,237)(17,85,194,134,359,73,335,238)(18,86,195,135,360,74,336,239)(19,87,196,136,361,75,337,240)(20,88,197,137,362,51,338,241)(21,89,198,138,363,52,339,242)(22,90,199,139,364,53,340,243)(23,91,200,140,365,54,341,244)(24,92,176,141,366,55,342,245)(25,93,177,142,367,56,343,246)(26,212,307,285,382,170,101,270)(27,213,308,286,383,171,102,271)(28,214,309,287,384,172,103,272)(29,215,310,288,385,173,104,273)(30,216,311,289,386,174,105,274)(31,217,312,290,387,175,106,275)(32,218,313,291,388,151,107,251)(33,219,314,292,389,152,108,252)(34,220,315,293,390,153,109,253)(35,221,316,294,391,154,110,254)(36,222,317,295,392,155,111,255)(37,223,318,296,393,156,112,256)(38,224,319,297,394,157,113,257)(39,225,320,298,395,158,114,258)(40,201,321,299,396,159,115,259)(41,202,322,300,397,160,116,260)(42,203,323,276,398,161,117,261)(43,204,324,277,399,162,118,262)(44,205,325,278,400,163,119,263)(45,206,301,279,376,164,120,264)(46,207,302,280,377,165,121,265)(47,208,303,281,378,166,122,266)(48,209,304,282,379,167,123,267)(49,210,305,283,380,168,124,268)(50,211,306,284,381,169,125,269), (1,29,368,385)(2,30,369,386)(3,31,370,387)(4,32,371,388)(5,33,372,389)(6,34,373,390)(7,35,374,391)(8,36,375,392)(9,37,351,393)(10,38,352,394)(11,39,353,395)(12,40,354,396)(13,41,355,397)(14,42,356,398)(15,43,357,399)(16,44,358,400)(17,45,359,376)(18,46,360,377)(19,47,361,378)(20,48,362,379)(21,49,363,380)(22,50,364,381)(23,26,365,382)(24,27,366,383)(25,28,367,384)(51,282,88,267)(52,283,89,268)(53,284,90,269)(54,285,91,270)(55,286,92,271)(56,287,93,272)(57,288,94,273)(58,289,95,274)(59,290,96,275)(60,291,97,251)(61,292,98,252)(62,293,99,253)(63,294,100,254)(64,295,76,255)(65,296,77,256)(66,297,78,257)(67,298,79,258)(68,299,80,259)(69,300,81,260)(70,276,82,261)(71,277,83,262)(72,278,84,263)(73,279,85,264)(74,280,86,265)(75,281,87,266)(101,341,307,200)(102,342,308,176)(103,343,309,177)(104,344,310,178)(105,345,311,179)(106,346,312,180)(107,347,313,181)(108,348,314,182)(109,349,315,183)(110,350,316,184)(111,326,317,185)(112,327,318,186)(113,328,319,187)(114,329,320,188)(115,330,321,189)(116,331,322,190)(117,332,323,191)(118,333,324,192)(119,334,325,193)(120,335,301,194)(121,336,302,195)(122,337,303,196)(123,338,304,197)(124,339,305,198)(125,340,306,199)(126,156,230,223)(127,157,231,224)(128,158,232,225)(129,159,233,201)(130,160,234,202)(131,161,235,203)(132,162,236,204)(133,163,237,205)(134,164,238,206)(135,165,239,207)(136,166,240,208)(137,167,241,209)(138,168,242,210)(139,169,243,211)(140,170,244,212)(141,171,245,213)(142,172,246,214)(143,173,247,215)(144,174,248,216)(145,175,249,217)(146,151,250,218)(147,152,226,219)(148,153,227,220)(149,154,228,221)(150,155,229,222)>;
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), (1,94,178,143,368,57,344,247)(2,95,179,144,369,58,345,248)(3,96,180,145,370,59,346,249)(4,97,181,146,371,60,347,250)(5,98,182,147,372,61,348,226)(6,99,183,148,373,62,349,227)(7,100,184,149,374,63,350,228)(8,76,185,150,375,64,326,229)(9,77,186,126,351,65,327,230)(10,78,187,127,352,66,328,231)(11,79,188,128,353,67,329,232)(12,80,189,129,354,68,330,233)(13,81,190,130,355,69,331,234)(14,82,191,131,356,70,332,235)(15,83,192,132,357,71,333,236)(16,84,193,133,358,72,334,237)(17,85,194,134,359,73,335,238)(18,86,195,135,360,74,336,239)(19,87,196,136,361,75,337,240)(20,88,197,137,362,51,338,241)(21,89,198,138,363,52,339,242)(22,90,199,139,364,53,340,243)(23,91,200,140,365,54