direct product, abelian, monomial, 2-elementary
Aliases: C23×C44, SmallGroup(352,188)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C23×C44 |
| C1 — C23×C44 |
| C1 — C23×C44 |
Generators and relations for C23×C44
G = < a,b,c,d | a2=b2=c2=d44=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
Subgroups: 236, all normal (8 characteristic)
C1, C2, C2, C4, C22, C2×C4, C23, C11, C22×C4, C24, C22, C22, C23×C4, C44, C2×C22, C2×C44, C22×C22, C22×C44, C23×C22, C23×C44
Quotients: C1, C2, C4, C22, C2×C4, C23, C11, C22×C4, C24, C22, C23×C4, C44, C2×C22, C2×C44, C22×C22, C22×C44, C23×C22, C23×C44
►Regular action on 352 points
Generators in S352
(1 297)(2 298)(3 299)(4 300)(5 301)(6 302)(7 303)(8 304)(9 305)(10 306)(11 307)(12 308)(13 265)(14 266)(15 267)(16 268)(17 269)(18 270)(19 271)(20 272)(21 273)(22 274)(23 275)(24 276)(25 277)(26 278)(27 279)(28 280)(29 281)(30 282)(31 283)(32 284)(33 285)(34 286)(35 287)(36 288)(37 289)(38 290)(39 291)(40 292)(41 293)(42 294)(43 295)(44 296)(45 138)(46 139)(47 140)(48 141)(49 142)(50 143)(51 144)(52 145)(53 146)(54 147)(55 148)(56 149)(57 150)(58 151)(59 152)(60 153)(61 154)(62 155)(63 156)(64 157)(65 158)(66 159)(67 160)(68 161)(69 162)(70 163)(71 164)(72 165)(73 166)(74 167)(75 168)(76 169)(77 170)(78 171)(79 172)(80 173)(81 174)(82 175)(83 176)(84 133)(85 134)(86 135)(87 136)(88 137)(89 233)(90 234)(91 235)(92 236)(93 237)(94 238)(95 239)(96 240)(97 241)(98 242)(99 243)(100 244)(101 245)(102 246)(103 247)(104 248)(105 249)(106 250)(107 251)(108 252)(109 253)(110 254)(111 255)(112 256)(113 257)(114 258)(115 259)(116 260)(117 261)(118 262)(119 263)(120 264)(121 221)(122 222)(123 223)(124 224)(125 225)(126 226)(127 227)(128 228)(129 229)(130 230)(131 231)(132 232)(177 313)(178 314)(179 315)(180 316)(181 317)(182 318)(183 319)(184 320)(185 321)(186 322)(187 323)(188 324)(189 325)(190 326)(191 327)(192 328)(193 329)(194 330)(195 331)(196 332)(197 333)(198 334)(199 335)(200 336)(201 337)(202 338)(203 339)(204 340)(205 341)(206 342)(207 343)(208 344)(209 345)(210 346)(211 347)(212 348)(213 349)(214 350)(215 351)(216 352)(217 309)(218 310)(219 311)(220 312)
(1 317)(2 318)(3 319)(4 320)(5 321)(6 322)(7 323)(8 324)(9 325)(10 326)(11 327)(12 328)(13 329)(14 330)(15 331)(16 332)(17 333)(18 334)(19 335)(20 336)(21 337)(22 338)(23 339)(24 340)(25 341)(26 342)(27 343)(28 344)(29 345)(30 346)(31 347)(32 348)(33 349)(34 350)(35 351)(36 352)(37 309)(38 310)(39 311)(40 312)(41 313)(42 314)(43 315)(44 316)(45 255)(46 256)(47 257)(48 258)(49 259)(50 260)(51 261)(52 262)(53 263)(54 264)(55 221)(56 222)(57 223)(58 224)(59 225)(60 226)(61 227)(62 228)(63 229)(64 230)(65 231)(66 232)(67 233)(68 234)(69 235)(70 236)(71 237)(72 238)(73 239)(74 240)(75 241)(76 242)(77 243)(78 244)(79 245)(80 246)(81 247)(82 248)(83 249)(84 250)(85 251)(86 252)(87 253)(88 254)(89 160)(90 161)(91 162)(92 163)(93 164)(94 165)(95 166)(96 167)(97 168)(98 169)(99 170)(100 171)(101 172)(102 173)(103 174)(104 175)(105 176)(106 133)(107 134)(108 135)(109 136)(110 137)(111 138)(112 139)(113 140)(114 141)(115 142)(116 143)(117 144)(118 145)(119 146)(120 147)(121 148)(122 149)(123 150)(124 151)(125 152)(126 153)(127 154)(128 155)(129 156)(130 157)(131 158)(132 159)(177 293)(178 294)(179 295)(180 296)(181 297)(182 298)(183 299)(184 300)(185 301)(186 302)(187 303)(188 304)(189 305)(190 306)(191 307)(192 308)(193 265)(194 266)(195 267)(196 268)(197 269)(198 270)(199 271)(200 272)(201 273)(202 274)(203 275)(204 276)(205 277)(206 278)(207 279)(208 280)(209 281)(210 282)(211 283)(212 284)(213 285)(214 286)(215 287)(216 288)(217 289)(218 290)(219 291)(220 292)
