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## G = C22×C102order 400 = 24·52

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

Aliases: C22×C102, SmallGroup(400,221)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C102
 Chief series C1 — C5 — C52 — C5×C10 — C102 — C2×C102 — C22×C102
 Lower central C1 — C22×C102
 Upper central C1 — C22×C102

Generators and relations for C22×C102
G = < a,b,c,d | a2=b2=c10=d10=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 536, all normal (4 characteristic)
C1, C2 [×15], C22 [×35], C5 [×6], C23 [×15], C10 [×90], C24, C2×C10 [×210], C52, C22×C10 [×90], C5×C10 [×15], C23×C10 [×6], C102 [×35], C2×C102 [×15], C22×C102
Quotients: C1, C2 [×15], C22 [×35], C5 [×6], C23 [×15], C10 [×90], C24, C2×C10 [×210], C52, C22×C10 [×90], C5×C10 [×15], C23×C10 [×6], C102 [×35], C2×C102 [×15], C22×C102

Smallest permutation representation of C22×C102
Regular action on 400 points
Generators in S400
(1 281)(2 282)(3 283)(4 284)(5 285)(6 286)(7 287)(8 288)(9 289)(10 290)(11 265)(12 266)(13 267)(14 268)(15 269)(16 270)(17 261)(18 262)(19 263)(20 264)(21 205)(22 206)(23 207)(24 208)(25 209)(26 210)(27 201)(28 202)(29 203)(30 204)(31 231)(32 232)(33 233)(34 234)(35 235)(36 236)(37 237)(38 238)(39 239)(40 240)(41 245)(42 246)(43 247)(44 248)(45 249)(46 250)(47 241)(48 242)(49 243)(50 244)(51 260)(52 251)(53 252)(54 253)(55 254)(56 255)(57 256)(58 257)(59 258)(60 259)(61 230)(62 221)(63 222)(64 223)(65 224)(66 225)(67 226)(68 227)(69 228)(70 229)(71 220)(72 211)(73 212)(74 213)(75 214)(76 215)(77 216)(78 217)(79 218)(80 219)(81 277)(82 278)(83 279)(84 280)(85 271)(86 272)(87 273)(88 274)(89 275)(90 276)(91 291)(92 292)(93 293)(94 294)(95 295)(96 296)(97 297)(98 298)(99 299)(100 300)(101 301)(102 302)(103 303)(104 304)(105 305)(106 306)(107 307)(108 308)(109 309)(110 310)(111 311)(112 312)(113 313)(114 314)(115 315)(116 316)(117 317)(118 318)(119 319)(120 320)(121 321)(122 322)(123 323)(124 324)(125 325)(126 326)(127 327)(128 328)(129 329)(130 330)(131 331)(132 332)(133 333)(134 334)(135 335)(136 336)(137 337)(138 338)(139 339)(140 340)(141 341)(142 342)(143 343)(144 344)(145 345)(146 346)(147 347)(148 348)(149 349)(150 350)(151 351)(152 352)(153 353)(154 354)(155 355)(156 356)(157 357)(158 358)(159 359)(160 360)(161 361)(162 362)(163 363)(164 364)(165 365)(166 366)(167 367)(168 368)(169 369)(170 370)(171 371)(172 372)(173 373)(174 374)(175 375)(176 376)(177 377)(178 378)(179 379)(180 380)(181 381)(182 382)(183 383)(184 384)(185 385)(186 386)(187 387)(188 388)(189 389)(190 390)(191 391)(192 392)(193 393)(194 394)(195 395)(196 396)(197 397)(198 398)(199 399)(200 400)
(1 181)(2 182)(3 183)(4 184)(5 185)(6 186)(7 187)(8 188)(9 189)(10 190)(11 365)(12 366)(13 367)(14 368)(15 369)(16 370)(17 361)(18 362)(19 363)(20 364)(21 305)(22 306)(23 307)(24 308)(25 309)(26 310)(27 301)(28 302)(29 303)(30 304)(31 131)(32 132)(33 133)(34 134)(35 135)(36 136)(37 137)(38 138)(39 139)(40 140)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 141)(48 142)(49 143)(50 144)(51 360)(52 351)(53 352)(54 353)(55 354)(56 355)(57 356)(58 357)(59 358)(60 359)(61 330)(62 321)(63 322)(64 323)(65 324)(66 325)(67 326)(68 327)(69 328)(70 329)(71 320)(72 311)(73 312)(74 313)(75 314)(76 315)(77 316)(78 317)(79 318)(80 319)(81 177)(82 178)(83 179)(84 180)(85 171)(86 172)(87 173)(88 174)(89 175)(90 176)(91 191)(92 192)(93 193)(94 194)(95 195)(96 196)(97 197)(98 198)(99 199)(100 200)(101 201)(102 202)(103 203)(104 204)(105 205)(106 206)(107 207)(108 208)(109 209)(110 210)(111 211)(112 212)(113 213)(114 214)(115 215)(116 216)(117 217)(118 218)(119 219)(120 220)(121 221)(122 222)(123 223)(124 224)(125 225)(126 226)(127 227)(128 228)(129 229)(130 230)(151 251)(152 252)(153 253)(154 254)(155 255)(156 256)(157 257)(158 258)(159 259)(160 260)(161 261)(162 262)(163 263)(164 264)(165 265)(166 266)(167 267)(168 268)(169 269)(170 270)(231 331)(232 332)(233 333)(234 334)(235 335)(236 336)(237 337)(238 338)(239 339)(240 340)(241 341)(242 342)(243 343)(244 344)(245 345)(246 346)(247 347)(248 348)(249 349)(250 350)(271 371)(272 372)(273 373)(274 374)(275 375)(276 376)(277 377)(278 378)(279 379)(280 380)(281 381)(282 382)(283 383)(284 384)(285 385)(286 386)(287 387)(288 388)(289 389)(290 390)(291 391)(292 392)(293 393)(294 394)(295 395)(296 396)(297 397)(298 398)(299 399)(300 400)
(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)
(1 169 151 104 41 95 117 123 83 32)(2 170 152 105 42 96 118 124 84 33)(3 161 153 106 43 97 119 125 85 34)(4 162 154 107 44 98 120 126 86 35)(5 163 155 108 45 99 111 127 87 36)(6 164 156 109 46 100 112 128 88 37)(7 165 157 110 47 91 113 129 89 38)(8 166 158 101 48 92 114 130 90 39)(9 167 159 102 49 93 115 121 81 40)(10 168 160 103 50 94 116 122 82 31)(11 58 26 341 391 74 70 375 338 387)(12 59 27 342 392 75 61 376 339 388)(13 60 28 343 393 76 62 377 340 389)(14 51 29 344 394 77 63 378 331 390)(15 52 30 345 395 78 64 379 332 381)(16 53 21 346 396 79 65 380 333 382)(17 54 22 347 397 80 66 371 334 383)(18 55 23 348 398 71 67 372 335 384)(19 56 24 349 399 72 68 373 336 385)(20 57 25 350 400 73 69 374 337 386)(131 190 268 260 203 144 194 216 222 178)(132 181 269 251 204 145 195 217 223 179)(133 182 270 252 205 146 196 218 224 180)(134 183 261 253 206 147 197 219 225 171)(135 184 262 254 207 148 198 220 226 172)(136 185 263 255 208 149 199 211 227 173)(137 186 264 256 209 150 200 212 228 174)(138 187 265 257 210 141 191 213 229 175)(139 188 266 258 201 142 192 214 230 176)(140 189 267 259 202 143 193 215 221 177)(231 290 368 360 303 244 294 316 322 278)(232 281 369 351 304 245 295 317 323 279)(233 282 370 352 305 246 296 318 324 280)(234 283 361 353 306 247 297 319 325 271)(235 284 362 354 307 248 298 320 326 272)(236 285 363 355 308 249 299 311 327 273)(237 286 364 356 309 250 300 312 328 274)(238 287 365 357 310 241 291 313 329 275)(239 288 366 358 301 242 292 314 330 276)(240 289 367 359 302 243 293 315 321 277)

