direct product, abelian, monomial, 2-elementary
Aliases: C22×C100, SmallGroup(400,45)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C100 |
| C1 — C22×C100 |
| 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, C4, C22, C5, C2×C4, C23, C10, C10, C22×C4, C20, C2×C10, C25, C2×C20, C22×C10, C50, C50, C22×C20, C100, C2×C50, C2×C100, C22×C50, C22×C100
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C22×C4, C20, C2×C10, C25, C2×C20, C22×C10, C50, C22×C20, C100, C2×C50, C2×C100, C22×C50, C22×C100
►Regular action on 400 points
Generators in S400
(1 272)(2 273)(3 274)(4 275)(5 276)(6 277)(7 278)(8 279)(9 280)(10 281)(11 282)(12 283)(13 284)(14 285)(15 286)(16 287)(17 288)(18 289)(19 290)(20 291)(21 292)(22 293)(23 294)(24 295)(25 296)(26 297)(27 298)(28 299)(29 300)(30 201)(31 202)(32 203)(33 204)(34 205)(35 206)(36 207)(37 208)(38 209)(39 210)(40 211)(41 212)(42 213)(43 214)(44 215)(45 216)(46 217)(47 218)(48 219)(49 220)(50 221)(51 222)(52 223)(53 224)(54 225)(55 226)(56 227)(57 228)(58 229)(59 230)(60 231)(61 232)(62 233)(63 234)(64 235)(65 236)(66 237)(67 238)(68 239)(69 240)(70 241)(71 242)(72 243)(73 244)(74 245)(75 246)(76 247)(77 248)(78 249)(79 250)(80 251)(81 252)(82 253)(83 254)(84 255)(85 256)(86 257)(87 258)(88 259)(89 260)(90 261)(91 262)(92 263)(93 264)(94 265)(95 266)(96 267)(97 268)(98 269)(99 270)(100 271)(101 328)(102 329)(103 330)(104 331)(105 332)(106 333)(107 334)(108 335)(109 336)(110 337)(111 338)(112 339)(113 340)(114 341)(115 342)(116 343)(117 344)(118 345)(119 346)(120 347)(121 348)(122 349)(123 350)(124 351)(125 352)(126 353)(127 354)(128 355)(129 356)(130 357)(131 358)(132 359)(133 360)(134 361)(135 362)(136 363)(137 364)(138 365)(139 366)(140 367)(141 368)(142 369)(143 370)(144 371)(145 372)(146 373)(147 374)(148 375)(149 376)(150 377)(151 378)(152 379)(153 380)(154 381)(155 382)(156 383)(157 384)(158 385)(159 386)(160 387)(161 388)(162 389)(163 390)(164 391)(165 392)(166 393)(167 394)(168 395)(169 396)(170 397)(171 398)(172 399)(173 400)(174 301)(175 302)(176 303)(177 304)(178 305)(179 306)(180 307)(181 308)(182 309)(183 310)(184 311)(185 312)(186 313)(187 314)(188 315)(189 316)(190 317)(191 318)(192 319)(193 320)(194 321)(195 322)(196 323)(197 324)(198 325)(199 326)(200 327)
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 121)(10 122)(11 123)(12 124)(13 125)(14 126)(15 127)(16 128)(17 129)(18 130)(19 131)(20 132)(21 133)(22 134)(23 135)(24 136)(25 137)(26 138)(27 139)(28 140)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 148)(37 149)(38 150)(39 151)(40 152)(41 153)(42 154)(43 155)(44 156)(45 157)(46 158)(47 159)(48 160)(49 161)(50 162)(51 163)(52 164)(53 165)(54 166)(55 167)(56 168)(57 169)(58 170)(59 171)(60 172)(61 173)(62 174)(63 175)(64 176)(65 177)(66 178)(67 179)(68 180)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 188)(77 189)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 196)(85 197)(86 198)(87 199)(88 200)(89 101)(90 102)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)(99 111)(100 112)(201 369)(202 370)(203 371)(204 372)(205 373)(206 374)(207 375)(208 376)(209 377)(210 378)(211 379)(212 380)(213 381)(214 382)(215 383)(216 384)(217 385)(218 386)(219 387)(220 388)(221 389)(222 390)(223 391)(224 392)(225 393)(226 394)(227 395)(228 396)(229 397)(230 398)(231 399)(232 400)(233 301)(234 302)(235 303)(236 304)(237 305)(238 306)(239 307)(240 308)(241 309)(242 