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## G = C24×C20order 320 = 26·5

### Abelian group of type [2,2,2,2,20]

Aliases: C24×C20, SmallGroup(320,1628)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24×C20
 Chief series C1 — C2 — C10 — C20 — C2×C20 — C22×C20 — C23×C20 — C24×C20
 Lower central C1 — C24×C20
 Upper central C1 — C24×C20

Generators and relations for C24×C20
G = < a,b,c,d,e | a2=b2=c2=d2=e20=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 1362, all normal (8 characteristic)
C1, C2, C2 [×30], C4 [×16], C22 [×155], C5, C2×C4 [×120], C23 [×155], C10, C10 [×30], C22×C4 [×140], C24 [×31], C20 [×16], C2×C10 [×155], C23×C4 [×30], C25, C2×C20 [×120], C22×C10 [×155], C24×C4, C22×C20 [×140], C23×C10 [×31], C23×C20 [×30], C24×C10, C24×C20
Quotients: C1, C2 [×31], C4 [×16], C22 [×155], C5, C2×C4 [×120], C23 [×155], C10 [×31], C22×C4 [×140], C24 [×31], C20 [×16], C2×C10 [×155], C23×C4 [×30], C25, C2×C20 [×120], C22×C10 [×155], C24×C4, C22×C20 [×140], C23×C10 [×31], C23×C20 [×30], C24×C10, C24×C20

