direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×Dic25, C100⋊4C4, C25⋊2C42, C20.9Dic5, C22.3D50, C5.(C4×Dic5), C2.2(C4×D25), (C2×C4).6D25, C50.10(C2×C4), (C2×C100).6C2, C10.14(C4×D5), (C2×C20).14D5, (C2×C10).18D10, (C2×C50).3C22, C2.2(C2×Dic25), (C2×Dic25).6C2, C10.15(C2×Dic5), SmallGroup(400,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C4×Dic25 |
Generators and relations for C4×Dic25
G = < a,b,c | a4=b50=1, c2=b25, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4×Dic25
►Regular action on 400 points
Generators in S400
(1 80 366 223)(2 81 367 224)(3 82 368 225)(4 83 369 226)(5 84 370 227)(6 85 371 228)(7 86 372 229)(8 87 373 230)(9 88 374 231)(10 89 375 232)(11 90 376 233)(12 91 377 234)(13 92 378 235)(14 93 379 236)(15 94 380 237)(16 95 381 238)(17 96 382 239)(18 97 383 240)(19 98 384 241)(20 99 385 242)(21 100 386 243)(22 51 387 244)(23 52 388 245)(24 53 389 246)(25 54 390 247)(26 55 391 248)(27 56 392 249)(28 57 393 250)(29 58 394 201)(30 59 395 202)(31 60 396 203)(32 61 397 204)(33 62 398 205)(34 63 399 206)(35 64 400 207)(36 65 351 208)(37 66 352 209)(38 67 353 210)(39 68 354 211)(40 69 355 212)(41 70 356 213)(42 71 357 214)(43 72 358 215)(44 73 359 216)(45 74 360 217)(46 75 361 218)(47 76 362 219)(48 77 363 220)(49 78 364 221)(50 79 365 222)(101 343 168 299)(102 344 169 300)(103 345 170 251)(104 346 171 252)(105 347 172 253)(106 348 173 254)(107 349 174 255)(108 350 175 256)(109 301 176 257)(110 302 177 258)(111 303 178 259)(112 304 179 260)(113 305 180 261)(114 306 181 262)(115 307 182 263)(116 308 183 264)(117 309 184 265)(118 310 185 266)(119 311 186 267)(120 312 187 268)(121 313 188 269)(122 314 189 270)(123 315 190 271)(124 316 191 272)(125 317 192 273)(126 318 193 274)(127 319 194 275)(128 320 195 276)(129 321 196 277)(130 322 197 278)(131 323 198 279)(132 324 199 280)(133 325 200 281)(134 326 151 282)(135 327 152 283)(136 328 153 284)(137 329 154 285)(138 330 155 286)(139 331 156 287)(140 332 157 288)(141 333 158 289)(142 334 159 290)(143 335 160 291)(144 336 161 292)(145 337 162 293)(146 338 163 294)(147 339 164 295)(148 340 165 296)(149 341 166 297)(150 342 167 298)
(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)
(1 179 26 154)(2 178 27 153)(3 177 28 152)(4 176 29 151)(5 175 30 200)(6 174 31 199)(7 173 32 198)(8 172 33 197)(9 171 34 196)(10 170 35 195)(11 169 36 194)(12 168 37 193)(13 167 38 192)(14 166 39 191)(15 165 40 190)(16 164 41 189)(17 163 42 188)(18 162 43 187)(19 161 44 186)(20 160 45 185)(21 159 46 184)(22 158 47 183)(23 157 48 182)(24 156 49 181)(25 155 50 180)(51 289 76 264)(52 288 77 263)(53 287 78 262)(54 286 79 261)(55 285 80 260)(56 284 81 259)(57 283 82 258)(58 282 83 257)(59 281 84 256)(60 280 85 255)(61 279 86 254)(62 278 87 253)(63 277 88 252)(64 276 89 251)(65 275 90 300)(66 274 91 299)(67 273 92 298)(68 272 93 297)(69 271 94 296)(70 270 95 295)(71 269 96 294)(72 268 97 293)(73 267 98 292)(74 266 99 291)(75 265 100 290)(101 352 126 377)(102 351 127 