C2^3xC5:C8

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C₂³×C₅⋊C₈, C₂⁴.₆F₅, Dic₅.₂₃C₂⁴, C₅⋊₂(C₂³×C₈), C₁₀⋊₂(C₂²×C₈), (C₂²×C₁₀)⋊₅C₈, C₂.₃(C₂³×F₅), (C₂³×C₁₀).₉C₄, C₂³.₆₈(C₂×F₅), C₁₀.₁₉(C₂³×C₄), C₂².₆₀(C₂²×F₅), (C₂²×Dic₅).₃₈C₄, (C₂³×Dic₅).₁₄C₂, Dic₅.₅₀(C₂²×C₄), (C₂×Dic₅).₃₆₆C₂³, (C₂²×Dic₅).₂₈₄C₂², (C₂×C₁₀)⋊₉(C₂×C₈), (C₂²×C₁₀).₈₃(C₂×C₄), (C₂×C₁₀).₁₀₅(C₂²×C₄), (C₂×Dic₅).₂₀₀(C₂×C₄), SmallGroup(320,1605)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅ — C₂³×C₅⋊C₈

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₅⋊C₈ — C₂×C₅⋊C₈ — C₂²×C₅⋊C₈ — C₂³×C₅⋊C₈

Lower central

C₅ — C₂³×C₅⋊C₈

Upper central

C₁ — C₂⁴

Subgroups: 746 in 338 conjugacy classes, 236 normal (9 characteristic)
C₁, C₂, C₂^[×14], C₄^[×8], C₂²^[×35], C₅, C₈^[×8], C₂×C₄^[×28], C₂³^[×15], C₁₀, C₁₀^[×14], C₂×C₈^[×28], C₂²×C₄^[×14], C₂⁴, Dic₅, Dic₅^[×7], C₂×C₁₀^[×35], C₂²×C₈^[×14], C₂³×C₄, C₅⋊C₈^[×8], C₂×Dic₅^[×28], C₂²×C₁₀^[×15], C₂³×C₈, C₂×C₅⋊C₈^[×28], C₂²×Dic₅^[×14], C₂³×C₁₀, C₂²×C₅⋊C₈^[×14], C₂³×Dic₅, C₂³×C₅⋊C₈

Quotients:
C₁, C₂^[×15], C₄^[×8], C₂²^[×35], C₈^[×8], C₂×C₄^[×28], C₂³^[×15], C₂×C₈^[×28], C₂²×C₄^[×14], C₂⁴, F₅, C₂²×C₈^[×14], C₂³×C₄, C₅⋊C₈^[×8], C₂×F₅^[×7], C₂³×C₈, C₂×C₅⋊C₈^[×28], C₂²×F₅^[×7], C₂²×C₅⋊C₈^[×14], C₂³×F₅, C₂³×C₅⋊C₈

Generators and relations
G = < a,b,c,d,e | a²=b²=c²=d⁵=e⁸=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d³ >

