metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C50.5D4, C50.1Q8, Dic25⋊1C4, C2.1Dic50, C22.4D50, C10.3Dic10, C25⋊2(C4⋊C4), (C2×C20).1D5, (C2×C4).1D25, C2.4(C4×D25), (C2×C100).1C2, C50.11(C2×C4), C10.15(C4×D5), (C2×C10).19D10, C2.1(C25⋊D4), (C2×C50).4C22, C5.(C10.D4), C10.12(C5⋊D4), (C2×Dic25).1C2, SmallGroup(400,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C50.D4
G = < a,b,c | a50=b4=1, c2=a25, bab-1=cac-1=a-1, cbc-1=b-1 >
Smallest permutation representation of C50.D4
►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)
(1 287 55 249)(2 286 56 248)(3 285 57 247)(4 284 58 246)(5 283 59 245)(6 282 60 244)(7 281 61 243)(8 280 62 242)(9 279 63 241)(10 278 64 240)(11 277 65 239)(12 276 66 238)(13 275 67 237)(14 274 68 236)(15 273 69 235)(16 272 70 234)(17 271 71 233)(18 270 72 232)(19 269 73 231)(20 268 74 230)(21 267 75 229)(22 266 76 228)(23 265 77 227)(24 264 78 226)(25 263 79 225)(26 262 80 224)(27 261 81 223)(28 260 82 222)(29 259 83 221)(30 258 84 220)(31 257 85 219)(32 256 86 218)(33 255 87 217)(34 254 88 216)(35 253 89 215)(36 252 90 214)(37 251 91 213)(38 300 92 212)(39 299 93 211)(40 298 94 210)(41 297 95 209)(42 296 96 208)(43 295 97 207)(44 294 98 206)(45 293 99 205)(46 292 100 204)(47 291 51 203)(48 290 52 202)(49 289 53 201)(50 288 54 250)(101 183 370 348)(102 182 371 347)(103 181 372 346)(104 180 373 345)(105 179 374 344)(106 178 375 343)(107 177 376 342)(108 176 377 341)(109 175 378 340)(110 174 379 339)(111 173 380 338)(112 172 381 337)(113 171 382 336)(114 170 383 335)(115 169 384 334)(116 168 385 333)(117 167 386 332)(118 166 387 331)(119 165 388 330)(120 164 389 329)(121 163 390 328)(122 162 391 327)(123 161 392 326)(124 160 393 325)(125 159 394 324)(126 158 395 323)(127 157 396 322)(128 156 397 321)(129 155 398 320)(130 154 399 319)(131 153 400 318)(132 152 351 317)(133 151 352 316)(134 200 353 315)(135 199 354 314)(136 198 355 313)(137 197 356 312)(138 196 357 311)(139 195 358 310)(140 194 359 309)(141 193 360 308)(142 192 361 307)(143 191 362 306)(144 190 363 305)(145 189 364 304)(146 188 365 303)(147 187 366 302)(148 186 367 301)(149 185 368 350)(150 184 369 349)
(1 326 26 301)(2 325 27 350)(3 324 28 349)(4 323 29 348)(5 322 30 347)(6 321 31 346)(7 320 32 345)(8 319 33 344)(9 318 34 343)(10 317 35 342)(11 316 36 341)(12 315 37 340)(13 314 38 339)(14 313 39 338)(15 312 40 337)(16 311 41 336)(17 310 42 335)(18 309 43 334)(19 308 44 333)(20 307 45 332)(21 306 46 331)(22 305 47 330)(23 304 48 329)(24 303 49 328)(25 302 50 327)(51 165 76 190)(52 164 77 189)(53 163 78 188)(54 162 79 187)(55 161 80 186)(56 160 81 185)(57 159 82 184)(58 158 83 183)(59 157 84 182)(60 156 85 181)(61 155 86 180)(62 154 87 179)(63 153 88 178)(64 152 89 177)(65 151 90 176)(66 200 91 175)(67 199 92 174)(68 198 93 173)(69 197 94 172)(70 196 95 171)(71 195 96 170)(72 194 97 169)(73 193 98 168)(74 192 99 167)(75 191 100 166)(101 246 126 221)(102 245 127 220)(103 244 128 219)(104 243 129 218)(105 242 130 217)(106 241 131 216)(107 240 132 215)(108 239 