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G = C23×C40order 320 = 26·5

Abelian group of type [2,2,2,40]

direct product, abelian, monomial, 2-elementary

Aliases: C23×C40, SmallGroup(320,1567)

Series: Derived Chief Lower central Upper central

C1 — C23×C40
C1C2C4C20C40C2×C40C22×C40 — C23×C40
C1 — C23×C40
C1 — C23×C40

Generators and relations for C23×C40
 G = < a,b,c,d | a2=b2=c2=d40=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 338, all normal (12 characteristic)
C1, C2, C2 [×14], C4, C4 [×7], C22 [×35], C5, C8 [×8], C2×C4 [×28], C23 [×15], C10, C10 [×14], C2×C8 [×28], C22×C4 [×14], C24, C20, C20 [×7], C2×C10 [×35], C22×C8 [×14], C23×C4, C40 [×8], C2×C20 [×28], C22×C10 [×15], C23×C8, C2×C40 [×28], C22×C20 [×14], C23×C10, C22×C40 [×14], C23×C20, C23×C40
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C5, C8 [×8], C2×C4 [×28], C23 [×15], C10 [×15], C2×C8 [×28], C22×C4 [×14], C24, C20 [×8], C2×C10 [×35], C22×C8 [×14], C23×C4, C40 [×8], C2×C20 [×28], C22×C10 [×15], C23×C8, C2×C40 [×28], C22×C20 [×14], C23×C10, C22×C40 [×14], C23×C20, C23×C40