Smallest permutation representation
►Regular action on 320 points

Generators in S₃₂₀

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

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

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

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

(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)| (1,315)(2,316)(3,317)(4,318)(5,319)(6,320)(7,313)(8,314)(9,283)(10,284)(11,285)(12,286)(13,287)(14,288)(15,281)(16,282)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,277)(34,278)(35,279)(36,280)(37,273)(38,274)(39,275)(40,276)(41,195)(42,196)(43,197)(44,198)(45,199)(46,200)(47,193)(48,194)(65,125)(66,126)(67,127)(68,128)(69,121)(70,122)(71,123)(72,124)(73,133)(74,134)(75,135)(76,136)(77,129)(78,130)(79,131)(80,132)(81,205)(82,206)(83,207)(84,208)(85,201)(86,202)(87,203)(88,204)(89,213)(90,214)(91,215)(92,216)(93,209)(94,210)(95,211)(96,212)(97,141)(98,142)(99,143)(100,144)(101,137)(102,138)(103,139)(104,140)(105,149)(106,150)(107,151)(108,152)(109,145)(110,146)(111,147)(112,148)(113,157)(114,158)(115,159)(116,160)(117,153)(118,154)(119,155)(120,156)(161,221)(162,222)(163,223)(164,224)(165,217)(166,218)(167,219)(168,220)(169,227)(170,228)(171,229)(172,230)(173,231)(174,232)(175,225)(176,226)(177,239)(178,240)(179,233)(180,234)(181,235)(182,236)(183,237)(184,238)(185,245)(186,246)(187,247)(188,248)(189,241)(190,242)(191,243)(192,244)(249,293)(250,294)(251,295)(252,296)(253,289)(254,290)(255,291)(256,292)(257,301)(258,302)(259,303)(260,304)(261,297)(262,298)(263,299)(264,300)(265,309)(266,310)(267,311)(268,312)(269,305)(270,306)(271,307)(272,308), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,73)(8,74)(9,235)(10,236)(11,237)(12,238)(13,239)(14,240)(15,233)(16,234)(17,111)(18,112)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,295)(26,296)(27,289)(28,290)(29,291)(30,292)(31,293)(32,294)(33,261)(34,262)(35,263)(36,264)(37,257)(38,258)(39,259)(40,260)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,81)(48,82)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,249)(58,250)(59,251)(60,252)(61,253)(62,254)(63,255)(64,256)(65,95)(66,96)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94)(97,229)(98,230)(99,231)(100,232)(101,225)(102,226)(103,227)(104,228)(113,243)(114,244)(115,245)(116,246)(117,247)(118,248)(119,241)(120,242)(121,215)(122,216)(123,209)(124,210)(125,211)(126,212)(127,213)(128,214)(129,317)(130,318)(131,319)(132,320)(133,313)(134,314)(135,315)(136,316)(137,175)(138,176)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(153,187)(154,188)(155,189)(156,190)(157,191)(158,192)(159,185)(160,186)(161,269)(162,270)(163,271)(164,272)(165,265)(166,266)(167,267)(168,268)(177,287)(178,288)(179,281)(180,282)(181,283)(182,284)(183,285)(184,286)(193,205)(194,206)(195,207)(196,208)(197,201)(198,202)(199,203)(200,204)(217,309)(218,310)(219,311)(220,312)(221,305)(222,306)(223,307)(224,308)(273,301)(274,302)(275,303)(276,304)(277,297)(278,298)(279,299)(280,300), (1,175)(2,176)(3,169)(4,170)(5,171)(6,172)(7,173)(8,174)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,113)(32,114)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,263)(42,264)(43,257)(44,258)(45,259)(46,260)(47,261)(48,262)(49,285)(50,286)(51,287)(52,288)(53,281)(54,282)(55,283)(56,284)(57,157)(58,158)(59,159)(60,160)(61,153)(62,154)(63,155)(64,156)(65,265)(66,266)(67,267)(68,268)(69,269)(70,270)(71,271)(72,272)(73,143)(74,144)(75,137)(76,138)(77,139)(78,140)(79,141)(80,142)(89,167)(90,168)(91,161)(92,162)(93,163)(94,164)(95,165)(96,166)(97,131)(98,132)(99,133)(100,134)(101,135)(102,136)(103,129)(104,130)(105,233)(106,234)(107,235)(108,236)(109,237)(110,238)(111,239)(112,240)(121,305)(122,306)(123,307)(124,308)(125,309)(126