133 214)(109 238 134 213)(110 237 135 212)(111 236 136 211)(112 235 137 210)(113 234 138 209)(114 233 139 208)(115 232 140 207)(116 231 141 206)(117 230 142 205)(118 229 143 204)(119 228 144 203)(120 227 145 202)(121 226 146 201)(122 225 147 250)(123 224 148 249)(124 223 149 248)(125 222 150 247)(251 378 276 353)(252 377 277 352)(253 376 278 351)(254 375 279 400)(255 374 280 399)(256 373 281 398)(257 372 282 397)(258 371 283 396)(259 370 284 395)(260 369 285 394)(261 368 286 393)(262 367 287 392)(263 366 288 391)(264 365 289 390)(265 364 290 389)(266 363 291 388)(267 362 292 387)(268 361 293 386)(269 360 294 385)(270 359 295 384)(271 358 296 383)(272 357 297 382)(273 356 298 381)(274 355 299 380)(275 354 300 379)
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), (1,287,55,249)(2,286,56,248)(3,285,57,247)(4,284,58,246)(5,283,59,245)(6,282,60,244)(7,281,61,243)(8,280,62,242)(9,279,63,241)(10,278,64,240)(11,277,65,239)(12,276,66,238)(13,275,67,237)(14,274,68,236)(15,273,69,235)(16,272,70,234)(17,271,71,233)(18,270,72,232)(19,269,73,231)(20,268,74,230)(21,267,75,229)(22,266,76,228)(23,265,77,227)(24,264,78,226)(25,263,79,225)(26,262,80,224)(27,261,81,223)(28,260,82,222)(29,259,83,221)(30,258,84,220)(31,257,85,219)(32,256,86,218)(33,255,87,217)(34,254,88,216)(35,253,89,215)(36,252,90,214)(37,251,91,213)(38,300,92,212)(39,299,93,211)(40,298,94,210)(41,297,95,209)(42,296,96,208)(43,295,97,207)(44,294,98,206)(45,293,99,205)(46,292,100,204)(47,291,51,203)(48,290,52,202)(49,289,53,201)(50,288,54,250)(101,183,370,348)(102,182,371,347)(103,181,372,346)(104,180,373,345)(105,179,374,344)(106,178,375,343)(107,177,376,342)(108,176,377,341)(109,175,378,340)(110,174,379,339)(111,173,380,338)(112,172,381,337)(113,171,382,336)(114,170,383,335)(115,169,384,334)(116,168,385,333)(117,167,386,332)(118,166,387,331)(119,165,388,330)(120,164,389,329)(121,163,390,328)(122,162,391,327)(123,161,392,326)(124,160,393,325)(125,159,394,324)(126,158,395,323)(127,157,396,322)(128,156,397,321)(129,155,398,320)(130,154,399,319)(131,153,400,318)(132,152,351,317)(133,151,352,316)(134,200,353,315)(135,199,354,314)(136,198,355,313)(137,197,356,312)(138,196,357,311)(139,195,358,310)(140,194,359,309)(141,193,360,308)(142,192,361,307)(143,191,362,306)(144,190,363,305)(145,189,364,304)(146,188,365,303)(147,187,366,302)(148,186,367,301)(149,185,368,350)(150,184,369,349), (1,326,26,301)(2,325,27,350)(3,324,28,349)(4,323,29,348)(5,322,30,347)(6,321,31,346)(7,320,32,345)(8,319,33,344)(9,318,34,343)(10,317,35,342)(11,316,36,341)(12,315,37,340)(13,314,38,339)(14,313,39,338)(15,312,40,337)(16,311,41,336)(17,310,42,335)(18,309,43,334)(19,308,44,333)(20,307,45,332)(21,306,46,331)(22,305,47,330)(23,304,48,329)(24,303,49,328)(25,302,50,327)(51,165,76,190)(52,164,77,189)(53,163,78,188)(54,162,79,187)(55,161,80,186)(56,160,81,185)(57,159,82,184)(58,158,83,183)(59,157,84,182)(60,156,85,181)(61,155,86,180)(62,154,87,179)(63,153,88,178)(64,152,89,177)(65,151,90,176)(66,200,91,175)(67,199,92,174)(68,198,93,173