376)(103 400 128 375)(104 399 129 374)(105 398 130 373)(106 397 131 372)(107 396 132 371)(108 395 133 370)(109 394 134 369)(110 393 135 368)(111 392 136 367)(112 391 137 366)(113 390 138 365)(114 389 139 364)(115 388 140 363)(116 387 141 362)(117 386 142 361)(118 385 143 360)(119 384 144 359)(120 383 145 358)(121 382 146 357)(122 381 147 356)(123 380 148 355)(124 379 149 354)(125 378 150 353)(201 326 226 301)(202 325 227 350)(203 324 228 349)(204 323 229 348)(205 322 230 347)(206 321 231 346)(207 320 232 345)(208 319 233 344)(209 318 234 343)(210 317 235 342)(211 316 236 341)(212 315 237 340)(213 314 238 339)(214 313 239 338)(215 312 240 337)(216 311 241 336)(217 310 242 335)(218 309 243 334)(219 308 244 333)(220 307 245 332)(221 306 246 331)(222 305 247 330)(223 304 248 329)(224 303 249 328)(225 302 250 327)
G:=sub<Sym(400)| (1,80,366,223)(2,81,367,224)(3,82,368,225)(4,83,369,226)(5,84,370,227)(6,85,371,228)(7,86,372,229)(8,87,373,230)(9,88,374,231)(10,89,375,232)(11,90,376,233)(12,91,377,234)(13,92,378,235)(14,93,379,236)(15,94,380,237)(16,95,381,238)(17,96,382,239)(18,97,383,240)(19,98,384,241)(20,99,385,242)(21,100,386,243)(22,51,387,244)(23,52,388,245)(24,53,389,246)(25,54,390,247)(26,55,391,248)(27,56,392,249)(28,57,393,250)(29,58,394,201)(30,59,395,202)(31,60,396,203)(32,61,397,204)(33,62,398,205)(34,63,399,206)(35,64,400,207)(36,65,351,208)(37,66,352,209)(38,67,353,210)(39,68,354,211)(40,69,355,212)(41,70,356,213)(42,71,357,214)(43,72,358,215)(44,73,359,216)(45,74,360,217)(46,75,361,218)(47,76,362,219)(48,77,363,220)(49,78,364,221)(50,79,365,222)(101,343,168,299)(102,344,169,300)(103,345,170,251)(104,346,171,252)(105,347,172,253)(106,348,173,254)(107,349,174,255)(108,350,175,256)(109,301,176,257)(110,302,177,258)(111,303,178,259)(112,304,179,260)(113,305,180,261)(114,306,181,262)(115,307,182,263)(116,308,183,264)(117,309,184,265)(118,310,185,266)(119,311,186,267)(120,312,187,268)(121,313,188,269)(122,314,189,270)(123,315,190,271)(124,316,191,272)(125,317,192,273)(126,318,193,274)(127,319,194,275)(128,320,195,276)(129,321,196,277)(130,322,197,278)(131,323,198,279)(132,324,199,280)(133,325,200,281)(134,326,151,282)(135,327,152,283)(136,328,153,284)(137,329,154,285)(138,330,155,286)(139,331,156,287)(140,332,157,288)(141,333,158,289)(142,334,159,290)(143,335,160,291)(144,336,161,292)(145,337,162,293)(146,338,163,294)(147,339,164,295)(148,340,165,296)(149,341,166,297)(150,342,167,298), (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), (1,179,26,154)(2,178,27,153)(3,177,28,152)(4,176,29,151)(5,175,30,200)(6,174,31,199)(7,173,32,198)(8,172,33,197)(9,171,34,196)(10,170,35,195)(11,169,36,194)(12,168,37,193)(13,167,38,192)(14,166,39,191)(15,165,40,190)(16,164,41,189)(17,163,42,188)(18,162,43,187)(19,161,44,186)(20,160,45,185)(21,159,46,184)(22,158,47,183)(23,157,48,182)(24,156,49,181)(25,155,50,180)(51,289,76,264)(52,288,77,263)(53,287,78,262)(54,286,79,261)(55,285,80,260)(56,284,81,259)(57,283,82,258)(58,282,83,257