,341,244)(24,92,176,141,366,55,342,245)(25,93,177,142,367,56,343,246)(26,212,307,285,382,170,101,270)(27,213,308,286,383,171,102,271)(28,214,309,287,384,172,103,272)(29,215,310,288,385,173,104,273)(30,216,311,289,386,174,105,274)(31,217,312,290,387,175,106,275)(32,218,313,291,388,151,107,251)(33,219,314,292,389,152,108,252)(34,220,315,293,390,153,109,253)(35,221,316,294,391,154,110,254)(36,222,317,295,392,155,111,255)(37,223,318,296,393,156,112,256)(38,224,319,297,394,157,113,257)(39,225,320,298,395,158,114,258)(40,201,321,299,396,159,115,259)(41,202,322,300,397,160,116,260)(42,203,323,276,398,161,117,261)(43,204,324,277,399,162,118,262)(44,205,325,278,400,163,119,263)(45,206,301,279,376,164,120,264)(46,207,302,280,377,165,121,265)(47,208,303,281,378,166,122,266)(48,209,304,282,379,167,123,267)(49,210,305,283,380,168,124,268)(50,211,306,284,381,169,125,269), (1,29,368,385)(2,30,369,386)(3,31,370,387)(4,32,371,388)(5,33,372,389)(6,34,373,390)(7,35,374,391)(8,36,375,392)(9,37,351,393)(10,38,352,394)(11,39,353,395)(12,40,354,396)(13,41,355,397)(14,42,356,398)(15,43,357,399)(16,44,358,400)(17,45,359,376)(18,46,360,377)(19,47,361,378)(20,48,362,379)(21,49,363,380)(22,50,364,381)(23,26,365,382)(24,27,366,383)(25,28,367,384)(51,282,88,267)(52,283,89,268)(53,284,90,269)(54,285,91,270)(55,286,92,271)(56,287,93,272)(57,288,94,273)(58,289,95,274)(59,290,96,275)(60,291,97,251)(61,292,98,252)(62,293,99,253)(63,294,100,254)(64,295,76,255)(65,296,77,256)(66,297,78,257)(67,298,79,258)(68,299,80,259)(69,300,81,260)(70,276,82,261)(71,277,83,262)(72,278,84,263)(73,279,85,264)(74,280,86,265)(75,281,87,266)(101,341,307,200)(102,342,308,176)(103,343,309,177)(104,344,310,178)(105,345,311,179)(106,346,312,180)(107,347,313,181)(108,348,314,182)(109,349,315,183)(110,350,316,184)(111,326,317,185)(112,327,318,186)(113,328,319,187)(114,329,320,188)(115,330,321,189)(116,331,322,190)(117,332,323,191)(118,333,324,192)(119,334,325,193)(120,335,301,194)(121,336,302,195)(122,337,303,196)(123,338,304,197)(124,339,305,198)(125,340,306,199)(126,156,230,223)(127,157,231,224)(128,158,232,225)(129,159,233,201)(130,160,234,202)(131,161,235,203)(132,162,236,204)(133,163,237,205)(134,164,238,206)(135,165,239,207)(136,166,240,208)(137,167,241,209)(138,168,242,210)(139,169,243,211)(140,170,244,212)(141,171,245,213)(142,172,246,214)(143,173,247,215)(144,174,248,216)(145,175,249,217)(146,151,250,218)(147,152,226,219)(148,153,227,220)(149,154,228,221)(150,155,229,222) );
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)], [(1,94,178,143,368,57,344,247),(2,95,179,144,369,58,345,248),(3,96,180,145,370,59,346,249),(4,97,181,146,371,60,347,250),(5,98,182,147,372,61,348,226),(6,99,183,148,373,62,349,227),(7,100,184,149,374,63,350,228),(8,76,185,150,375,64,326,229),(9,77,186,126,351,65,327,230),(10,78,187,127,352,66,328,231),(11,79,188,128,353,67,329,232),(12,80,189,129,354,68,330,233),(13,81,190,130,355,69,331,234),(14,82,191,131,356,70,332,235),(15,83,192,132,357,71,333,236),(16,84,193,133,358,72,334,237),(17,85,194,134,359,73,335,238),(18,86,195,135,360,74,336,239),(19,87,196,136,361,75,337,240),(20,88,197,137,362,51,338,241),(21,89,198,138,363,52,339,242),(22,90,199,139,364,53,340,243),(23,91,200,140,365,54,341,244),(24,92,176,141,366,55,342,245),(25,93,177,142,367,56,343,246),(26,212,307,285,382,170,101,270),(27,213,308,286,383,171,102,271),(28,214,309,287,384,172,103,272),(29,215,310