Smallest permutation representation of C23×C40
Regular action on 320 points
Generators in S320
(1 214)(2 215)(3 216)(4 217)(5 218)(6 219)(7 220)(8 221)(9 222)(10 223)(11 224)(12 225)(13 226)(14 227)(15 228)(16 229)(17 230)(18 231)(19 232)(20 233)(21 234)(22 235)(23 236)(24 237)(25 238)(26 239)(27 240)(28 201)(29 202)(30 203)(31 204)(32 205)(33 206)(34 207)(35 208)(36 209)(37 210)(38 211)(39 212)(40 213)(41 142)(42 143)(43 144)(44 145)(45 146)(46 147)(47 148)(48 149)(49 150)(50 151)(51 152)(52 153)(53 154)(54 155)(55 156)(56 157)(57 158)(58 159)(59 160)(60 121)(61 122)(62 123)(63 124)(64 125)(65 126)(66 127)(67 128)(68 129)(69 130)(70 131)(71 132)(72 133)(73 134)(74 135)(75 136)(76 137)(77 138)(78 139)(79 140)(80 141)(81 299)(82 300)(83 301)(84 302)(85 303)(86 304)(87 305)(88 306)(89 307)(90 308)(91 309)(92 310)(93 311)(94 312)(95 313)(96 314)(97 315)(98 316)(99 317)(100 318)(101 319)(102 320)(103 281)(104 282)(105 283)(106 284)(107 285)(108 286)(109 287)(110 288)(111 289)(112 290)(113 291)(114 292)(115 293)(116 294)(117 295)(118 296)(119 297)(120 298)(161 250)(162 251)(163 252)(164 253)(165 254)(166 255)(167 256)(168 257)(169 258)(170 259)(171 260)(172 261)(173 262)(174 263)(175 264)(176 265)(177 266)(178 267)(179 268)(180 269)(181 270)(182 271)(183 272)(184 273)(185 274)(186 275)(187 276)(188 277)(189 278)(190 279)(191 280)(192 241)(193 242)(194 243)(195 244)(196 245)(197 246)(198 247)(199 248)(200 249)
(1 193)(2 194)(3 195)(4 196)(5 197)(6 198)(7 199)(8 200)(9 161)(10 162)(11 163)(12 164)(13 165)(14 166)(15 167)(16 168)(17 169)(18 170)(19 171)(20 172)(21 173)(22 174)(23 175)(24 176)(25 177)(26 178)(27 179)(28 180)(29 181)(30 182)(31 183)(32 184)(33 185)(34 186)(35 187)(36 188)(37 189)(38 190)(39 191)(40 192)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 81)(72 82)(73 83)(74 84)(75 85)(76 86)(77 87)(78 88)(79 89)(80 90)(121 288)(122 289)(123 290)(124 291)(125 292)(126 293)(127 294)(128 295)(129 296)(130 297)(131 298)(132 299)(133 300)(134 301)(135 302)(136 303)(137 304)(138 305)(139 306)(140 307)(141 308)(142 309)(143 310)(144 311)(145 312)(146 313)(147 314)(148 315)(149 316)(150 317)(151 318)(152 319)(153 320)(154 281)(155 282)(156 283)(157 284)(158 285)(159 286)(160 287)(201 269)(202 270)(203 271)(204 272)(205 273)(206 274)(207 275)(208 276)(209 277)(210 278)(211 279)(212 280)(213 241)(214 242)(215 243)(216 244)(217 245)(218 246)(219 247)(220 248)(221 249)(222 250)(223 251)(224 252)(225 253)(226 254)(227 255)(228 256)(229 257)(230 258)(231 259)(232 260)(233 261)(234 262)(235 263)(236 264)(237 265)(238 266)(239 267)(240 268)
(1 315)(2 316)(3 317)(4 318)(5 319)(6 320)(7 281)(8 282)(9 283)(10 284)(11 285)(12 286)(13 287)(14 288)(15 289)(16 290)(17 291)(18 292)(19 293)(20 294)(21 295)(22 296)(23 297)(24 298)(25 299)(26 300)(27 301)(28 302)(29 303)(30 304)(31 305)(32 306)(33 307)(34 308)(35 309)(36 310)(37 311)(38 312)(39 313)(40 314)(41 276)(42 277)(43 278)(44 279)(45 280)(46 241)(47 242)(48 243)(49 244)(50 245)(51 246)(52 247)(53 248)(54 249)(55 250)(56 251)(57 252)(58 253)(59 254)(60 255)(61 256)(62 257)(63 258)(64 259)(65 260)(66 261)(67 262)(68 263)(69 264)(70 265)(71 266)(72 267)(73 268)(74 269)(75 270)(76 271)(77 272)(78 273)(79 274)(80 275)(81 238)(82 239)(83 240)(84 201)(85 202)(86 203)(87 204)(88 205)(89 206)(90 207)(91 208)(92 209)(93 210)(94 211)(95 212)(96 213)(97 214)(98 215)(99 216)(100 217)(101 218)(102 219)(103 220)(104 221)(105 222)(106 223)(107 224)(108 225)(109 226)(110 227)(111 228)(112 229)(113 230)(114 231)(115 232)(116 233)(117 234)(118 235)(119 236)(120 237)(121 166)(122 167)(123 168)(124 169)(125 170)(126 171)(127 172)(128 173)(129 174)(130 175)(131 176)(132 177)(133 178)(134 179)(135 180)(136 181)(137 182)(138 183)(139 184)(140 185)(141 186)(142 187)(143 188)(144 189)(145 190)(146 191)(147 192)(148 193)(149 194)(150 195)(151 196)(152 197)(153 198)(154 199)(155 200)(156 161)(157 162)(158 163)(159 164)(160 165)
(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)

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

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

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

320 conjugacy classes

class 1 2A···2O4A···4P5A5B5C5D8A···8AF10A···10BH20A···20BL40A···40DX
order12···24···455558···810···1020···2040···40
size11···11···111111···11···11···11···1

320 irreducible representations

dim111111111111
type+++
imageC1C2C2C4C4C5C8C10C10C20C20C40
kernelC23×C40C22×C40C23×C20C22×C20C23×C10C23×C8C22×C10C22×C8C23×C4C22×C4C24C23
# reps1141142432564568128

Matrix representation of C23×C40 in GL4(𝔽41) generated by

1000
04000
0010
0001
,
40000
0100
0010
0001
,
40000
0100
00400
0001
,
1000
01600
00310
00029
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,31,0,0,0,0,29] >;

C23×C40 in GAP, Magma, Sage, TeX

C_2^3\times C_{40}
% in TeX

G:=Group("C2^3xC40");
// GroupNames label

G:=SmallGroup(320,1567);
// by ID

G=gap.SmallGroup(320,1567);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^2=d^40=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