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 85)(13 86)(14 87)(15 88)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)(41 70)(42 71)(43 72)(44 73)(89 218)(90 219)(91 220)(92 177)(93 178)(94 179)(95 180)(96 181)(97 182)(98 183)(99 184)(100 185)(101 186)(102 187)(103 188)(104 189)(105 190)(106 191)(107 192)(108 193)(109 194)(110 195)(111 196)(112 197)(113 198)(114 199)(115 200)(116 201)(117 202)(118 203)(119 204)(120 205)(121 206)(122 207)(123 208)(124 209)(125 210)(126 211)(127 212)(128 213)(129 214)(130 215)(131 216)(132 217)(133 307)(134 308)(135 265)(136 266)(137 267)(138 268)(139 269)(140 270)(141 271)(142 272)(143 273)(144 274)(145 275)(146 276)(147 277)(148 278)(149 279)(150 280)(151 281)(152 282)(153 283)(154 284)(155 285)(156 286)(157 287)(158 288)(159 289)(160 290)(161 291)(162 292)(163 293)(164 294)(165 295)(166 296)(167 297)(168 298)(169 299)(170 300)(171 301)(172 302)(173 303)(174 304)(175 305)(176 306)(221 342)(222 343)(223 344)(224 345)(225 346)(226 347)(227 348)(228 349)(229 350)(230 351)(231 352)(232 309)(233 310)(234 311)(235 312)(236 313)(237 314)(238 315)(239 316)(240 317)(241 318)(242 319)(243 320)(244 321)(245 322)(246 323)(247 324)(248 325)(249 326)(250 327)(251 328)(252 329)(253 330)(254 331)(255 332)(256 333)(257 334)(258 335)(259 336)(260 337)(261 338)(262 339)(263 340)(264 341)
(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)
G:=sub<Sym(352)| (1,297)(2,298)(3,299)(4,300)(5,301)(6,302)(7,303)(8,304)(9,305)(10,306)(11,307)(12,308)(13,265)(14,266)(15,267)(16,268)(17,269)(18,270)(19,271)(20,272)(21,273)(22,274)(23,275)(24,276)(25,277)(26,278)(27,279)(28,280)(29,281)(30,282)(31,283)(32,284)(33,285)(34,286)(35,287)(36,288)(37,289)(38,290)(39,291)(40,292)(41,293)(42,294)(43,295)(44,296)(45,138)(46,139)(47,140)(48,141)(49,142)(50,143)(51,144)(52,145)(53,146)(54,147)(55,148)(56,149)(57,150)(58,151)(59,152)(60,153)(61,154)(62,155)(63,156)(64,157)(65,158)(66,159)(67,160)(68,161)(69,162)(70,163)(71,164)(72,165)(73,166)(74,167)(75,168)(76,169)(77,170)(78,171)(79,172)(80,173)(81,174)(82,175)(83,176)(84,133)(85,134)(86,135)(87,136)(88,137)(89,233)(90,234)(91,235)(92,236)(93,237)(94,238)(95,239)(96,240)(97,241)(98,242)(99,243)(100,244)(101,245)(102,246)(103,247)(104,248)(105,249)(106,250)(107,251)(108,252)(109,253)(110,254)(111,255)(112,256)(113,257)(114,258)(115,259)(116,260)(117,261)(118,262)(119,263)(120,264)(121,221)(122,222)(123,223)(124,224)(125,225)(126,226)(127,227)(128,228)(129,229)(130,230)(131,231)(132,232)(177,313)(178,314)(179,315)(180,316)(181,317)(182,318)(183,319)(184,320)(185,321)(186,322)(187,323)(188,324)(189,325)(190,326)(191,327)(192,328)(193,329)(194,330)(195,331)(196,332)(197,333)(198,334)(199,335)(200,336)(201,337)(202,338)(203,339)(204,340)(205,341)(206,342)(207,343)(208,344)(209,345)(210,346)(211,347)(212,348)(213,349)(214,350)(215,351)(216,352)(217,309)(218,310)(219,311)(220,312), (1,317)(2,318)(3,319)(4