,310)(127,311)(128,312)(145,183)(146,184)(147,177)(148,178)(149,179)(150,180)(151,181)(152,182)(185,251)(186,252)(187,253)(188,254)(189,255)(190,256)(191,249)(192,250)(193,297)(194,298)(195,299)(196,300)(197,301)(198,302)(199,303)(200,304)(201,273)(202,274)(203,275)(204,276)(205,277)(206,278)(207,279)(208,280)(209,223)(210,224)(211,217)(212,218)(213,219)(214,220)(215,221)(216,222)(225,315)(226,316)(227,317)(228,318)(229,319)(230,320)(231,313)(232,314)(241,291)(242,292)(243,293)(244,294)(245,295)(246,296)(247,289)(248,290), (1,49,157,35,161)(2,36,50,162,158)(3,163,37,159,51)(4,160,164,52,38)(5,53,153,39,165)(6,40,54,166,154)(7,167,33,155,55)(8,156,168,56,34)(9,231,213,205,29)(10,206,232,30,214)(11,31,207,215,225)(12,216,32,226,208)(13,227,209,201,25)(14,202,228,26,210)(15,27,203,211,229)(16,212,28,230,204)(17,317,223,273,115)(18,274,318,116,224)(19,117,275,217,319)(20,218,118,320,276)(21,313,219,277,119)(22,278,314,120,220)(23,113,279,221,315)(24,222,114,316,280)(41,69,137,183,249)(42,184,70,250,138)(43,251,177,139,71)(44,140,252,72,178)(45,65,141,179,253)(46,180,66,254,142)(47,255,181,143,67)(48,144,256,68,182)(57,83,91,175,285)(58,176,84,286,92)(59,287,169,93,85)(60,94,288,86,170)(61,87,95,171,281)(62,172,88,282,96)(63,283,173,89,81)(64,90,284,82,174)(73,267,261,189,151)(74,190,268,152,262)(75,145,191,263,269)(76,264,146,270,192)(77,271,257,185,147)(78,186,272,148,258)(79,149,187,259,265)(80,260,150,266,188)(97,233,289,199,125)(98,200,234,126,290)(99,127,193,291,235)(100,292,128,236,194)(101,237,293,195,121)(102,196,238,122,294)(103,123,197,295,239)(104,296,124,240,198)(105,247,303,309,131)(106,310,248,132,304)(107,133,311,297,241)(108,298,134,242,312)(109,243,299,305,135)(110,306,244,136,300)(111,129,307,301,245)(112,302,130,246,308), (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( (1,315)(2,316)(3,317)(4,318)(5,319)(6,320)(7,313)(8,314)(9,283)(10,284)(11,285)(12,286)(13,287)(14,288)(15,281)(16,282)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,277)(34,278)(35,279)(36,280)(37,273)(38,274)(39,275)(40,276)(41,195)(42,196)(43,197)(44,198)(45,199)(46,200)(47,193)(48,194)(65,125)(66,126)(67,127)(68,128)(69,121)(70,122)(71,123)(72,124)(73,133)(74,134)(75,135)(76,136)(77,129)(78,130)(79,131)(80,132)(81,205)(82,206)(83,207)(84,208)(85,201)(86,202)(87,203)(88,204)(89,213)(90,214)(91,215)(92,216)(93,209)(94,210)(95,211)(96,212)(97,141)(98,142)(99,143)(100,144)(101,137)(102,138)(103,139)(104,140)(105,149)(106,150)(107,151)(108,152)(109,145)(110,146)(111,147)(112,148)(113,157)(114,158)(115,159)(116,160)(117,153)(118,154)(119,155)(120,156)(161,221)(162,222)(163,223)(164,224)(165,217)(166,218)(167,219)(168,220)(169,227)(170,228)(171,229)(172,230)(173,231)(174,232)(175,225)(176,226)(177,239)(178,240)(179,233)(180,234)(181,235)(182,236)(183,237)(184,238)(185,245)(186,246)(187,247)(188,248)(189,241)(190,242)(191,243)(192,244)(249,293)(250,294)(251,295)(252,296)(253,289)(254,290)(255,291)(256,292)(257,301)(258,302)(259,303)(260,304)(261,297)(262,298)(263,299)(264,300)(265,309)(266,310)(267,311)(268,312)(269,305)(270,306)(271,307)(272,308), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,73)(8,74)(9,235)(10,236)(11,237)(12,238)(13,239)(14,240)(15,233)(16,234)(17,111)(18,112)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,295)(26,296)(27,289)(28,290)(29,291)(30,292)(31,293)(32,294)(33,261)(34,262)(35,263)(36,264)(37,257)(38,258)(39,259)(40,260)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,81)(48,82)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,249)(58,250)(59,251)(60,252)(61,253)(62,254)(63,255)(64,256)(65,95)(66,96)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94)(97,229)(98,230)(99,231)(100,232)(101,225)(102,226)(103,227)(104,228)(113,243