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

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

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

400 conjugacy classes

 class 1 2A ··· 2O 5A ··· 5X 10A ··· 10MV order 1 2 ··· 2 5 ··· 5 10 ··· 10 size 1 1 ··· 1 1 ··· 1 1 ··· 1

400 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C5 C10 kernel C22×C102 C2×C102 C23×C10 C22×C10 # reps 1 15 24 360

Matrix representation of C22×C102 in GL4(𝔽11) generated by

 1 0 0 0 0 10 0 0 0 0 10 0 0 0 0 1
,
 10 0 0 0 0 10 0 0 0 0 10 0 0 0 0 1
,
 5 0 0 0 0 6 0 0 0 0 4 0 0 0 0 2
,
 3 0 0 0 0 9 0 0 0 0 9 0 0 0 0 7
G:=sub<GL(4,GF(11))| [1,0,0,0,0,10,0,0,0,0,10,0,0,0,0,1],[10,0,0,0,0,10,0,0,0,0,10,0,0,0,0,1],[5,0,0,0,0,6,0,0,0,0,4,0,0,0,0,2],[3,0,0,0,0,9,0,0,0,0,9,0,0,0,0,7] >;

C22×C102 in GAP, Magma, Sage, TeX

C_2^2\times C_{10}^2
% in TeX

G:=Group("C2^2xC10^2");
// GroupNames label

G:=SmallGroup(400,221);
// by ID

G=gap.SmallGroup(400,221);
# by ID

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

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

׿
×
𝔽