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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

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, C4, C22, C5, C2×C4, C23, C10, C10, C22×C4, C24, C20, C2×C10, C23×C4, C25, C2×C20, C22×C10, C24×C4, C22×C20, C23×C10, C23×C20, C24×C10, C24×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C22×C4, C24, C20, C2×C10, C23×C4, C25, C2×C20, C22×C10, C24×C4, C22×C20, C23×C10, C23×C20, C24×C10, C24×C20

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

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

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

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

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

Matrix representation of C24×C20 ►in 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] >;
 

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
 

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