)(114,244)(115,245)(116,246)(117,247)(118,248)(119,241)(120,242)(121,215)(122,216)(123,209)(124,210)(125,211)(126,212)(127,213)(128,214)(129,317)(130,318)(131,319)(132,320)(133,313)(134,314)(135,315)(136,316)(137,175)(138,176)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(153,187)(154,188)(155,189)(156,190)(157,191)(158,192)(159,185)(160,186)(161,269)(162,270)(163,271)(164,272)(165,265)(166,266)(167,267)(168,268)(177,287)(178,288)(179,281)(180,282)(181,283)(182,284)(183,285)(184,286)(193,205)(194,206)(195,207)(196,208)(197,201)(198,202)(199,203)(200,204)(217,309)(218,310)(219,311)(220,312)(221,305)(222,306)(223,307)(224,308)(273,301)(274,302)(275,303)(276,304)(277,297)(278,298)(279,299)(280,300), (1,175)(2,176)(3,169)(4,170)(5,171)(6,172)(7,173)(8,174)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,113)(32,114)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,263)(42,264)(43,257)(44,258)(45,259)(46,260)(47,261)(48,262)(49,285)(50,286)(51,287)(52,288)(53,281)(54,282)(55,283)(56,284)(57,157)(58,158)(59,159)(60,160)(61,153)(62,154)(63,155)(64,156)(65,265)(66,266)(67,267)(68,268)(69,269)(70,270)(71,271)(72,272)(73,143)(74,144)(75,137)(76,138)(77,139)(78,140)(79,141)(80,142)(89,167)(90,168)(91,161)(92,162)(93,163)(94,164)(95,165)(96,166)(97,131)(98,132)(99,133)(100,134)(101,135)(102,136)(103,129)(104,130)(105,233)(106,234)(107,235)(108,236)(109,237)(110,238)(111,239)(112,240)(121,305)(122,306)(123,307)(124,308)(125,309)(126,310)(127,311)(128,312)(145,183)(146,184)(147,177)(148,178)(149,179)(150,180)(151,181)(152,182)(185,251)(186,252)(187,253)(188,254)(189,255)(190,256)(191,249)(192,250)(193,297)(194,298)(195,299)(196,300)(197,301)(198,302)(199,303)(200,304)(201,273)(202,274)(203,275)(204,276)(205,277)(206,278)(207,279)(208,280)(209,223)(210,224)(211,217)(212,218)(213,219)(214,220)(215,221)(216,222)(225,315)(226,316)(227,317)(228,318)(229,319)(230,320)(231,313)(232,314)(241,291)(242,292)(243,293)(244,294)(245,295)(246,296)(247,289)(248,290), (1,49,157,35,161)(2,36,50,162,158)(3,163,37,159,51)(4,160,164,52,38)(5,53,153,39,165)(6,40,54,166,154)(7,167,33,155,55)(8,156,168,56,34)(9,231,213,205,29)(10,206,232,30,214)(11,31,207,215,225)(12,216,32,226,208)(13,227,209,201,25)(14,202,228,26,210)(15,27,203,211,229)(16,212,28,230,204)(17,317,223,273,115)(18,274,318,116,224)(19,117,275,217,319)(20,218,118,320,276)(21,313,219,277,119)(22,278,314,120,220)(23,113,279,221,315)(24,222,114,316,280)(41,69,137,183,249)(42,184,70,250,138)(43,251,177,139,71)(44,140,252,72,178)(45,65,141,179,253)(46,180,66,254,142)(47,255,181,143,67)(48,144,256,68,182)(57,83,91,175,285)(58,176,84,286,92)(59,287,169,93,85)(60,94,288,86,170)(61,87,95,171,281)(62,172,88,282,96)(63,283,173,89,81)(64,90,284,82,174)(73,267,261,189,151)(74,190,268,152,262)(75,145,191,263,269)(76,264,146,270,192)(77,271,257,185,147)(78,186,272,148,258)(79,149,187,259,265)(80,260,150,266,188)(97,233,289,199,125)(98,200,234,126,290)(99,127,193,291,235)(100,292,128,236,194)(101,237,293,195,121)(102,196,238,122,294)(103,123,197,295,239)(104,296,124,240,198)(105,247,303,309,131)(106,310,248,132,304)(107,133,311,297,241)(108,298,134,242,312)(109,243,299,305,135)(110,306,244,136,300)(111,129,307,301,245)(112,302,130,246,308), (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([(1,315),(2,316),(3,317),(4,318),(5,319),(6,320),(7,313),(8,314),(9,283),(10,284),(11,285),(12,286),(13,287),(14,288),(15,281),(16,282),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,49),(24,50),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,57),(32,58),(33,277),(34,278),(35,279),(36,280),(37,273),(38,274),(39,275),(40,276),(41,195),(42,196),(43,197),(44,198),(45,199),(46,200),(47,193),(48,194),(65,125),(66,126),(67,127),(68,128),(69,121