Smallest permutation representation of C24×C20
Regular action on 320 points
Generators in S320
(1 169)(2 170)(3 171)(4 172)(5 173)(6 174)(7 175)(8 176)(9 177)(10 178)(11 179)(12 180)(13 161)(14 162)(15 163)(16 164)(17 165)(18 166)(19 167)(20 168)(21 135)(22 136)(23 137)(24 138)(25 139)(26 140)(27 121)(28 122)(29 123)(30 124)(31 125)(32 126)(33 127)(34 128)(35 129)(36 130)(37 131)(38 132)(39 133)(40 134)(41 118)(42 119)(43 120)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)(59 116)(60 117)(61 143)(62 144)(63 145)(64 146)(65 147)(66 148)(67 149)(68 150)(69 151)(70 152)(71 153)(72 154)(73 155)(74 156)(75 157)(76 158)(77 159)(78 160)(79 141)(80 142)(81 245)(82 246)(83 247)(84 248)(85 249)(86 250)(87 251)(88 252)(89 253)(90 254)(91 255)(92 256)(93 257)(94 258)(95 259)(96 260)(97 241)(98 242)(99 243)(100 244)(181 230)(182 231)(183 232)(184 233)(185 234)(186 235)(187 236)(188 237)(189 238)(190 239)(191 240)(192 221)(193 222)(194 223)(195 224)(196 225)(197 226)(198 227)(199 228)(200 229)(201 310)(202 311)(203 312)(204 313)(205 314)(206 315)(207 316)(208 317)(209 318)(210 319)(211 320)(212 301)(213 302)(214 303)(215 304)(216 305)(217 306)(218 307)(219 308)(220 309)(261 296)(262 297)(263 298)(264 299)(265 300)(266 281)(267 282)(268 283)(269 284)(270 285)(271 286)(272 287)(273 288)(274 289)(275 290)(276 291)(277 292)(278 293)(279 294)(280 295)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 41)(19 42)(20 43)(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(81 221)(82 222)(83 223)(84 224)(85 225)(86 226)(87 227)(88 228)(89 229)(90 230)(91 231)(92 232)(93 233)(94 234)(95 235)(96 236)(97 237)(98 238)(99 239)(100 240)(101 169)(102 170)(103 171)(104 172)(105 173)(106 174)(107 175)(108 176)(109 177)(110 178)(111 179)(112 180)(113 161)(114 162)(115 163)(116 164)(117 165)(118 166)(119 167)(120 168)(121 156)(122 157)(123 158)(124 159)(125 160)(126 141)(127 142)(128 143)(129 144)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(136 151)(137 152)(138 153)(139 154)(140 155)(181 254)(182 255)(183 256)(184 257)(185 258)(186 259)(187 260)(188 241)(189 242)(190 243)(191 244)(192 245)(193 246)(194 247)(195 248)(196 249)(197 250)(198 251)(199 252)(200 253)(201 300)(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)(261 306)(262 307)(263 308)(264 309)(265 310)(266 311)(267 312)(268 313)(269 314)(270 315)(271 316)(272 317)(273 318)(274 319)(275 320)(276 301)(277 302)(278 303)(279 304)(280 305)
(1 270)(2 271)(3 272)(4 273)(5 274)(6 275)(7 276)(8 277)(9 278)(10 279)(11 280)(12 261)(13 262)(14 263)(15 264)(16 265)(17 266)(18 267)(19 268)(20 269)(21 185)(22 186)(23 187)(24 188)(25 189)(26 190)(27 191)(28 192)(29 193)(30 194)(31 195)(32 196)(33 197)(34 198)(35 199)(36 200)(37 181)(38 182)(39 183)(40 184)(41 312)(42 313)(43 314)(44 315)(45 316)(46 317)(47 318)(48 319)(49 320)(50 301)(51 302)(52 303)(53 304)(54 305)(55 306)(56 307)(57 308)(58 309)(59 310)(60 311)(61 251)(62 252)(63 253)(64 254)(65 255)(66 256)(67 257)(68 258)(69 259)(70 260)(71 241)(72 242)(73 243)(74 244)(75 245)(76 246)(77 247)(78 248)(79 249)(80 250)(81 157)(82 158)(83 159)(84 160)(85 141)(86 142)(87 143)(88 144)(89 145)(90 146)(91 147)(92 148)(93 149)(94 150)(95 151)(96 152)(97 153)(98 154)(99 155)(100 156)(101 206)(102 207)(103 208)(104 209)(105 210)(106 211)(107 212)(108 213)(109 214)(110 215)(111 216)(112 217)(113 218)(114 219)(115 220)(116 201)(117 202)(118 203)(119 204)(120 205)(121 240)(122 221)(123 222)(124 223)(125 224)(126 225)(127 226)(128 227)(129 228)(130 229)(131 230)(132 231)(133 232)(134 233)(135 234)(136 235)(137 236)(138 237)(139 238)(140 239)(161 297)(162 298)(163 299)(164 300)(165 281)(166 282)(167 283)(168 284)(169 285)(170 286)(171 287)(172 288)(173 289)(174 290)(175 291)(176 292)(177 293)(178 294)(179 295)(180 296)