310)(243 311)(244 312)(245 313)(246 314)(247 315)(248 316)(249 317)(250 318)(251 319)(252 320)(253 321)(254 322)(255 323)(256 324)(257 325)(258 326)(259 327)(260 328)(261 329)(262 330)(263 331)(264 332)(265 333)(266 334)(267 335)(268 336)(269 337)(270 338)(271 339)(272 340)(273 341)(274 342)(275 343)(276 344)(277 345)(278 346)(279 347)(280 348)(281 349)(282 350)(283 351)(284 352)(285 353)(286 354)(287 355)(288 356)(289 357)(290 358)(291 359)(292 360)(293 361)(294 362)(295 363)(296 364)(297 365)(298 366)(299 367)(300 368)
(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)| (1,272)(2,273)(3,274)(4,275)(5,276)(6,277)(7,278)(8,279)(9,280)(10,281)(11,282)(12,283)(13,284)(14,285)(15,286)(16,287)(17,288)(18,289)(19,290)(20,291)(21,292)(22,293)(23,294)(24,295)(25,296)(26,297)(27,298)(28,299)(29,300)(30,201)(31,202)(32,203)(33,204)(34,205)(35,206)(36,207)(37,208)(38,209)(39,210)(40,211)(41,212)(42,213)(43,214)(44,215)(45,216)(46,217)(47,218)(48,219)(49,220)(50,221)(51,222)(52,223)(53,224)(54,225)(55,226)(56,227)(57,228)(58,229)(59,230)(60,231)(61,232)(62,233)(63,234)(64,235)(65,236)(66,237)(67,238)(68,239)(69,240)(70,241)(71,242)(72,243)(73,244)(74,245)(75,246)(76,247)(77,248)(78,249)(79,250)(80,251)(81,252)(82,253)(83,254)(84,255)(85,256)(86,257)(87,258)(88,259)(89,260)(90,261)(91,262)(92,263)(93,264)(94,265)(95,266)(96,267)(97,268)(98,269)(99,270)(100,271)(101,328)(102,329)(103,330)(104,331)(105,332)(106,333)(107,334)(108,335)(109,336)(110,337)(111,338)(112,339)(113,340)(114,341)(115,342)(116,343)(117,344)(118,345)(119,346)(120,347)(121,348)(122,349)(123,350)(124,351)(125,352)(126,353)(127,354)(128,355)(129,356)(130,357)(131,358)(132,359)(133,360)(134,361)(135,362)(136,363)(137,364)(138,365)(139,366)(140,367)(141,368)(142,369)(143,370)(144,371)(145,372)(146,373)(147,374)(148,375)(149,376)(150,377)(151,378)(152,379)(153,380)(154,381)(155,382)(156,383)(157,384)(158,385)(159,386)(160,387)(161,388)(162,389)(163,390)(164,391)(165,392)(166,393)(167,394)(168,395)(169,396)(170,397)(171,398)(172,399)(173,400)(174,301)(175,302)(176,303)(177,304)(178,305)(179,306)(180,307)(181,308)(182,309)(183,310)(184,311)(185,312)(186,313)(187,314)(188,315)(189,316)(190,317)(191,318)(192,319)(193,320)(194,321)(195,322)(196,323)(197,324)(198,325)(199,326)(200,327), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,127)(16,128)(17,129)(18,130)(19,131)(20,132)(21,133)(22,134)(23,135)(24,136)(25,137)(26,138)(27,139)(28,140)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,111)(100,112)(201,369)(202,370)(203,371)(204,372)(205,373)(206,374)(207,375)(208,376)(209,377)(210,378)(211,379)(212,380)(213,381)(214,382)(215,383)(216,384)(217,385)(218,386)(219,387)(220,388)(221,389)(222,390)(223,391)(224,392)(225,393)(226,394)(227,395)(228,396)(229,397)(230,398)(231,399)(232,400)(233,301)(234,302)(235,303)(236,304)(237,305)(238,306)(239,307)(240,308)(241,309)(242,310)(243,311)(244,312)(245,313)(246,314)(247,315)(248,316)(249,317)(250,318)(251,319)(252,320)(253,321)(254,322)(255,323)(256,324)(257,325)(258,326)(259,327)(260,328)(261,329)(262,330)(263,331)(264,332)(265,333)(266,334)(267,335)(268,336)(269,337)(270,338)(271,339)(272,340)(273,341)(274,342)(275,343)(276,344)(277,345)(278,346)(279,347)(280,348)(281,349)(282,350)(283,351)(284,352)(285,353)(286,354)(287,355)(288,356)(289,357)(290,358)(291,359)(292,360)(293,361)(294,362)(295,363)(296,364)(297,365)(298,366)(299,367)(