)(59,281,84,256)(60,280,85,255)(61,279,86,254)(62,278,87,253)(63,277,88,252)(64,276,89,251)(65,275,90,300)(66,274,91,299)(67,273,92,298)(68,272,93,297)(69,271,94,296)(70,270,95,295)(71,269,96,294)(72,268,97,293)(73,267,98,292)(74,266,99,291)(75,265,100,290)(101,352,126,377)(102,351,127,376)(103,400,128,375)(104,399,129,374)(105,398,130,373)(106,397,131,372)(107,396,132,371)(108,395,133,370)(109,394,134,369)(110,393,135,368)(111,392,136,367)(112,391,137,366)(113,390,138,365)(114,389,139,364)(115,388,140,363)(116,387,141,362)(117,386,142,361)(118,385,143,360)(119,384,144,359)(120,383,145,358)(121,382,146,357)(122,381,147,356)(123,380,148,355)(124,379,149,354)(125,378,150,353)(201,326,226,301)(202,325,227,350)(203,324,228,349)(204,323,229,348)(205,322,230,347)(206,321,231,346)(207,320,232,345)(208,319,233,344)(209,318,234,343)(210,317,235,342)(211,316,236,341)(212,315,237,340)(213,314,238,339)(214,313,239,338)(215,312,240,337)(216,311,241,336)(217,310,242,335)(218,309,243,334)(219,308,244,333)(220,307,245,332)(221,306,246,331)(222,305,247,330)(223,304,248,329)(224,303,249,328)(225,302,250,327)>;
G:=Group( (1,80,366,223)(2,81,367,224)(3,82,368,225)(4,83,369,226)(5,84,370,227)(6,85,371,228)(7,86,372,229)(8,87,373,230)(9,88,374,231)(10,89,375,232)(11,90,376,233)(12,91,377,234)(13,92,378,235)(14,93,379,236)(15,94,380,237)(16,95,381,238)(17,96,382,239)(18,97,383,240)(19,98,384,241)(20,99,385,242)(21,100,386,243)(22,51,387,244)(23,52,388,245)(24,53,389,246)(25,54,390,247)(26,55,391,248)(27,56,392,249)(28,57,393,250)(29,58,394,201)(30,59,395,202)(31,60,396,203)(32,61,397,204)(33,62,398,205)(34,63,399,206)(35,64,400,207)(36,65,351,208)(37,66,352,209)(38,67,353,210)(39,68,354,211)(40,69,355,212)(41,70,356,213)(42,71,357,214)(43,72,358,215)(44,73,359,216)(45,74,360,217)(46,75,361,218)(47,76,362,219)(48,77,363,220)(49,78,364,221)(50,79,365,222)(101,343,168,299)(102,344,169,300)(103,345,170,251)(104,346,171,252)(105,347,172,253)(106,348,173,254)(107,349,174,255)(108,350,175,256)(109,301,176,257)(110,302,177,258)(111,303,178,259)(112,304,179,260)(113,305,180,261)(114,306,181,262)(115,307,182,263)(116,308,183,264)(117,309,184,265)(118,310,185,266)(119,311,186,267)(120,312,187,268)(121,313,188,269)(122,314,189,270)(123,315,190,271)(124,316,191,272)(125,317,192,273)(126,318,193,274)(127,319,194,275)(128,320,195,276)(129,321,196,277)(130,322,197,278)(131,323,198,279)(132,324,199,280)(133,325,200,281)(134,326,151,282)(135,327,152,283)(136,328,153,284)(137,329,154,285)(138,330,155,286)(139,331,156,287)(140,332,157,288)(141,333,158,289)(142,334,159,290)(143,335,160,291)(144,336,161,292)(145,337,162,293)(146,338,163,294)(147,339,164,295)(148,340,165,296)(149,341,166,297)(150,342,167,298), (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), (1,179,26,154)(2,178,27,153)(3,177,28,152)(4,176,29,151)(5,175,30,200)(6,174,31,199)(7,173,32,198)(8,172,33,197)(9,171,34,196)(10,170,35,195)(11,169,36,194)(12,168,37,193)(13,167,38,192)(14,166,39,191