(1 137)(2 138)(3 139)(4 140)(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)(21 167)(22 168)(23 169)(24 170)(25 171)(26 172)(27 173)(28 174)(29 175)(30 176)(31 177)(32 178)(33 179)(34 180)(35 161)(36 162)(37 163)(38 164)(39 165)(40 166)(41 149)(42 150)(43 151)(44 152)(45 153)(46 154)(47 155)(48 156)(49 157)(50 158)(51 159)(52 160)(53 141)(54 142)(55 143)(56 144)(57 145)(58 146)(59 147)(60 148)(61 112)(62 113)(63 114)(64 115)(65 116)(66 117)(67 118)(68 119)(69 120)(70 101)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(81 320)(82 301)(83 302)(84 303)(85 304)(86 305)(87 306)(88 307)(89 308)(90 309)(91 310)(92 311)(93 312)(94 313)(95 314)(96 315)(97 316)(98 317)(99 318)(100 319)(181 299)(182 300)(183 281)(184 282)(185 283)(186 284)(187 285)(188 286)(189 287)(190 288)(191 289)(192 290)(193 291)(194 292)(195 293)(196 294)(197 295)(198 296)(199 297)(200 298)(201 255)(202 256)(203 257)(204 258)(205 259)(206 260)(207 241)(208 242)(209 243)(210 244)(211 245)(212 246)(213 247)(214 248)(215 249)(216 250)(217 251)(218 252)(219 253)(220 254)(221 275)(222 276)(223 277)(224 278)(225 279)(226 280)(227 261)(228 262)(229 263)(230 264)(231 265)(232 266)(233 267)(234 268)(235 269)(236 270)(237 271)(238 272)(239 273)(240 274)
(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)| (1,169)(2,170)(3,171)(4,172)(5,173)(6,174)(7,175)(8,176)(9,177)(10,178)(11,179)(12,180)(13,161)(14,162)(15,163)(16,164)(17,165)(18,166)(19,167)(20,168)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,127)(34,128)(35,129)(36,130)(37,131)(38,132)(39,133)(40,134)(41,118)(42,119)(43,120)(44,101)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,153)(72,154)(73,155)(74,156)(75,157)(76,158)(77,159)(78,160)(79,141)(80,142)(81,245)(82,246)(83,247)(84,248)(85,249)(86,250)(87,251)(88,252)(89,253)(90,254)(91,255)(92,256)(93,257)(94,258)(95,259)(96,260)(97,241)(98,242)(99,243)(100,244)(181,230)(182,231)(183,232)(184,233)(185,234)(186,235)(187,236)(188,237)(189,238)(190,239)(191,240)(192,221)(193,222)(194,223)(195,224)(196,225)(197,226)(198,227)(199,228)(200,229)(201,310)(202,311)(203,312)(204,313)(205,314)(206,315)(207,316)(208,317)(209,318)(210,319)(211,320)(212,301)(213,302)(214,303)(215,304)(216,305)(217,306)(218,307)(219,308)(220,309)(261,296)(262,297)(263,298)(264,299)(265,300)(266,281)(267,282)(268,283)(269,284)(270,285)(271,286)(272,287)(273,288)(274,289)(275,290)(276,291)(277,292)(278,293)(279,294)(280,295), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,41)(19,42)(20,43)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(81,221)(82,222)(83,223)(84,224)(85,225)(86,226)(87,227)(88,228)(89,229)(90,230)(91,231)(92,232)(93,233)(94,234)(95,235)(96,236)(97,237)(98,238)(99,239)(100,240)(101,169)(102,170)(103,171)(104,172)(105,173)(106,174)(107,175)(108,176)(109,177)(110,178)(111,179)(112,180)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,156)(122,157)(123,158)(124,159)(125,160)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155)(181,254)(182,255)(183,256)(184,257)(185,258)(186,259)(187,260)(188,241)(189,242)(190,243)(191,244)(192,245)(193,246)(194,247)(195,248)(196,249)(197,250)(198,251)(199,252)(200,253)(201,300)(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)(261,306