,320)(5,321)(6,322)(7,323)(8,324)(9,325)(10,326)(11,327)(12,328)(13,329)(14,330)(15,331)(16,332)(17,333)(18,334)(19,335)(20,336)(21,337)(22,338)(23,339)(24,340)(25,341)(26,342)(27,343)(28,344)(29,345)(30,346)(31,347)(32,348)(33,349)(34,350)(35,351)(36,352)(37,309)(38,310)(39,311)(40,312)(41,313)(42,314)(43,315)(44,316)(45,255)(46,256)(47,257)(48,258)(49,259)(50,260)(51,261)(52,262)(53,263)(54,264)(55,221)(56,222)(57,223)(58,224)(59,225)(60,226)(61,227)(62,228)(63,229)(64,230)(65,231)(66,232)(67,233)(68,234)(69,235)(70,236)(71,237)(72,238)(73,239)(74,240)(75,241)(76,242)(77,243)(78,244)(79,245)(80,246)(81,247)(82,248)(83,249)(84,250)(85,251)(86,252)(87,253)(88,254)(89,160)(90,161)(91,162)(92,163)(93,164)(94,165)(95,166)(96,167)(97,168)(98,169)(99,170)(100,171)(101,172)(102,173)(103,174)(104,175)(105,176)(106,133)(107,134)(108,135)(109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)(177,293)(178,294)(179,295)(180,296)(181,297)(182,298)(183,299)(184,300)(185,301)(186,302)(187,303)(188,304)(189,305)(190,306)(191,307)(192,308)(193,265)(194,266)(195,267)(196,268)(197,269)(198,270)(199,271)(200,272)(201,273)(202,274)(203,275)(204,276)(205,277)(206,278)(207,279)(208,280)(209,281)(210,282)(211,283)(212,284)(213,285)(214,286)(215,287)(216,288)(217,289)(218,290)(219,291)(220,292), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,85)(13,86)(14,87)(15,88)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(89,218)(90,219)(91,220)(92,177)(93,178)(94,179)(95,180)(96,181)(97,182)(98,183)(99,184)(100,185)(101,186)(102,187)(103,188)(104,189)(105,190)(106,191)(107,192)(108,193)(109,194)(110,195)(111,196)(112,197)(113,198)(114,199)(115,200)(116,201)(117,202)(118,203)(119,204)(120,205)(121,206)(122,207)(123,208)(124,209)(125,210)(126,211)(127,212)(128,213)(129,214)(130,215)(131,216)(132,217)(133,307)(134,308)(135,265)(136,266)(137,267)(138,268)(139,269)(140,270)(141,271)(142,272)(143,273)(144,274)(145,275)(146,276)(147,277)(148,278)(149,279)(150,280)(151,281)(152,282)(153,283)(154,284)(155,285)(156,286)(157,287)(158,288)(159,289)(160,290)(161,291)(162,292)(163,293)(164,294)(165,295)(166,296)(167,297)(168,298)(169,299)(170,300)(171,301)(172,302)(173,303)(174,304)(175,305)(176,306)(221,342)(222,343)(223,344)(224,345)(225,346)(226,347)(227,348)(228,349)(229,350)(230,351)(231,352)(232,309)(233,310)(234,311)(235,312)(236,313)(237,314)(238,315)(239,316)(240,317)(241,318)(242,319)(243,320)(244,321)(245,322)(246,323)(247,324)(248,325)(249,326)(250,327)(251,328)(252,329)(253,330)(254,331)(255,332)(256,333)(257,334)(258,335)(259,336)(260,337)(261,338)(262,339)(263,340)(264,341), (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)>;
G:=Group( (1,297)(2,298)(3,299)(4,300)(5,301)(6,302)(7,303)(8,304)(9,305)(10,306)(11,307)(12,308)(13,265)(14,266)(15,267)(16,268)(17,269)(18,270)(19,271)(20,272)(21,273)(22,274)(23,275)(24,276)(25,277)(26,278)(27,279)(28,280)(29,281)(30,282)(31,283)(32,284)(33,285)(34,286)(35,287)(36,288)(37,289)(38,290)(39,291)(40,292)(41,293)(42,294)(43,295)(44,296)(45,138)(46,139)(47,140)(48,141)(49,142)(50,143)(51,144)(52,145)(53,146)(54,147)(55,148)(56,149)(57,150)(58,151)(59,152)(60,153)(61,154)(62,155)(63,156)(64,157)(65,158)(66,159)(67,160)(68,161)(69,162)(70,163)(71,164)(72,165)(73,166)(74,167)