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## G = C22×Dic25order 400 = 24·52

### Direct product of C22 and Dic25

Series: Derived Chief Lower central Upper central

 Derived series C1 — C25 — C22×Dic25
 Chief series C1 — C5 — C25 — C50 — Dic25 — C2×Dic25 — C22×Dic25
 Lower central C25 — C22×Dic25
 Upper central C1 — C23

Generators and relations for C22×Dic25
G = < a,b,c,d | a2=b2=c50=1, d2=c25, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 389 in 81 conjugacy classes, 59 normal (10 characteristic)
C1, C2, C2, C4, C22, C5, C2×C4, C23, C10, C10, C22×C4, Dic5, C2×C10, C25, C2×Dic5, C22×C10, C50, C50, C22×Dic5, Dic25, C2×C50, C2×Dic25, C22×C50, C22×Dic25
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, Dic5, D10, C2×Dic5, C22×D5, D25, C22×Dic5, Dic25, D50, C2×Dic25, C22×D25, C22×Dic25

Smallest permutation representation of C22×Dic25
Regular action on 400 points
Generators in S400
(1 123)(2 124)(3 125)(4 126)(5 127)(6 128)(7 129)(8 130)(9 131)(10 132)(11 133)(12 134)(13 135)(14 136)(15 137)(16 138)(17 139)(18 140)(19 141)(20 142)(21 143)(22 144)(23 145)(24 146)(25 147)(26 148)(27 149)(28 150)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 184)(52 185)(53 186)(54 187)(55 188)(56 189)(57 190)(58 191)(59 192)(60 193)(61 194)(62 195)(63 196)(64 197)(65 198)(66 199)(67 200)(68 151)(69 152)(70 153)(71 154)(72 155)(73 156)(74 157)(75 158)(76 159)(77 160)(78 161)(79 162)(80 163)(81 164)(82 165)(83 166)(84 167)(85 168)(86 169)(87 170)(88 171)(89 172)(90 173)(91 174)(92 175)(93 176)(94 177)(95 178)(96 179)(97 180)(98 181)(99 182)(100 183)(201 331)(202 332)(203 333)(204 334)(205 335)(206 336)(207 337)(208 338)(209 339)(210 340)(211 341)(212 342)(213 343)(214 344)(215 345)(216 346)(217 347)(218 348)(219 349)(220 350)(221 301)(222 302)(223 303)(224 304)(225 305)(226 306)(227 307)(228 308)(229 309)(230 310)(231 311)(232 312)(233 313)(234 314)(235 315)(236 316)(237 317)(238 318)(239 319)(240 320)(241 321)(242 322)(243 323)(244 324)(245 325)(246 326)(247 327)(248 328)(249 329)(250 330)(251 365)(252 366)(253 367)(254 368)(255 369)(256 370)(257 371)(258 372)(259 373)(260 374)(261 375)(262 376)(263 377)(264 378)(265 379)(266 380)(267 381)(268 382)(269 383)(270 384)(271 385)(272 386)(273 387)(274 388)(275 389)(276 390)(277 391)(278 392)(279 393)(280 394)(281 395)(282 396)(283 397)(284 398)(285 399)(286 400)(287 351)(288 352)(289 353)(290 354)(291 355)(292 356)(293 357)(294 358)(295 359)(296 360)(297 361)(298 362)(299 363)(300 364)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(101 184)(102 185)(103 186)(104 187)(105 188)(106 189)(107 190)(108 191)(109 192)(110 193)(111 194)(112 195)(113 196)(114 197)(115 198)(116 199)(117 200)(118 151)(119 152)(120 153)(121 154)(122 155)(123 156)(124 157)(125 158)(126 159)(127 160)(128 161)(129 162)(130 163)(131 164)(132 165)(133 166)(134 167)(135 168)(136 169)(137 170)(138 171)(139 172)(140 173)(141 174)(142 175)(143 176)(144 177)(145 178)(146 179)(147 180)(148 181)(149 182)(150 183)(201 283)(202 284)(203 285)(204 286)(205 287)(206 288)(207 289)(208 290)(209 291)(210 292)(211 293)(212 294)(213 295)(214 296)(215 297)(216 298)(217 299)(218 300)(219 251)(220 252)(221 253)(222 254)(223 255)(224 256)(225 257)(226 258)(227 259)(228 260)(229 261)(230 262)(231 263)(232 264)(233 265)(234 266)(235 267)(236 268)(237 269)(238 270)(239 271)(240 272)(241 273)(242 274)(243 275)(244 276)(245 277)(246 278)(247 279)(248 280)(249 281)(250 282)(301 367)(302 368)(303 369)(304 370)(305 371)(306 372)(307 373)(308 374)(309 375)(310 376)(311 377)(312 378)(313 379)(314 380)(315 381)(316 382)(317 383)(318 384)(319 385)(320 386)(321 387)(322 388)(323 389)(324 390)(325 391)(326 392)(327 393)(328 394)(329 395)(330 396)(331 397)(332 398)(333 399)(334 400)(335 351)(336 352)(337 353)(338 354)(339 355)(340 356)(341 357)(342 358)(343 359)(344 360)(345 361)(346 362)(347 363)(348 364)(349 365)(350 366)
(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)
(1 265 26 290)(2 264 27 289)(3 263 28 288)(4 262 29 287)(5 261 30 286)(6 260 31 285)(7 259 32 284)(8 258 33 283)(9 257 34 282)(10 256 35 281)(11 255 36 280)(12 254 37 279)(13 253 38 278)(14 252 39 277)(15 251 40 276)(16 300 41 275)(17 299 42 274)(18 298 43 273)(19 297 44 272)(20 296 45 271)(21 295 46 270)(22 294 47 269)(23 293 48 268)(24 292 49 267)(25 291 50 266)(51 205 76 230)(52 204 77 229)(53 203 78 228)(54 202 79 227)(55 201 80 226)(56 250 81 225)(57 249 82 224)(58 248 83 223)(59 247 84 222)(60 246 85 221)(61 245 86 220)(62 244 87 219)(63 243 88 218)(64 242 89 217)(65 241 90 216)(66 240 91 215)(67 239 92 214)(68 238 93 213)(69 237 94 212)(70 236 95 211)(71 235 96 210)(72 234 97 209)(73 233 98 208)(74 232 99 207)(75 231 100 206)(101 351 126 376)(102 400 127 375)(103 399 128 374)(104 398 129 373)(105 397 130 372)(106 396 131 371)(107 395 132 370)(108 394 133 369)(109 393 134 368)(110 392 135 367)(111 391 136 366)(112 390 137 365)(113 389 138 364)(114 388 139 363)(115 387 140 362)(116 386 141 361)(117 385 142 360)(118 384 143 359)(119 383 144 358)(120 382 145 357)(121 381 146 356)(122 380 147 355)(123 379 148 354)(124 378 149 353)(125 377 150 352)(151 318 176 343)(152 317 177 342)(153 316 178 341)(154 315 179 340)(155 314 180 339)(156 313 181 338)(157 312 182 337)(158 311 183 336)(159 310 184 335)(160 309 185 334)(161 308 186 333)(162 307 187 332)(163 306 188 331)(164 305 189 330)(165 304 190 329)(166 303 191 328)(167 302 192 327)(168 301 193 326)(169 350 194 325)(170 349 195 324)(171 348 196 323)(172 347 197 322)(173 346 198 321)(174 345 199 320)(175 344 200 319)