)(262,307)(263,308)(264,309)(265,310)(266,311)(267,312)(268,313)(269,314)(270,315)(271,316)(272,317)(273,318)(274,319)(275,320)(276,301)(277,302)(278,303)(279,304)(280,305), (1,270)(2,271)(3,272)(4,273)(5,274)(6,275)(7,276)(8,277)(9,278)(10,279)(11,280)(12,261)(13,262)(14,263)(15,264)(16,265)(17,266)(18,267)(19,268)(20,269)(21,185)(22,186)(23,187)(24,188)(25,189)(26,190)(27,191)(28,192)(29,193)(30,194)(31,195)(32,196)(33,197)(34,198)(35,199)(36,200)(37,181)(38,182)(39,183)(40,184)(41,312)(42,313)(43,314)(44,315)(45,316)(46,317)(47,318)(48,319)(49,320)(50,301)(51,302)(52,303)(53,304)(54,305)(55,306)(56,307)(57,308)(58,309)(59,310)(60,311)(61,251)(62,252)(63,253)(64,254)(65,255)(66,256)(67,257)(68,258)(69,259)(70,260)(71,241)(72,242)(73,243)(74,244)(75,245)(76,246)(77,247)(78,248)(79,249)(80,250)(81,157)(82,158)(83,159)(84,160)(85,141)(86,142)(87,143)(88,144)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,153)(98,154)(99,155)(100,156)(101,206)(102,207)(103,208)(104,209)(105,210)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217)(113,218)(114,219)(115,220)(116,201)(117,202)(118,203)(119,204)(120,205)(121,240)(122,221)(123,222)(124,223)(125,224)(126,225)(127,226)(128,227)(129,228)(130,229)(131,230)(132,231)(133,232)(134,233)(135,234)(136,235)(137,236)(138,237)(139,238)(140,239)(161,297)(162,298)(163,299)(164,300)(165,281)(166,282)(167,283)(168,284)(169,285)(170,286)(171,287)(172,288)(173,289)(174,290)(175,291)(176,292)(177,293)(178,294)(179,295)(180,296), (1,137)(2,138)(3,139)(4,140)(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)(21,167)(22,168)(23,169)(24,170)(25,171)(26,172)(27,173)(28,174)(29,175)(30,176)(31,177)(32,178)(33,179)(34,180)(35,161)(36,162)(37,163)(38,164)(39,165)(40,166)(41,149)(42,150)(43,151)(44,152)(45,153)(46,154)(47,155)(48,156)(49,157)(50,158)(51,159)(52,160)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,320)(82,301)(83,302)(84,303)(85,304)(86,305)(87,306)(88,307)(89,308)(90,309)(91,310)(92,311)(93,312)(94,313)(95,314)(96,315)(97,316)(98,317)(99,318)(100,319)(181,299)(182,300)(183,281)(184,282)(185,283)(186,284)(187,285)(188,286)(189,287)(190,288)(191,289)(192,290)(193,291)(194,292)(195,293)(196,294)(197,295)(198,296)(199,297)(200,298)(201,255)(202,256)(203,257)(204,258)(205,259)(206,260)(207,241)(208,242)(209,243)(210,244)(211,245)(212,246)(213,247)(214,248)(215,249)(216,250)(217,251)(218,252)(219,253)(220,254)(221,275)(222,276)(223,277)(224,278)(225,279)(226,280)(227,261)(228,262)(229,263)(230,264)(231,265)(232,266)(233,267)(234,268)(235,269)(236,270)(237,271)(238,272)(239,273)(240,274), (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( (1,169)(2,170)(3,171)(4,172)(5,173)(6,174)(7,175)(8,176)(9,177)(10,178)(11,179)(12,180)(13,161)(14,162)(15,163)(16,164)(17,165)(18,166)(19,167)(20,168)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,127)(34,128)(35,129)(36,130)(37,131)(38,132)(39,133)(40,134)(41,118)(42,119)(43,120)(44,101)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,153)(72,154)(73,155)(74,156)(75,157)(76,158)(77,159)(78,160)(79,141)(80,142)(81,245)(82,246)(83,247)(84,248)(85,249)(86,250)(87,251)(88,252)(89,253)(90,254)(91,255)(92,256)(93,257)(94,258)(95,259)(96,260)(97,241)(98,242)(99,243)(100,244)(181,230)(182,231)(183,232)(184,233)(185,234)(186,235)(187,236)(188,237)(189,238)(190,239)(191,240)(192,221)(193,222)(194,223)(195,224)(196,225)(197,226)(198,227)(199,228)(200,229)(201,310)(202,311)(203,312)(204,313)(205,314)(206,315)(207,316)(208,317)(209,318)(210,319