(75,168)(76,169)(77,170)(78,171)(79,172)(80,173)(81,174)(82,175)(83,176)(84,133)(85,134)(86,135)(87,136)(88,137)(89,233)(90,234)(91,235)(92,236)(93,237)(94,238)(95,239)(96,240)(97,241)(98,242)(99,243)(100,244)(101,245)(102,246)(103,247)(104,248)(105,249)(106,250)(107,251)(108,252)(109,253)(110,254)(111,255)(112,256)(113,257)(114,258)(115,259)(116,260)(117,261)(118,262)(119,263)(120,264)(121,221)(122,222)(123,223)(124,224)(125,225)(126,226)(127,227)(128,228)(129,229)(130,230)(131,231)(132,232)(177,313)(178,314)(179,315)(180,316)(181,317)(182,318)(183,319)(184,320)(185,321)(186,322)(187,323)(188,324)(189,325)(190,326)(191,327)(192,328)(193,329)(194,330)(195,331)(196,332)(197,333)(198,334)(199,335)(200,336)(201,337)(202,338)(203,339)(204,340)(205,341)(206,342)(207,343)(208,344)(209,345)(210,346)(211,347)(212,348)(213,349)(214,350)(215,351)(216,352)(217,309)(218,310)(219,311)(220,312), (1,317)(2,318)(3,319)(4,320)(5,321)(6,322)(7,323)(8,324)(9,325)(10,326)(11,327)(12,328)(13,329)(14,330)(15,331)(16,332)(17,333)(18,334)(19,335)(20,336)(21,337)(22,338)(23,339)(24,340)(25,341)(26,342)(27,343)(28,344)(29,345)(30,346)(31,347)(32,348)(33,349)(34,350)(35,351)(36,352)(37,309)(38,310)(39,311)(40,312)(41,313)(42,314)(43,315)(44,316)(45,255)(46,256)(47,257)(48,258)(49,259)(50,260)(51,261)(52,262)(53,263)(54,264)(55,221)(56,222)(57,223)(58,224)(59,225)(60,226)(61,227)(62,228)(63,229)(64,230)(65,231)(66,232)(67,233)(68,234)(69,235)(70,236)(71,237)(72,238)(73,239)(74,240)(75,241)(76,242)(77,243)(78,244)(79,245)(80,246)(81,247)(82,248)(83,249)(84,250)(85,251)(86,252)(87,253)(88,254)(89,160)(90,161)(91,162)(92,163)(93,164)(94,165)(95,166)(96,167)(97,168)(98,169)(99,170)(100,171)(101,172)(102,173)(103,174)(104,175)(105,176)(106,133)(107,134)(108,135)(109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)(177,293)(178,294)(179,295)(180,296)(181,297)(182,298)(183,299)(184,300)(185,301)(186,302)(187,303)(188,304)(189,305)(190,306)(191,307)(192,308)(193,265)(194,266)(195,267)(196,268)(197,269)(198,270)(199,271)(200,272)(201,273)(202,274)(203,275)(204,276)(205,277)(206,278)(207,279)(208,280)(209,281)(210,282)(211,283)(212,284)(213,285)(214,286)(215,287)(216,288)(217,289)(218,290)(219,291)(220,292), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,85)(13,86)(14,87)(15,88)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(89,218)(90,219)(91,220)(92,177)(93,178)(94,179)(95,180)(96,181)(97,182)(98,183)(99,184)(100,185)(101,186)(102,187)(103,188)(104,189)(105,190)(106,191)(107,192)(108,193)(109,194)(110,195)(111,196)(112,197)(113,198)(114,199)(115,200)(116,201)(117,202)(118,203)(119,204)(120,205)(121,206)(122,207)(123,208)(124,209)(125,210)(126,211)(127,212)(128,213)(129,214)(130,215)(131,216)(132,217)(133,307)(134,308)(135,265)(136,266)(137,267)(138,268)(139,269)(140,270)(141,271)(142,272)(143,273)(144,274)(145,275)(146,276)(147,277)(148,278)(149,279)(150,280)(151,281)(152,282)(153,283)(154,284)(155,285)(156,286)(157,287)(158,288)(159,289)(160,290)(161,291)(162,292)(163,293)(164,294)(165,295)(166,296)(167,297)(168,298)(169,299)(170,300)(171,301)(172,302)(173,303)(174,304)(175,305)(176,306)(221,342)(222,343)(223,344)(224,345)(225,346)(226,347)(227,348)(228,349)(229,350)(230,351)(231,352)(232,309)(233,310)(234,311)(235,312)(236,313)(237,314)(238,315)(239,316)(240,317)(241,318)(242,319)(243,320)(244,321)(245,322)(246,323)(247,324)(248,325)(249,326)(250,327)(251,328)(252,329)(253,330)(254,331)(255,332)(256,333)(257,334)(258,335)(259,336)(260,337)(261,338)(262,339)(263,340)(264,341), (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) );
