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G = C24×C20  order 320 = 26·5

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

direct product, abelian, monomial, 2-elementary

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

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

Generators and relations
 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 >

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

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽41)

400000
01000
004000
00010
00001
,
10000
01000
004000
000400
000040
,
10000
040000
004000
000400
00001
,
10000
01000
00100
000400
000040
,
100000
05000
00100
000310
000023

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] >;
 

320 conjugacy classes

class 1 2A···2AE4A···4AF5A5B5C5D10A···10DT20A···20DX
order12···24···4555510···1020···20
size11···11···111111···11···1

320 irreducible representations

dim11111111
type+++
imageC1C2C2C4C5C10C10C20
kernelC24×C20C23×C20C24×C10C23×C10C24×C4C23×C4C25C24
# reps13013241204128

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
 

׿
×
⋊
ℤ
𝔽
○
≀