)(211,320)(212,301)(213,302)(214,303)(215,304)(216,305)(217,306)(218,307)(219,308)(220,309)(261,296)(262,297)(263,298)(264,299)(265,300)(266,281)(267,282)(268,283)(269,284)(270,285)(271,286)(272,287)(273,288)(274,289)(275,290)(276,291)(277,292)(278,293)(279,294)(280,295), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,41)(19,42)(20,43)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(81,221)(82,222)(83,223)(84,224)(85,225)(86,226)(87,227)(88,228)(89,229)(90,230)(91,231)(92,232)(93,233)(94,234)(95,235)(96,236)(97,237)(98,238)(99,239)(100,240)(101,169)(102,170)(103,171)(104,172)(105,173)(106,174)(107,175)(108,176)(109,177)(110,178)(111,179)(112,180)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,156)(122,157)(123,158)(124,159)(125,160)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155)(181,254)(182,255)(183,256)(184,257)(185,258)(186,259)(187,260)(188,241)(189,242)(190,243)(191,244)(192,245)(193,246)(194,247)(195,248)(196,249)(197,250)(198,251)(199,252)(200,253)(201,300)(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)(261,306)(262,307)(263,308)(264,309)(265,310)(266,311)(267,312)(268,313)(269,314)(270,315)(271,316)(272,317)(273,318)(274,319)(275,320)(276,301)(277,302)(278,303)(279,304)(280,305), (1,270)(2,271)(3,272)(4,273)(5,274)(6,275)(7,276)(8,277)(9,278)(10,279)(11,280)(12,261)(13,262)(14,263)(15,264)(16,265)(17,266)(18,267)(19,268)(20,269)(21,185)(22,186)(23,187)(24,188)(25,189)(26,190)(27,191)(28,192)(29,193)(30,194)(31,195)(32,196)(33,197)(34,198)(35,199)(36,200)(37,181)(38,182)(39,183)(40,184)(41,312)(42,313)(43,314)(44,315)(45,316)(46,317)(47,318)(48,319)(49,320)(50,301)(51,302)(52,303)(53,304)(54,305)(55,306)(56,307)(57,308)(58,309)(59,310)(60,311)(61,251)(62,252)(63,253)(64,254)(65,255)(66,256)(67,257)(68,258)(69,259)(70,260)(71,241)(72,242)(73,243)(74,244)(75,245)(76,246)(77,247)(78,248)(79,249)(80,250)(81,157)(82,158)(83,159)(84,160)(85,141)(86,142)(87,143)(88,144)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,153)(98,154)(99,155)(100,156)(101,206)(102,207)(103,208)(104,209)(105,210)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217)(113,218)(114,219)(115,220)(116,201)(117,202)(118,203)(119,204)(120,205)(121,240)(122,221)(123,222)(124,223)(125,224)(126,225)(127,226)(128,227)(129,228)(130,229)(131,230)(132,231)(133,232)(134,233)(135,234)(136,235)(137,236)(138,237)(139,238)(140,239)(161,297)(162,298)(163,299)(164,300)(165,281)(166,282)(167,283)(168,284)(169,285)(170,286)(171,287)(172,288)(173,289)(174,290)(175,291)(176,292)(177,293)(178,294)(179,295)(180,296), (1,137)(2,138)(3,139)(4,140)(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)(21,167)(22,168)(23,169)(24,170)(25,171)(26,172)(27,173)(28,174)(29,175)(30,176)(31,177)(32,178)(33,179)(34,180)(35,161)(36,162)(37,163)(38,164)(39,165)(40,166)(41,149)(42,150)(43,151)(44,152)(45,153)(46,154)(47,155)(48,156)(49,157)(50,158)(51,159)(52,160)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,320)(82,301)(83,302)(84,303)(85,304)(86,305)(87,306)(88,307)(89,308)(90,309)(91,310)(92,311)(93,312)(94,313)(95,314)(96,315)(97,316)(98,317)(99,318)(100,319)(181,299)(182,300)(183,281)(184,282)(185,283)(186,284)(187,285)(188,286)(189,287)(190,288)(191,289)(192,290)(193,291)(194,292)(195,293)(196,294)(197,295)(198,296)(199,297)(200,298)(201,255)(202,256)(203,257)(204,258)(205,259)(206,260)(207,241)(208,242)(209,243)(210,244)(211,245)(212,246)(213,247)(214,248)(215,249)(216,250)(217,251)(218,252)(219,253)(220,254)(221,275)(222,276)(223,277