G=PermutationGroup([[(1,297),(2,298),(3,299),(4,300),(5,301),(6,302),(7,303),(8,304),(9,305),(10,306),(11,307),(12,308),(13,265),(14,266),(15,267),(16,268),(17,269),(18,270),(19,271),(20,272),(21,273),(22,274),(23,275),(24,276),(25,277),(26,278),(27,279),(28,280),(29,281),(30,282),(31,283),(32,284),(33,285),(34,286),(35,287),(36,288),(37,289),(38,290),(39,291),(40,292),(41,293),(42,294),(43,295),(44,296),(45,138),(46,139),(47,140),(48,141),(49,142),(50,143),(51,144),(52,145),(53,146),(54,147),(55,148),(56,149),(57,150),(58,151),(59,152),(60,153),(61,154),(62,155),(63,156),(64,157),(65,158),(66,159),(67,160),(68,161),(69,162),(70,163),(71,164),(72,165),(73,166),(74,167),(75,168),(76,169),(77,170),(78,171),(79,172),(80,173),(81,174),(82,175),(83,176),(84,133),(85,134),(86,135),(87,136),(88,137),(89,233),(90,234),(91,235),(92,236),(93,237),(94,238),(95,239),(96,240),(97,241),(98,242),(99,243),(100,244),(101,245),(102,246),(103,247),(104,248),(105,249),(106,250),(107,251),(108,252),(109,253),(110,254),(111,255),(112,256),(113,257),(114,258),(115,259),(116,260),(117,261),(118,262),(119,263),(120,264),(121,221),(122,222),(123,223),(124,224),(125,225),(126,226),(127,227),(128,228),(129,229),(130,230),(131,231),(132,232),(177,313),(178,314),(179,315),(180,316),(181,317),(182,318),(183,319),(184,320),(185,321),(186,322),(187,323),(188,324),(189,325),(190,326),(191,327),(192,328),(193,329),(194,330),(195,331),(196,332),(197,333),(198,334),(199,335),(200,336),(201,337),(202,338),(203,339),(204,340),(205,341),(206,342),(207,343),(208,344),(209,345),(210,346),(211,347),(212,348),(213,349),(214,350),(215,351),(216,352),(217,309),(218,310),(219,311),(220,312)], [(1,317),(2,318),(3,319),(4,320),(5,321),(6,322),(7,323),(8,324),(9,325),(10,326),(11,327),(12,328),(13,329),(14,330),(15,331),(16,332),(17,333),(18,334),(19,335),(20,336),(21,337),(22,338),(23,339),(24,340),(25,341),(26,342),(27,343),(28,344),(29,345),(30,346),(31,347),(32,348),(33,349),(34,350),(35,351),(36,352),(37,309),(38,310),(39,311),(40,312),(41,313),(42,314),(43,315),(44,316),(45,255),(46,256),(47,257),(48,258),(49,259),(50,260),(51,261),(52,262),(53,263),(54,264),(55,221),(56,222),(57,223),(58,224),(59,225),(60,226),(61,227),(62,228),(63,229),(64,230),(65,231),(66,232),(67,233),(68,234),(69,235),(70,236),(71,237),(72,238),(73,239),(74,240),(75,241),(76,242),(77,243),(78,244),(79,245),(80,246),(81,247),(82,248),(83,249),(84,250),(85,251),(86,252),(87,253),(88,254),(89,160),(90,161),(91,162),(92,163),(93,164),(94,165),(95,166),(96,167),(97,168),(98,169),(99,170),(100,171),(101,172),(102,173),(103,174),(104,175),(105,176),(106,133),(107,134),(108,135),(109,136),(110,137),(111,138),(112,139),(113,140),(114,141),(115,142),(116,143),(117,144),(118,145),(119,146),(120,147),(121,148),(122,149),(123,150),(124,151),(125,152),(126,153),(127,154),(128,155),(129,156),(130,157),(131,158),(132,159),(177,293),(178,294),(179,295),(180,296),(181