,288,385,173,104,273),(30,216,311,289,386,174,105,274),(31,217,312,290,387,175,106,275),(32,218,313,291,388,151,107,251),(33,219,314,292,389,152,108,252),(34,220,315,293,390,153,109,253),(35,221,316,294,391,154,110,254),(36,222,317,295,392,155,111,255),(37,223,318,296,393,156,112,256),(38,224,319,297,394,157,113,257),(39,225,320,298,395,158,114,258),(40,201,321,299,396,159,115,259),(41,202,322,300,397,160,116,260),(42,203,323,276,398,161,117,261),(43,204,324,277,399,162,118,262),(44,205,325,278,400,163,119,263),(45,206,301,279,376,164,120,264),(46,207,302,280,377,165,121,265),(47,208,303,281,378,166,122,266),(48,209,304,282,379,167,123,267),(49,210,305,283,380,168,124,268),(50,211,306,284,381,169,125,269)], [(1,29,368,385),(2,30,369,386),(3,31,370,387),(4,32,371,388),(5,33,372,389),(6,34,373,390),(7,35,374,391),(8,36,375,392),(9,37,351,393),(10,38,352,394),(11,39,353,395),(12,40,354,396),(13,41,355,397),(14,42,356,398),(15,43,357,399),(16,44,358,400),(17,45,359,376),(18,46,360,377),(19,47,361,378),(20,48,362,379),(21,49,363,380),(22,50,364,381),(23,26,365,382),(24,27,366,383),(25,28,367,384),(51,282,88,267),(52,283,89,268),(53,284,90,269),(54,285,91,270),(55,286,92,271),(56,287,93,272),(57,288,94,273),(58,289,95,274),(59,290,96,275),(60,291,97,251),(61,292,98,252),(62,293,99,253),(63,294,100,254),(64,295,76,255),(65,296,77,256),(66,297,78,257),(67,298,79,258),(68,299,80,259),(69,300,81,260),(70,276,82,261),(71,277,83,262),(72,278,84,263),(73,279,85,264),(74,280,86,265),(75,281,87,266),(101,341,307,200),(102,342,308,176),(103,343,309,177),(104,344,310,178),(105,345,311,179),(106,346,312,180),(107,347,313,181),(108,348,314,182),(109,349,315,183),(110,350,316,184),(111,326,317,185),(112,327,318,186),(113,328,319,187),(114,329,320,188),(115,330,321,189),(116,331,322,190),(117,332,323,191),(118,333,324,192),(119,334,325,193),(120,335,301,194),(121,336,302,195),(122,337,303,196),(123,338,304,197),(124,339,305,198),(125,340,306,199),(126,156,230,223),(127,157,231,224),(128,158,232,225),(129,159,233,201),(130,160,234,202),(131,161,235,203),(132,162,236,204),(133,163,237,205),(134,164,238,206),(135,165,239,207),(136,166,240,208),(137,167,241,209),(138,168,242,210),(139,169,243,211),(140,170,244,212),(141,171,245,213),(142,172,246,214),(143,173,247,215),(144,174,248,216),(145,175,249,217),(146,151,250,218),(147,152,226,219),(148,153,227,220),(149,154,228,221),(150,155,229,222)]])
175 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 8A | 8B | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 25A | ··· | 25T | 40A | ··· | 40H | 50A | ··· | 50T | 100A | ··· | 100T | 100U | ··· | 100BH | 200A | ··· | 200AN |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 25 | ··· | 25 | 40 | ··· | 40 | 50 | ··· | 50 | 100 | ··· | 100 | 100 | ··· | 100 | 200 | ··· | 200 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 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 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | C25 | C50 | C50 | D4 | Q16 | C5×D4 | C5×Q16 | D4×C25 | Q16×C25 |
| kernel | Q16×C25 | C200 | Q8×C25 | C5×Q16 | C40 | C5×Q8 | Q16 | C8 | Q8 | C50 | C25 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 20 | 20 | 40 | 1 | 2 | 4 | 8 | 20 | 40 |
Matrix representation of Q16×C25 ►in GL2(𝔽401) generated by
| 125 | 0 |
| 0 | 125 |
| 0 | 53 |
| 174 | 53 |
| 334 | 263 |
| 265 | 67 |
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