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## G = C2×C10×C20order 400 = 24·52

### Abelian group of type [2,10,20]

Aliases: C2×C10×C20, SmallGroup(400,201)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10×C20
 Chief series C1 — C2 — C10 — C5×C10 — C5×C20 — C10×C20 — C2×C10×C20
 Lower central C1 — C2×C10×C20
 Upper central C1 — C2×C10×C20

Generators and relations for C2×C10×C20
G = < a,b,c | a2=b10=c20=1, ab=ba, ac=ca, bc=cb >

Subgroups: 216, all normal (8 characteristic)
C1, C2, C2 [×6], C4 [×4], C22 [×7], C5 [×6], C2×C4 [×6], C23, C10 [×42], C22×C4, C20 [×24], C2×C10 [×42], C52, C2×C20 [×36], C22×C10 [×6], C5×C10, C5×C10 [×6], C22×C20 [×6], C5×C20 [×4], C102 [×7], C10×C20 [×6], C2×C102, C2×C10×C20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5 [×6], C2×C4 [×6], C23, C10 [×42], C22×C4, C20 [×24], C2×C10 [×42], C52, C2×C20 [×36], C22×C10 [×6], C5×C10 [×7], C22×C20 [×6], C5×C20 [×4], C102 [×7], C10×C20 [×6], C2×C102, C2×C10×C20

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

400 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C5 C10 C10 C20 kernel C2×C10×C20 C10×C20 C2×C102 C102 C22×C20 C2×C20 C22×C10 C2×C10 # reps 1 6 1 8 24 144 24 192

Matrix representation of C2×C10×C20 in GL3(𝔽41) generated by

 40 0 0 0 1 0 0 0 1
,
 23 0 0 0 23 0 0 0 4
,
 4 0 0 0 4 0 0 0 5
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,1],[23,0,0,0,23,0,0,0,4],[4,0,0,0,4,0,0,0,5] >;

C2×C10×C20 in GAP, Magma, Sage, TeX

C_2\times C_{10}\times C_{20}
% in TeX

G:=Group("C2xC10xC20");
// GroupNames label

G:=SmallGroup(400,201);
// by ID

G=gap.SmallGroup(400,201);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-5,-2,1200]);
// Polycyclic

G:=Group<a,b,c|a^2=b^10=c^20=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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