C2^2xC10^2 - GroupNames

G = C₂²×C₁₀² order 400 = 2⁴·5²

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

direct product, abelian, monomial

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, C₂², C₅, C₂³, C₁₀, C₂⁴, C₂×C₁₀, C₅², C₂²×C₁₀, C₅×C₁₀, C₂³×C₁₀, C₁₀², C₂×C₁₀², C₂²×C₁₀²
Quotients: C₁, C₂, C₂², C₅, C₂³, C₁₀, C₂⁴, C₂×C₁₀, C₅², C₂²×C₁₀, C₅×C₁₀, C₂³×C₁₀, C₁₀², C₂×C₁₀², C₂²×C₁₀²

Smallest permutation representation of C₂²×C₁₀²
►Regular action on 400 points

Generators in S₄₀₀

(1 247)(2 248)(3 249)(4 250)(5 241)(6 242)(7 243)(8 244)(9 245)(10 246)(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)(41 272)(42 273)(43 274)(44 275)(45 276)(46 277)(47 278)(48 279)(49 280)(50 271)(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)(81 281)(82 282)(83 283)(84 284)(85 285)(86 286)(87 287)(88 288)(89 289)(90 290)(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)

(1 147)(2 148)(3 149)(4 150)(5 141)(6 142)(7 143)(8 144)(9 145)(10 146)(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)(41 172)(42 173)(43 174)(44 175)(45 176)(46 177)(47 178)(48 179)(49 180)(50 171)(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)(81 181)(82 182)(83 183)(84 184)(85 185)(86 186)(87 187)(88 188)(89 189)(90 190)(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)

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

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

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

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

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

G = C22×C102 order 400 = 24·52

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

G = C₂²×C₁₀² order 400 = 2⁴·5²