)(69,197,94,172)(70,196,95,171)(71,195,96,170)(72,194,97,169)(73,193,98,168)(74,192,99,167)(75,191,100,166)(101,246,126,221)(102,245,127,220)(103,244,128,219)(104,243,129,218)(105,242,130,217)(106,241,131,216)(107,240,132,215)(108,239,133,214)(109,238,134,213)(110,237,135,212)(111,236,136,211)(112,235,137,210)(113,234,138,209)(114,233,139,208)(115,232,140,207)(116,231,141,206)(117,230,142,205)(118,229,143,204)(119,228,144,203)(120,227,145,202)(121,226,146,201)(122,225,147,250)(123,224,148,249)(124,223,149,248)(125,222,150,247)(251,378,276,353)(252,377,277,352)(253,376,278,351)(254,375,279,400)(255,374,280,399)(256,373,281,398)(257,372,282,397)(258,371,283,396)(259,370,284,395)(260,369,285,394)(261,368,286,393)(262,367,287,392)(263,366,288,391)(264,365,289,390)(265,364,290,389)(266,363,291,388)(267,362,292,387)(268,361,293,386)(269,360,294,385)(270,359,295,384)(271,358,296,383)(272,357,297,382)(273,356,298,381)(274,355,299,380)(275,354,300,379)>;
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), (1,287,55,249)(2,286,56,248)(3,285,57,247)(4,284,58,246)(5,283,59,245)(6,282,60,244)(7,281,61,243)(8,280,62,242)(9,279,63,241)(10,278,64,240)(11,277,65,239)(12,276,66,238)(13,275,67,237)(14,274,68,236)(15,273,69,235)(16,272,70,234)(17,271,71,233)(18,270,72,232)(19,269,73,231)(20,268,74,230)(21,267,75,229)(22,266,76,228)(23,265,77,227)(24,264,78,226)(25,263,79,225)(26,262,80,224)(27,261,81,223)(28,260,82,222)(29,259,83,221)(30,258,84,220)(31,257,85,219)(32,256,86,218)(33,255,87,217)(34,254,88,216)(35,253,89,215)(36,252,90,214)(37,251,91,213)(38,300,92,212)(39,299,93,211)(40,298,94,210)(41,297,95,209)(42,296,96,208)(43,295,97,207)(44,294,98,206)(45,293,99,205)(46,292,100,204)(47,291,51,203)(48,290,52,202)(49,289,53,201)(50,288,54,250)(101,183,370,348)(102,182,371,347)(103,181,372,346)(104,180,373,345)(105,179,374,344)(106,178,375,343)(107,177,376,342)(108,176,377,341)(109,175,378,340)(110,174,379,339)(111,173,380,338)(112,172,381,337)(113,171,382,336)(114,170,383,335)(115,169,384,334)(116,168,385,333)(117,167,386,332)(118,166,387,331)(119,165,388,330)(120,164,389,329)(121,163,390,328)(122,162,391,327)(123,161,392,326)(124,160,393,325)(125,159,394,324)(126,158,395,323)(127,157,396,322)(128,156,397,321)(129,155,398,320)(130,154,399,319)(131,153,400,318)(132,152,351,317)(133,151,352,316)(134,200,353,315)(135,199,354,314)(136,198,355,313)(137,197,356,312)(138,196,357,311)(139,195,358,310)(140,194,359,309)(141,193,360,308)(142,192,361,307)(143,191,362,306)(144,190,363,305)(145,189,364,304)(146,188,365,303)(147,187,366,302)(148,186,367,301)(149,185,368,350)(150,184,369,349), (1,326,26,301)(2,325,27,350)(3,324,28,349)(4,323,29,348)(5,322,30,347)(6,321,31,346)(7,320,32,345)(8,319,33,344)(9,318,34,343)(10,317,35,342)(11,316,36,341)(12,315,37,340)(13,314,38,339)(14,313,39,338)(15,312,40,337)(16,311,41,336)(17,310,42,335)(18,309,43,334)(19,308,44,333)(20,307,45,332)(21,306,46,331)(22,305,47,330)(23,304,48,329)(24,303,49,328)(25,302,50,327)(51,165,76,190)(52,164,77,189)(53