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## G = C22×C100order 400 = 24·52

### Abelian group of type [2,2,100]

Aliases: C22×C100, SmallGroup(400,45)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C100
 Chief series C1 — C5 — C10 — C50 — C100 — C2×C100 — C22×C100
 Lower central C1 — C22×C100
 Upper central C1 — C22×C100

Generators and relations for C22×C100
G = < a,b,c | a2=b2=c100=1, ab=ba, ac=ca, bc=cb >

Subgroups: 81, all normal (12 characteristic)
C1, C2, C2 [×6], C4 [×4], C22 [×7], C5, C2×C4 [×6], C23, C10, C10 [×6], C22×C4, C20 [×4], C2×C10 [×7], C25, C2×C20 [×6], C22×C10, C50, C50 [×6], C22×C20, C100 [×4], C2×C50 [×7], C2×C100 [×6], C22×C50, C22×C100
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], C23, C10 [×7], C22×C4, C20 [×4], C2×C10 [×7], C25, C2×C20 [×6], C22×C10, C50 [×7], C22×C20, C100 [×4], C2×C50 [×7], C2×C100 [×6], C22×C50, C22×C100

Smallest permutation representation of C22×C100
Regular action on 400 points
Generators in S400
(1 264)(2 265)(3 266)(4 267)(5 268)(6 269)(7 270)(8 271)(9 272)(10 273)(11 274)(12 275)(13 276)(14 277)(15 278)(16 279)(17 280)(18 281)(19 282)(20 283)(21 284)(22 285)(23 286)(24 287)(25 288)(26 289)(27 290)(28 291)(29 292)(30 293)(31 294)(32 295)(33 296)(34 297)(35 298)(36 299)(37 300)(38 201)(39 202)(40 203)(41 204)(42 205)(43 206)(44 207)(45 208)(46 209)(47 210)(48 211)(49 212)(50 213)(51 214)(52 215)(53 216)(54 217)(55 218)(56 219)(57 220)(58 221)(59 222)(60 223)(61 224)(62 225)(63 226)(64 227)(65 228)(66 229)(67 230)(68 231)(69 232)(70 233)(71 234)(72 235)(73 236)(74 237)(75 238)(76 239)(77 240)(78 241)(79 242)(80 243)(81 244)(82 245)(83 246)(84 247)(85 248)(86 249)(87 250)(88 251)(89 252)(90 253)(91 254)(92 255)(93 256)(94 257)(95 258)(96 259)(97 260)(98 261)(99 262)(100 263)(101 313)(102 314)(103 315)(104 316)(105 317)(106 318)(107 319)(108 320)(109 321)(110 322)(111 323)(112 324)(113 325)(114 326)(115 327)(116 328)(117 329)(118 330)(119 331)(120 332)(121 333)(122 334)(123 335)(124 336)(125 337)(126 338)(127 339)(128 340)(129 341)(130 342)(131 343)(132 344)(133 345)(134 346)(135 347)(136 348)(137 349)(138 350)(139 351)(140 352)(141 353)(142 354)(143 355)(144 356)(145 357)(146 358)(147 359)(148 360)(149 361)(150 362)(151 363)(152 364)(153 365)(154 366)(155 367)(156 368)(157 369)(158 370)(159 371)(160 372)(161 373)(162 374)(163 375)(164 376)(165 377)(166 378)(167 379)(168 380)(169 381)(170 382)(171 383)(172 384)(173 385)(174 386)(175 387)(176 388)(177 389)(178 390)(179 391)(180 392)(181 393)(182 394)(183 395)(184 396)(185 397)(186 398)(187 399)(188 400)(189 301)(190 302)(191 303)(192 304)(193 305)(194 306)(195 307)(196 308)(197 309)(198 310)(199 311)(200 312)
(1 149)(2 150)(3 151)(4 152)(5 153)(6 154)(7 155)(8 156)(9 157)(10 158)(11 159)(12 160)(13 161)(14 162)(15 163)(16 164)(17 165)(18 166)(19 167)(20 168)(21 169)(22 170)(23 171)(24 172)(25 173)(26 174)(27 175)(28 176)(29 177)(30 178)(31 179)(32 180)(33 181)(34 182)(35 183)(36 184)(37 185)(38 186)(39 187)(40 188)(41 189)(42 190)(43 191)(44 192)(45 193)(46 194)(47 195)(48 196)(49 197)(50 198)(51 199)(52 200)(53 101)(54 102)(55 103)(56 104)(57 105)(58 106)(59 107)(60 108)(61 109)(62 110)(63 111)(64 112)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)(81 129)(82 130)(83 131)(84 132)(85 133)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 141)(94 142)(95 143)(96 144)(97 145)(98 146)(99 147)(100 148)(201 398)(202 399)(203 400)(204 301)(205 302)(206 303)(207 304)(208 305)(209 306)(210 307)(211 308)(212 309)(213 310)(214 311)(215 312)(216 313)(217 314)(218 315)(219 316)(220 317)(221 318)(222 319)(223 320)(224 321)(225 322)(226 323)(227 324)(228 325)(229 326)(230 327)(231 328)(232 329)(233 330)(234 331)(235 332)(236 333)(237 334)(238 335)(239 336)(240 337)(241 338)(242 339)(243 340)(244 341)(245 342)(246 343)(247 344)(248 345)(249 346)(250 347)(251 348)(252 349)(253 350)(254 351)(255 352)(256 353)(257 354)(258 355)(259 356)(260 357)(261 358)(262 359)(263 360)(264 361)(265 362)(266 363)(267 364)(268 365)(269 366)(270 367)(271 368)(272 369)(273 370)(274 371)(275 372)(276 373)(277 374)(278 375)(279 376)(280 377)(281 378)(282 379)(283 380)(284 381)(285 382)(286 383)(287 384)(288 385)(289 386)(290 387)(291 388)(292 389)(293 390)(294 391)(295 392)(296 393)(297 394)(298 395)(299 396)(300 397)
(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)

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

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

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

400 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 5A 5B 5C 5D 10A ··· 10AB 20A ··· 20AF 25A ··· 25T 50A ··· 50EJ 100A ··· 100FD order 1 2 ··· 2 4 ··· 4 5 5 5 5 10 ··· 10 20 ··· 20 25 ··· 25 50 ··· 50 100 ··· 100 size 1 1 ··· 1 1 ··· 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

400 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C5 C10 C10 C20 C25 C50 C50 C100 kernel C22×C100 C2×C100 C22×C50 C2×C50 C22×C20 C2×C20 C22×C10 C2×C10 C22×C4 C2×C4 C23 C22 # reps 1 6 1 8 4 24 4 32 20 120 20 160

Matrix representation of C22×C100 in GL3(𝔽101) generated by

 100 0 0 0 1 0 0 0 100
,
 100 0 0 0 100 0 0 0 1
,
 93 0 0 0 6 0 0 0 76
G:=sub<GL(3,GF(101))| [100,0,0,0,1,0,0,0,100],[100,0,0,0,100,0,0,0,1],[93,0,0,0,6,0,0,0,76] >;

C22×C100 in GAP, Magma, Sage, TeX

C_2^2\times C_{100}
% in TeX

G:=Group("C2^2xC100");
// GroupNames label

G:=SmallGroup(400,45);
// by ID

G=gap.SmallGroup(400,45);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,240,261]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^100=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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