,297),(182,298),(183,299),(184,300),(185,301),(186,302),(187,303),(188,304),(189,305),(190,306),(191,307),(192,308),(193,265),(194,266),(195,267),(196,268),(197,269),(198,270),(199,271),(200,272),(201,273),(202,274),(203,275),(204,276),(205,277),(206,278),(207,279),(208,280),(209,281),(210,282),(211,283),(212,284),(213,285),(214,286),(215,287),(216,288),(217,289),(218,290),(219,291),(220,292)], [(1,74),(2,75),(3,76),(4,77),(5,78),(6,79),(7,80),(8,81),(9,82),(10,83),(11,84),(12,85),(13,86),(14,87),(15,88),(16,45),(17,46),(18,47),(19,48),(20,49),(21,50),(22,51),(23,52),(24,53),(25,54),(26,55),(27,56),(28,57),(29,58),(30,59),(31,60),(32,61),(33,62),(34,63),(35,64),(36,65),(37,66),(38,67),(39,68),(40,69),(41,70),(42,71),(43,72),(44,73),(89,218),(90,219),(91,220),(92,177),(93,178),(94,179),(95,180),(96,181),(97,182),(98,183),(99,184),(100,185),(101,186),(102,187),(103,188),(104,189),(105,190),(106,191),(107,192),(108,193),(109,194),(110,195),(111,196),(112,197),(113,198),(114,199),(115,200),(116,201),(117,202),(118,203),(119,204),(120,205),(121,206),(122,207),(123,208),(124,209),(125,210),(126,211),(127,212),(128,213),(129,214),(130,215),(131,216),(132,217),(133,307),(134,308),(135,265),(136,266),(137,267),(138,268),(139,269),(140,270),(141,271),(142,272),(143,273),(144,274),(145,275),(146,276),(147,277),(148,278),(149,279),(150,280),(151,281),(152,282),(153,283),(154,284),(155,285),(156,286),(157,287),(158,288),(159,289),(160,290),(161,291),(162,292),(163,293),(164,294),(165,295),(166,296),(167,297),(168,298),(169,299),(170,300),(171,301),(172,302),(173,303),(174,304),(175,305),(176,306),(221,342),(222,343),(223,344),(224,345),(225,346),(226,347),(227,348),(228,349),(229,350),(230,351),(231,352),(232,309),(233,310),(234,311),(235,312),(236,313),(237,314),(238,315),(239,316),(240,317),(241,318),(242,319),(243,320),(244,321),(245,322),(246,323),(247,324),(248,325),(249,326),(250,327),(251,328),(252,329),(253,330),(254,331),(255,332),(256,333),(257,334),(258,335),(259,336),(260,337),(261,338),(262,339),(263,340),(264,341)], [(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)]])
352 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4P | 11A | ··· | 11J | 22A | ··· | 22ET | 44A | ··· | 44FD |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
352 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C11 | C22 | C22 | C44 |
| kernel | C23×C44 | C22×C44 | C23×C22 | C22×C22 | C23×C4 | C22×C4 | C24 | C23 |
| # reps | 1 | 14 | 1 | 16 | 10 | 140 | 10 | 160 |
Matrix representation of C23×C44 ►in GL4(𝔽89) generated by
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 1 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 88 |
| 81 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 10 |
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,88,0,0,0,0,88],[1,0,0,0,0,88,0,0,0,0,88,0,0,0,0,1],[1,0,0,0,0,88,0,0,0,0,1,0,0,0,0,88],[81,0,0,0,0,1,0,0,0,0,50,0,0,0,0,10] >;
C23×C44 in GAP, Magma, Sage, TeX
C_2^3\times C_{44} % in TeX
G:=Group("C2^3xC44"); // GroupNames label
G:=SmallGroup(352,188);
// by ID
G=gap.SmallGroup(352,188);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-11,-2,1056]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^2=d^44=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