)(224,278)(225,279)(226,280)(227,261)(228,262)(229,263)(230,264)(231,265)(232,266)(233,267)(234,268)(235,269)(236,270)(237,271)(238,272)(239,273)(240,274), (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([(1,169),(2,170),(3,171),(4,172),(5,173),(6,174),(7,175),(8,176),(9,177),(10,178),(11,179),(12,180),(13,161),(14,162),(15,163),(16,164),(17,165),(18,166),(19,167),(20,168),(21,135),(22,136),(23,137),(24,138),(25,139),(26,140),(27,121),(28,122),(29,123),(30,124),(31,125),(32,126),(33,127),(34,128),(35,129),(36,130),(37,131),(38,132),(39,133),(40,134),(41,118),(42,119),(43,120),(44,101),(45,102),(46,103),(47,104),(48,105),(49,106),(50,107),(51,108),(52,109),(53,110),(54,111),(55,112),(56,113),(57,114),(58,115),(59,116),(60,117),(61,143),(62,144),(63,145),(64,146),(65,147),(66,148),(67,149),(68,150),(69,151),(70,152),(71,153),(72,154),(73,155),(74,156),(75,157),(76,158),(77,159),(78,160),(79,141),(80,142),(81,245),(82,246),(83,247),(84,248),(85,249),(86,250),(87,251),(88,252),(89,253),(90,254),(91,255),(92,256),(93,257),(94,258),(95,259),(96,260),(97,241),(98,242),(99,243),(100,244),(181,230),(182,231),(183,232),(184,233),(185,234),(186,235),(187,236),(188,237),(189,238),(190,239),(191,240),(192,221),(193,222),(194,223),(195,224),(196,225),(197,226),(198,227),(199,228),(200,229),(201,310),(202,311),(203,312),(204,313),(205,314),(206,315),(207,316),(208,317),(209,318),(210,319),(211,320),(212,301),(213,302),(214,303),(215,304),(216,305),(217,306),(218,307),(219,308),(220,309),(261,296),(262,297),(263,298),(264,299),(265,300),(266,281),(267,282),(268,283),(269,284),(270,285),(271,286),(272,287),(273,288),(274,289),(275,290),(276,291),(277,292),(278,293),(279,294),(280,295)], [(1,44),(2,45),(3,46),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,56),(14,57),(15,58),(16,59),(17,60),(18,41),(19,42),(20,43),(21,68),(22,69),(23,70),(24,71),(25,72),(26,73),(27,74),(28,75),(29,76),(30,77),(31,78),(32,79),(33,80),(34,61),(35,62),(36,63),(37,64),(38,65),(39,66),(40,67),(81,221),(82,222),(83,223),(84,224),(85,225),(86,226),(87,227),(88,228),(89,229),(90,230),(91,231),(92,232),(93,233),(94,234),(95,235),(96,236),(97,237),(98,238),(99,239),(100,240),(101,169),(102,170),(103,171),(104,172),(105,173),(106,174),(107,175),(108,176),(109,177),(110,178),(111,179),(112,180),(113,161),(114,162),(115,163),(116,164),(117,165),(118,166),(119,167),(120,168),(121,156),(122,157),(123,158),(124,159),(125,160),(126,141),(127,142),(128,143),(129,144),(130,145),(131,146),(132,147),(133,148),(134,149),(135,150),(136,151),(137,152),(138,153),(139,154),(140,155),(181,254),(182,255),(183,256),(184,257),(185,258),(186,259),(187,260),(188,241),(189,242),(190,243),(191,244),(192,245),(193,246),(194,247),(195,248),(196,249),(197,250),(198,251),(199,252),(200,253),(201,300),(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),(261,306),(262,307),(263,308),(264,309),(265,310),(266,311),(267,312),(268,313),(269,314),(270,315),(271,316),(272,317),(273,318),(274,319),(275,320),(276,301),(277,302),(278,303),(279,304),(280,305)], [(1,270),(2,271),(3,272),(4,273),(5,274),(6,275),(7,276),(8,277),(9,278),(10,279),(11,280),(12,261),(13,262),(14,263),(15,264),(16,265),(17,266),(18,267),(19,268),(20,269),(21,185),(22,186),(23,187),(24,188),(25,189),(26,190),(27,191),(28,192),(29,193),(30,194),(31,195),(32,196),(33,197),(34,198),(35,199),(36,200),(37,181),(38,182),(39,183),(40,184),(41,312),(42,313),(43,314),(44,315),(45,316),(46,317),(47,318),(48,319),(49,320),(50,301