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

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

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

112 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 5A 5B 10A ··· 10N 25A ··· 25J 50A ··· 50BR order 1 2 ··· 2 4 ··· 4 5 5 10 ··· 10 25 ··· 25 50 ··· 50 size 1 1 ··· 1 25 ··· 25 2 2 2 ··· 2 2 ··· 2 2 ··· 2

112 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + - + + - + image C1 C2 C2 C4 D5 Dic5 D10 D25 Dic25 D50 kernel C22×Dic25 C2×Dic25 C22×C50 C2×C50 C22×C10 C2×C10 C2×C10 C23 C22 C22 # reps 1 6 1 8 2 8 6 10 40 30

Matrix representation of C22×Dic25 in GL4(𝔽101) generated by

 100 0 0 0 0 1 0 0 0 0 100 0 0 0 0 100
,
 1 0 0 0 0 100 0 0 0 0 100 0 0 0 0 100
,
 100 0 0 0 0 1 0 0 0 0 8 69 0 0 32 11
,
 10 0 0 0 0 1 0 0 0 0 0 100 0 0 100 0
G:=sub<GL(4,GF(101))| [100,0,0,0,0,1,0,0,0,0,100,0,0,0,0,100],[1,0,0,0,0,100,0,0,0,0,100,0,0,0,0,100],[100,0,0,0,0,1,0,0,0,0,8,32,0,0,69,11],[10,0,0,0,0,1,0,0,0,0,0,100,0,0,100,0] >;

C22×Dic25 in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_{25}
% in TeX

G:=Group("C2^2xDic25");
// GroupNames label

G:=SmallGroup(400,43);
// by ID

G=gap.SmallGroup(400,43);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,4324,628,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^50=1,d^2=c^25,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