)(15,165,40,190)(16,164,41,189)(17,163,42,188)(18,162,43,187)(19,161,44,186)(20,160,45,185)(21,159,46,184)(22,158,47,183)(23,157,48,182)(24,156,49,181)(25,155,50,180)(51,289,76,264)(52,288,77,263)(53,287,78,262)(54,286,79,261)(55,285,80,260)(56,284,81,259)(57,283,82,258)(58,282,83,257)(59,281,84,256)(60,280,85,255)(61,279,86,254)(62,278,87,253)(63,277,88,252)(64,276,89,251)(65,275,90,300)(66,274,91,299)(67,273,92,298)(68,272,93,297)(69,271,94,296)(70,270,95,295)(71,269,96,294)(72,268,97,293)(73,267,98,292)(74,266,99,291)(75,265,100,290)(101,352,126,377)(102,351,127,376)(103,400,128,375)(104,399,129,374)(105,398,130,373)(106,397,131,372)(107,396,132,371)(108,395,133,370)(109,394,134,369)(110,393,135,368)(111,392,136,367)(112,391,137,366)(113,390,138,365)(114,389,139,364)(115,388,140,363)(116,387,141,362)(117,386,142,361)(118,385,143,360)(119,384,144,359)(120,383,145,358)(121,382,146,357)(122,381,147,356)(123,380,148,355)(124,379,149,354)(125,378,150,353)(201,326,226,301)(202,325,227,350)(203,324,228,349)(204,323,229,348)(205,322,230,347)(206,321,231,346)(207,320,232,345)(208,319,233,344)(209,318,234,343)(210,317,235,342)(211,316,236,341)(212,315,237,340)(213,314,238,339)(214,313,239,338)(215,312,240,337)(216,311,241,336)(217,310,242,335)(218,309,243,334)(219,308,244,333)(220,307,245,332)(221,306,246,331)(222,305,247,330)(223,304,248,329)(224,303,249,328)(225,302,250,327) );
G=PermutationGroup([[(1,80,366,223),(2,81,367,224),(3,82,368,225),(4,83,369,226),(5,84,370,227),(6,85,371,228),(7,86,372,229),(8,87,373,230),(9,88,374,231),(10,89,375,232),(11,90,376,233),(12,91,377,234),(13,92,378,235),(14,93,379,236),(15,94,380,237),(16,95,381,238),(17,96,382,239),(18,97,383,240),(19,98,384,241),(20,99,385,242),(21,100,386,243),(22,51,387,244),(23,52,388,245),(24,53,389,246),(25,54,390,247),(26,55,391,248),(27,56,392,249),(28,57,393,250),(29,58,394,201),(30,59,395,202),(31,60,396,203),(32,61,397,204),(33,62,398,205),(34,63,399,206),(35,64,400,207),(36,65,351,208),(37,66,352,209),(38,67,353,210),(39,68,354,211),(40,69,355,212),(41,70,356,213),(42,71,357,214),(43,72,358,215),(44,73,359,216),(45,74,360,217),(46,75,361,218),(47,76,362,219),(48,77,363,220),(49,78,364,221),(50,79,365,222),(101,343,168,299),(102,344,169,300),(103,345,170,251),(104,346,171,252),(105,347,172,253),(106,348,173,254),(107,349,174,255),(108,350,175,256),(109,301,176,257),(110,302,177,258),(111,303,178,259),(112,304,179,260),(113,305,180,261),(114,306,181,262),(115,307,182,263),(116,308,183,264),(117,309,184,265),(118,310,185,266),(119,311,186,267),(120,312,187,268),(121,313,188,269),(122,314,189,270),(123,315,190,271),(124,316,191,272),(125,317,192,273),(126,318,193,274),(127,319,194,275),(128,320,195,276),(129,321,196,277),(130,322,197,278),(131,323,198,279),(132,324,199,280),(133,325,200,281),(134,326,151,282),(135,327,152,283),(136,328,153,284),(137,329,154,285),(138,330,155,286),(139,331,156,287),(140,332,157,288),(141,333,158,289),(142,334,159,290),(143