),(51,302),(52,303),(53,304),(54,305),(55,306),(56,307),(57,308),(58,309),(59,310),(60,311),(61,251),(62,252),(63,253),(64,254),(65,255),(66,256),(67,257),(68,258),(69,259),(70,260),(71,241),(72,242),(73,243),(74,244),(75,245),(76,246),(77,247),(78,248),(79,249),(80,250),(81,157),(82,158),(83,159),(84,160),(85,141),(86,142),(87,143),(88,144),(89,145),(90,146),(91,147),(92,148),(93,149),(94,150),(95,151),(96,152),(97,153),(98,154),(99,155),(100,156),(101,206),(102,207),(103,208),(104,209),(105,210),(106,211),(107,212),(108,213),(109,214),(110,215),(111,216),(112,217),(113,218),(114,219),(115,220),(116,201),(117,202),(118,203),(119,204),(120,205),(121,240),(122,221),(123,222),(124,223),(125,224),(126,225),(127,226),(128,227),(129,228),(130,229),(131,230),(132,231),(133,232),(134,233),(135,234),(136,235),(137,236),(138,237),(139,238),(140,239),(161,297),(162,298),(163,299),(164,300),(165,281),(166,282),(167,283),(168,284),(169,285),(170,286),(171,287),(172,288),(173,289),(174,290),(175,291),(176,292),(177,293),(178,294),(179,295),(180,296)], [(1,137),(2,138),(3,139),(4,140),(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),(21,167),(22,168),(23,169),(24,170),(25,171),(26,172),(27,173),(28,174),(29,175),(30,176),(31,177),(32,178),(33,179),(34,180),(35,161),(36,162),(37,163),(38,164),(39,165),(40,166),(41,149),(42,150),(43,151),(44,152),(45,153),(46,154),(47,155),(48,156),(49,157),(50,158),(51,159),(52,160),(53,141),(54,142),(55,143),(56,144),(57,145),(58,146),(59,147),(60,148),(61,112),(62,113),(63,114),(64,115),(65,116),(66,117),(67,118),(68,119),(69,120),(70,101),(71,102),(72,103),(73,104),(74,105),(75,106),(76,107),(77,108),(78,109),(79,110),(80,111),(81,320),(82,301),(83,302),(84,303),(85,304),(86,305),(87,306),(88,307),(89,308),(90,309),(91,310),(92,311),(93,312),(94,313),(95,314),(96,315),(97,316),(98,317),(99,318),(100,319),(181,299),(182,300),(183,281),(184,282),(185,283),(186,284),(187,285),(188,286),(189,287),(190,288),(191,289),(192,290),(193,291),(194,292),(195,293),(196,294),(197,295),(198,296),(199,297),(200,298),(201,255),(202,256),(203,257),(204,258),(205,259),(206,260),(207,241),(208,242),(209,243),(210,244),(211,245),(212,246),(213,247),(214,248),(215,249),(216,250),(217,251),(218,252),(219,253),(220,254),(221,275),(222,276),(223,277),(224,278),(225,279),(226,280),(227,261),(228,262),(229,263),(230,264),(231,265),(232,266),(233,267),(234,268),(235,269),(236,270),(237,271),(238,272),(239,273),(240,274)], [(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 ··· 2AE 4A ··· 4AF 5A 5B 5C 5D 10A ··· 10DT 20A ··· 20DX order 1 2 ··· 2 4 ··· 4 5 5 5 5 10 ··· 10 20 ··· 20 size 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 type + + + image C1 C2 C2 C4 C5 C10 C10 C20 kernel C24×C20 C23×C20 C24×C10 C23×C10 C24×C4 C23×C4 C25 C24 # reps 1 30 1 32 4 120 4 128

Matrix representation of C24×C20 in GL5(𝔽41)

 40 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 40
,
 10 0 0 0 0 0 5 0 0 0 0 0 1 0 0 0 0 0 31 0 0 0 0 0 23

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[10,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,23] >;

C24×C20 in GAP, Magma, Sage, TeX

C_2^4\times C_{20}
% in TeX

G:=Group("C2^4xC20");
// GroupNames label

G:=SmallGroup(320,1628);
// by ID

G=gap.SmallGroup(320,1628);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-5,-2,1120]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^20=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,d*e=e*d>;
// generators/relations

׿
×
𝔽