300,368), (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( (1,272)(2,273)(3,274)(4,275)(5,276)(6,277)(7,278)(8,279)(9,280)(10,281)(11,282)(12,283)(13,284)(14,285)(15,286)(16,287)(17,288)(18,289)(19,290)(20,291)(21,292)(22,293)(23,294)(24,295)(25,296)(26,297)(27,298)(28,299)(29,300)(30,201)(31,202)(32,203)(33,204)(34,205)(35,206)(36,207)(37,208)(38,209)(39,210)(40,211)(41,212)(42,213)(43,214)(44,215)(45,216)(46,217)(47,218)(48,219)(49,220)(50,221)(51,222)(52,223)(53,224)(54,225)(55,226)(56,227)(57,228)(58,229)(59,230)(60,231)(61,232)(62,233)(63,234)(64,235)(65,236)(66,237)(67,238)(68,239)(69,240)(70,241)(71,242)(72,243)(73,244)(74,245)(75,246)(76,247)(77,248)(78,249)(79,250)(80,251)(81,252)(82,253)(83,254)(84,255)(85,256)(86,257)(87,258)(88,259)(89,260)(90,261)(91,262)(92,263)(93,264)(94,265)(95,266)(96,267)(97,268)(98,269)(99,270)(100,271)(101,328)(102,329)(103,330)(104,331)(105,332)(106,333)(107,334)(108,335)(109,336)(110,337)(111,338)(112,339)(113,340)(114,341)(115,342)(116,343)(117,344)(118,345)(119,346)(120,347)(121,348)(122,349)(123,350)(124,351)(125,352)(126,353)(127,354)(128,355)(129,356)(130,357)(131,358)(132,359)(133,360)(134,361)(135,362)(136,363)(137,364)(138,365)(139,366)(140,367)(141,368)(142,369)(143,370)(144,371)(145,372)(146,373)(147,374)(148,375)(149,376)(150,377)(151,378)(152,379)(153,380)(154,381)(155,382)(156,383)(157,384)(158,385)(159,386)(160,387)(161,388)(162,389)(163,390)(164,391)(165,392)(166,393)(167,394)(168,395)(169,396)(170,397)(171,398)(172,399)(173,400)(174,301)(175,302)(176,303)(177,304)(178,305)(179,306)(180,307)(181,308)(182,309)(183,310)(184,311)(185,312)(186,313)(187,314)(188,315)(189,316)(190,317)(191,318)(192,319)(193,320)(194,321)(195,322)(196,323)(197,324)(198,325)(199,326)(200,327), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,127)(16,128)(17,129)(18,130)(19,131)(20,132)(21,133)(22,134)(23,135)(24,136)(25,137)(26,138)(27,139)(28,140)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,111)(100,112)(201,369)(202,370)(203,371)(204,372)(205,373)(206,374)(207,375)(208,376)(209,377)(210,378)(211,379)(212,380)(213,381)(214,382)(215,383)(216,384)(217,385)(218,386)(219,387)(220,388)(221,389)(222,390)(223,391)(224,392)(225,393)(226,394)(227,395)(228,396)(229,397)(230,398)(231,399)(232,400)(233,301)(234,302)(235,303)(236,304)(237,305)(238,306)(239,307)(240,308)(241,309)(242,310)(243,311)(244,312)(245,313)(246,314)(247,315)(248,316)(249,317)(250,318)(251,319)(252,320)(253,321)(254,322)(255,323)(256,324)(257,325)(258,326)(259,327)(260,328)(261,329)(262,330)(263,331)(264,332)(265,333)(266,334)(267,335)(268,336)(269,337)(270,338)(271,339)(272,340)(273,341)(274,342)(275,343)(276,344)(277,345)(278,346)(279,347)(280,348)(281,349)(282,350)(283,351)(284,352)(285,353)(286,354)(287,355)(288,356)(289,357)(290,358)(291,359)(292,360)(293,361)(294,362)(295,363)(296,364)(297,365)(298,366)(299,367)(300,368), (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([[(1,272),(2,273),(3,274),(4,275),(5,276),(6,277),(7,278),(8,279),(9,280),(10,281),(11,282),(12,283),(13,284),(14,285),(15,286),(16,287),(17,288),(18,289),(19,290),(20,291),(21,292),(22,293),(23,294),(24,295),(25,296),(26,297),(27,298),(28,299),(29,300),(30,201),(31,202),(32,203),(33,204),(34,205),(35,206),(36,207),(37,208),(38,209),(39,210),(40,211),(41,212),(42,213),(43,214),(44,215),(45,216),(46,217),(47,218),(48,219),(49,220),(50,221),(51,222),