,163,78,188)(54,162,79,187)(55,161,80,186)(56,160,81,185)(57,159,82,184)(58,158,83,183)(59,157,84,182)(60,156,85,181)(61,155,86,180)(62,154,87,179)(63,153,88,178)(64,152,89,177)(65,151,90,176)(66,200,91,175)(67,199,92,174)(68,198,93,173)(69,197,94,172)(70,196,95,171)(71,195,96,170)(72,194,97,169)(73,193,98,168)(74,192,99,167)(75,191,100,166)(101,246,126,221)(102,245,127,220)(103,244,128,219)(104,243,129,218)(105,242,130,217)(106,241,131,216)(107,240,132,215)(108,239,133,214)(109,238,134,213)(110,237,135,212)(111,236,136,211)(112,235,137,210)(113,234,138,209)(114,233,139,208)(115,232,140,207)(116,231,141,206)(117,230,142,205)(118,229,143,204)(119,228,144,203)(120,227,145,202)(121,226,146,201)(122,225,147,250)(123,224,148,249)(124,223,149,248)(125,222,150,247)(251,378,276,353)(252,377,277,352)(253,376,278,351)(254,375,279,400)(255,374,280,399)(256,373,281,398)(257,372,282,397)(258,371,283,396)(259,370,284,395)(260,369,285,394)(261,368,286,393)(262,367,287,392)(263,366,288,391)(264,365,289,390)(265,364,290,389)(266,363,291,388)(267,362,292,387)(268,361,293,386)(269,360,294,385)(270,359,295,384)(271,358,296,383)(272,357,297,382)(273,356,298,381)(274,355,299,380)(275,354,300,379) );
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)], [(1,287,55,249),(2,286,56,248),(3,285,57,247),(4,284,58,246),(5,283,59,245),(6,282,60,244),(7,281,61,243),(8,280,62,242),(9,279,63,241),(10,278,64,240),(11,277,65,239),(12,276,66,238),(13,275,67,237),(14,274,68,236),(15,273,69,235),(16,272,70,234),(17,271,71,233),(18,270,72,232),(19,269,73,231),(20,268,74,230),(21,267,75,229),(22,266,76,228),(23,265,77,227),(24,264,78,226),(25,263,79,225),(26,262,80,224),(27,261,81,223),(28,260,82,222),(29,259,83,221),(30,258,84,220),(31,257,85,219),(32,256,86,218),(33,255,87,217),(34,254,88,216),(35,253,89,215),(36,252,90,214),(37,251,91,213),(38,300,92,212),(39,299,93,211),(40,298,94,210),(41,297,95,209),(42,296,96,208),(43,295,97,207),(44,294,98,206),(45,293,99,205),(46,292,100,204),(47,291,51,203),(48,290,52,202),(49,289,53,201),(50,288,54,250),(101,183,370,348),(102,182,371,347),(103,181,372,346),(104,180,373,345),(105,179,374,344),(106,178,375,343),(107,177,376,342),(108,176,377,341),(109,175,378,340),(110,174,379,339),(111,173,380,338),(112,172,381,337),(113,171,382,336),(114,170,383,335),(115,169,384,334),(116,168,385,333),(117,167,386,332),(118,166,387,331),(119,165,388,330),(120,164,389,329),(121,163,390,328),(122,162,391,327),(123,161,392,326),(124,160,393,325),(125,159,394,324),(126,158,395,323),(127,157,396,322),(128,156,397,321),(129,155,398,320),(130,154,399,319),(131,153,400,318),(132,152,351,317),(133,151,352,316),(134,200,353,315),(135,199,354,314),(136,198,355,313),(137,197,356,312),(138,196,357,311),(139,195,358,310),(140,194,359,309),(141,193,360,308),(142,192,361,307),(143,191,362,306),(144,190,363,305),(145,189,364,304),(146,188,365,303),(147,187,366,302),(148,186,367,301),(149,185,368,350),(150,184,369,349)], [(1,326,26,301),(2,325,27,350),(3,324,28,349),(4,323,29,348