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## G = C23×C40order 320 = 26·5

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

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23×C40
 Chief series C1 — C2 — C4 — C20 — C40 — C2×C40 — C22×C40 — C23×C40
 Lower central C1 — C23×C40
 Upper central 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, C4, C4, C22, C5, C8, C2×C4, C23, C10, C10, C2×C8, C22×C4, C24, C20, C20, C2×C10, C22×C8, C23×C4, C40, C2×C20, C22×C10, C23×C8, C2×C40, C22×C20, C23×C10, C22×C40, C23×C20, C23×C40
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, C23, C10, C2×C8, C22×C4, C24, C20, C2×C10, C22×C8, C23×C4, C40, C2×C20, C22×C10, C23×C8, C2×C40, C22×C20, C23×C10, C22×C40, C23×C20, C23×C40

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

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

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

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

320 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4P 5A 5B 5C 5D 8A ··· 8AF 10A ··· 10BH 20A ··· 20BL 40A ··· 40DX order 1 2 ··· 2 4 ··· 4 5 5 5 5 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 ··· 1 1 ··· 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

320 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C4 C5 C8 C10 C10 C20 C20 C40 kernel C23×C40 C22×C40 C23×C20 C22×C20 C23×C10 C23×C8 C22×C10 C22×C8 C23×C4 C22×C4 C24 C23 # reps 1 14 1 14 2 4 32 56 4 56 8 128

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

 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
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

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