,335,160,291),(144,336,161,292),(145,337,162,293),(146,338,163,294),(147,339,164,295),(148,340,165,296),(149,341,166,297),(150,342,167,298)], [(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)], [(1,179,26,154),(2,178,27,153),(3,177,28,152),(4,176,29,151),(5,175,30,200),(6,174,31,199),(7,173,32,198),(8,172,33,197),(9,171,34,196),(10,170,35,195),(11,169,36,194),(12,168,37,193),(13,167,38,192),(14,166,39,191),(15,165,40,190),(16,164,41,189),(17,163,42,188),(18,162,43,187),(19,161,44,186),(20,160,45,185),(21,159,46,184),(22,158,47,183),(23,157,48,182),(24,156,49,181),(25,155,50,180),(51,289,76,264),(52,288,77,263),(53,287,78,262),(54,286,79,261),(55,285,80,260),(56,284,81,259),(57,283,82,258),(58,282,83,257),(59,281,84,256),(60,280,85,255),(61,279,86,254),(62,278,87,253),(63,277,88,252),(64,276,89,251),(65,275,90,300),(66,274,91,299),(67,273,92,298),(68,272,93,297),(69,271,94,296),(70,270,95,295),(71,269,96,294),(72,268,97,293),(73,267,98,292),(74,266,99,291),(75,265,100,290),(101,352,126,377),(102,351,127,376),(103,400,128,375),(104,399,129,374),(105,398,130,373),(106,397,131,372),(107,396,132,371),(108,395,133,370),(109,394,134,369),(110,393,135,368),(111,392,136,367),(112,391,137,366),(113,390,138,365),(114,389,139,364),(115,388,140,363),(116,387,141,362),(117,386,142,361),(118,385,143,360),(119,384,144,359),(120,383,145,358),(121,382,146,357),(122,381,147,356),(123,380,148,355),(124,379,149,354),(125,378,150,353),(201,326,226,301),(202,325,227,350),(203,324,228,349),(204,323,229,348),(205,322,230,347),(206,321,231,346),(207,320,232,345),(208,319,233,344),(209,318,234,343),(210,317,235,342),(211,316,236,341),(212,315,237,340),(213,314,238,339),(214,313,239,338),(215,312,240,337),(216,311,241,336),(217,310,242,335),(218,309,243,334),(219,308,244,333),(220,307,245,332),(221,306,246,331),(222,305,247,330),(223,304,248,329),(224,303,249,328),(225,302,250,327)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 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 | 1 | 1 | 1 | 1 | 25 | ··· | 25 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | D5 | Dic5 | D10 | C4×D5 | D25 | Dic25 | D50 | C4×D25 |
| kernel | C4×Dic25 | C2×Dic25 | C2×C100 | Dic25 | C100 | C2×C20 | C20 | C2×C10 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 2 | 4 | 2 | 8 | 10 | 20 | 10 | 40 |
Matrix representation of C4×Dic25 ►in GL3(𝔽101) generated by
| 91 | 0 | 0 |
| 0 | 91 | 0 |
| 0 | 0 | 91 |
| 100 | 0 | 0 |
| 0 | 68 | 97 |
| 0 | 17 | 51 |
| 10 | 0 | 0 |
| 0 | 22 | 1 |
| 0 | 22 | 79 |
G:=sub<GL(3,GF(101))| [91,0,0,0,91,0,0,0,91],[100,0,0,0,68,17,0,97,51],[10,0,0,0,22,22,0,1,79] >;
C4×Dic25 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_{25} % in TeX
G:=Group("C4xDic25"); // GroupNames label
G:=SmallGroup(400,11);
// by ID
G=gap.SmallGroup(400,11);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,55,4324,628,11525]);
// Polycyclic
G:=Group<a,b,c|a^4=b^50=1,c^2=b^25,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export