),(5,322,30,347),(6,321,31,346),(7,320,32,345),(8,319,33,344),(9,318,34,343),(10,317,35,342),(11,316,36,341),(12,315,37,340),(13,314,38,339),(14,313,39,338),(15,312,40,337),(16,311,41,336),(17,310,42,335),(18,309,43,334),(19,308,44,333),(20,307,45,332),(21,306,46,331),(22,305,47,330),(23,304,48,329),(24,303,49,328),(25,302,50,327),(51,165,76,190),(52,164,77,189),(53,163,78,188),(54,162,79,187),(55,161,80,186),(56,160,81,185),(57,159,82,184),(58,158,83,183),(59,157,84,182),(60,156,85,181),(61,155,86,180),(62,154,87,179),(63,153,88,178),(64,152,89,177),(65,151,90,176),(66,200,91,175),(67,199,92,174),(68,198,93,173),(69,197,94,172),(70,196,95,171),(71,195,96,170),(72,194,97,169),(73,193,98,168),(74,192,99,167),(75,191,100,166),(101,246,126,221),(102,245,127,220),(103,244,128,219),(104,243,129,218),(105,242,130,217),(106,241,131,216),(107,240,132,215),(108,239,133,214),(109,238,134,213),(110,237,135,212),(111,236,136,211),(112,235,137,210),(113,234,138,209),(114,233,139,208),(115,232,140,207),(116,231,141,206),(117,230,142,205),(118,229,143,204),(119,228,144,203),(120,227,145,202),(121,226,146,201),(122,225,147,250),(123,224,148,249),(124,223,149,248),(125,222,150,247),(251,378,276,353),(252,377,277,352),(253,376,278,351),(254,375,279,400),(255,374,280,399),(256,373,281,398),(257,372,282,397),(258,371,283,396),(259,370,284,395),(260,369,285,394),(261,368,286,393),(262,367,287,392),(263,366,288,391),(264,365,289,390),(265,364,290,389),(266,363,291,388),(267,362,292,387),(268,361,293,386),(269,360,294,385),(270,359,295,384),(271,358,296,383),(272,357,297,382),(273,356,298,381),(274,355,299,380),(275,354,300,379)]])
106 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 25A | ··· | 25J | 50A | ··· | 50AD | 100A | ··· | 100AN |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 50 | 50 | 50 | 50 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
106 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | + | - | |||||
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D5 | D10 | Dic10 | C4×D5 | C5⋊D4 | D25 | D50 | Dic50 | C4×D25 | C25⋊D4 |
| kernel | C50.D4 | C2×Dic25 | C2×C100 | Dic25 | C50 | C50 | C2×C20 | C2×C10 | C10 | C10 | C10 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 |
Matrix representation of C50.D4 ►in GL4(𝔽101) generated by
| 2 | 51 | 0 | 0 |
| 50 | 13 | 0 | 0 |
| 0 | 0 | 13 | 68 |
| 0 | 0 | 33 | 95 |
| 16 | 70 | 0 | 0 |
| 18 | 85 | 0 | 0 |
| 0 | 0 | 5 | 72 |
| 0 | 0 | 81 | 96 |
| 42 | 7 | 0 | 0 |
| 22 | 59 | 0 | 0 |
| 0 | 0 | 98 | 57 |
| 0 | 0 | 92 | 3 |
G:=sub<GL(4,GF(101))| [2,50,0,0,51,13,0,0,0,0,13,33,0,0,68,95],[16,18,0,0,70,85,0,0,0,0,5,81,0,0,72,96],[42,22,0,0,7,59,0,0,0,0,98,92,0,0,57,3] >;
C50.D4 in GAP, Magma, Sage, TeX
C_{50}.D_4 % in TeX
G:=Group("C50.D4"); // GroupNames label
G:=SmallGroup(400,12);
// by ID
G=gap.SmallGroup(400,12);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,121,31,4324,628,11525]);
// Polycyclic
G:=Group<a,b,c|a^50=b^4=1,c^2=a^25,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
Export