(52,223),(53,224),(54,225),(55,226),(56,227),(57,228),(58,229),(59,230),(60,231),(61,232),(62,233),(63,234),(64,235),(65,236),(66,237),(67,238),(68,239),(69,240),(70,241),(71,242),(72,243),(73,244),(74,245),(75,246),(76,247),(77,248),(78,249),(79,250),(80,251),(81,252),(82,253),(83,254),(84,255),(85,256),(86,257),(87,258),(88,259),(89,260),(90,261),(91,262),(92,263),(93,264),(94,265),(95,266),(96,267),(97,268),(98,269),(99,270),(100,271),(101,328),(102,329),(103,330),(104,331),(105,332),(106,333),(107,334),(108,335),(109,336),(110,337),(111,338),(112,339),(113,340),(114,341),(115,342),(116,343),(117,344),(118,345),(119,346),(120,347),(121,348),(122,349),(123,350),(124,351),(125,352),(126,353),(127,354),(128,355),(129,356),(130,357),(131,358),(132,359),(133,360),(134,361),(135,362),(136,363),(137,364),(138,365),(139,366),(140,367),(141,368),(142,369),(143,370),(144,371),(145,372),(146,373),(147,374),(148,375),(149,376),(150,377),(151,378),(152,379),(153,380),(154,381),(155,382),(156,383),(157,384),(158,385),(159,386),(160,387),(161,388),(162,389),(163,390),(164,391),(165,392),(166,393),(167,394),(168,395),(169,396),(170,397),(171,398),(172,399),(173,400),(174,301),(175,302),(176,303),(177,304),(178,305),(179,306),(180,307),(181,308),(182,309),(183,310),(184,311),(185,312),(186,313),(187,314),(188,315),(189,316),(190,317),(191,318),(192,319),(193,320),(194,321),(195,322),(196,323),(197,324),(198,325),(199,326),(200,327)], [(1,113),(2,114),(3,115),(4,116),(5,117),(6,118),(7,119),(8,120),(9,121),(10,122),(11,123),(12,124),(13,125),(14,126),(15,127),(16,128),(17,129),(18,130),(19,131),(20,132),(21,133),(22,134),(23,135),(24,136),(25,137),(26,138),(27,139),(28,140),(29,141),(30,142),(31,143),(32,144),(33,145),(34,146),(35,147),(36,148),(37,149),(38,150),(39,151),(40,152),(41,153),(42,154),(43,155),(44,156),(45,157),(46,158),(47,159),(48,160),(49,161),(50,162),(51,163),(52,164),(53,165),(54,166),(55,167),(56,168),(57,169),(58,170),(59,171),(60,172),(61,173),(62,174),(63,175),(64,176),(65,177),(66,178),(67,179),(68,180),(69,181),(70,182),(71,183),(72,184),(73,185),(74,186),(75,187),(76,188),(77,189),(78,190),(79,191),(80,192),(81,193),(82,194),(83,195),(84,196),(85,197),(86,198),(87,199),(88,200),(89,101),(90,102),(91,103),(92,104),(93,105),(94,106),(95,107),(96,108),(97,109),(98,110),(99,111),(100,112),(201,369),(202,370),(203,371),(204,372),(205,373),(206,374),(207,375),(208,376),(209,377),(210,378),(211,379),(212,380),(213,381),(214,382),(215,383),(216,384),(217,385),(218,386),(219,387),(220,388),(221,389),(222,390),(223,391),(224,392),(225,393),(226,394),(227,395),(228,396),(229,397),(230,398),(231,399),(232,400),(233,301),(234,302),(235,303),(236,304),(237,305),(238,306),(239,307),(240,308),(241,309),(242,310),(243,311),(244,312),(245,313),(246,314),(247,315),(248,316),(249,317),(250,318),(251,319),(252,320),(253,321),(254,322),(255,323),(256,324),(257,325),(258,326),(259,327),(260,328),(261,329),(262,330),(263,331),(264,332),(265,333),(266,334),(267,335),(268,336),(269,337),(270,338),(271,339),(272,340),(273,341),(274,342),(275,343),(276,344),(277,345),(278,346),(279,347),(280,348),(281,349),(282,350),(283,351),(284,352),(285,353),(286,354),(287,355),(288,356),(289,357),(290,358),(291,359),(292,360),(293,361),(294,362),(295,363),(296,364),(297,365),(298,366),(299,367),(300,368)], [(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