),(70,122),(71,123),(72,124),(73,133),(74,134),(75,135),(76,136),(77,129),(78,130),(79,131),(80,132),(81,205),(82,206),(83,207),(84,208),(85,201),(86,202),(87,203),(88,204),(89,213),(90,214),(91,215),(92,216),(93,209),(94,210),(95,211),(96,212),(97,141),(98,142),(99,143),(100,144),(101,137),(102,138),(103,139),(104,140),(105,149),(106,150),(107,151),(108,152),(109,145),(110,146),(111,147),(112,148),(113,157),(114,158),(115,159),(116,160),(117,153),(118,154),(119,155),(120,156),(161,221),(162,222),(163,223),(164,224),(165,217),(166,218),(167,219),(168,220),(169,227),(170,228),(171,229),(172,230),(173,231),(174,232),(175,225),(176,226),(177,239),(178,240),(179,233),(180,234),(181,235),(182,236),(183,237),(184,238),(185,245),(186,246),(187,247),(188,248),(189,241),(190,242),(191,243),(192,244),(249,293),(250,294),(251,295),(252,296),(253,289),(254,290),(255,291),(256,292),(257,301),(258,302),(259,303),(260,304),(261,297),(262,298),(263,299),(264,300),(265,309),(266,310),(267,311),(268,312),(269,305),(270,306),(271,307),(272,308)], [(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,73),(8,74),(9,235),(10,236),(11,237),(12,238),(13,239),(14,240),(15,233),(16,234),(17,111),(18,112),(19,105),(20,106),(21,107),(22,108),(23,109),(24,110),(25,295),(26,296),(27,289),(28,290),(29,291),(30,292),(31,293),(32,294),(33,261),(34,262),(35,263),(36,264),(37,257),(38,258),(39,259),(40,260),(41,83),(42,84),(43,85),(44,86),(45,87),(46,88),(47,81),(48,82),(49,145),(50,146),(51,147),(52,148),(53,149),(54,150),(55,151),(56,152),(57,249),(58,250),(59,251),(60,252),(61,253),(62,254),(63,255),(64,256),(65,95),(66,96),(67,89),(68,90),(69,91),(70,92),(71,93),(72,94),(97,229),(98,230),(99,231),(100,232),(101,225),(102,226),(103,227),(104,228),(113,243),(114,244),(115,245),(116,246),(117,247),(118,248),(119,241),(120,242),(121,215),(122,216),(123,209),(124,210),(125,211),(126,212),(127,213),(128,214),(129,317),(130,318),(131,319),(132,320),(133,313),(134,314),(135,315),(136,316),(137,175),(138,176),(139,169),(140,170),(141,171),(142,172),(143,173),(144,174),(153,187),(154,188),(155,189),(156,190),(157,191),(158,192),(159,185),(160,186),(161,269),(162,270),(163,271),(164,272),(165,265),(166,266),(167,267),(168,268),(177,287),(178,288),(179,281),(180,282),(181,283),(182,284),(183,285),(184,286),(193,205),(194,206),(195,207),(196,208),(197,201),(198,202),(199,203),(200,204),(217,309),(218,310),(219,311),(220,312),(221,305),(222,306),(223,307),(224,308),(273,301),(274,302),(275,303),(276,304),(277,297),(278,298),(279,299),(280,300)], [(1,175),(2,176),(3,169),(4,170),(5,171),(6,172),(7,173),(8,174),(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20),(25,115),(26,116),(27,117),(28,118),(29,119),(30,120),(31,113),(32,114),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,263),(42,264),(43,257),(44,258),(45,259),(46,260),(47,261),(48,262),(49,285),(50,286),(51,287),(52,288),(53,281),(54,282),(55,283),(56,284),(57,157),(58,158),(59,159),(60,160),(61,153),(62,154),(63,155),(64,156),(65,265),(66,266),(67,267),(68,268),(69,269),(70,270),(71,271),(72,272),(73,143),(74,144),(75,137),(76,138),(77,139),(78,140),(79,141),(80,142),(89,167),(90,168),(91,161),(92,162),(93,163),(94,164),(95,165),(96,166),(97,131),(98,132),(99,133),(100,134),(101,135),(102,136),(103,129),(104,130),(105,233),(106,234),(107,235),(108,236),(109,237),(110,238),(111,239),(112,240),(121,305),(122,306),(123,307),(124,308),(125,309),(126,310),(127,311),(128,312),(145,183),(146,184),(147,177),(148,178),(149,179),(150,180),(151,181),(152,182),(185,251),(186,252),(187,253),(188,254),(189,255),(190,256),(191,249),(192,250),(193,297),(194,298),(195,299),(196,300),(197,301),(198,302),(199,303),(200,304),(201,273),(202,274),(203,275),(204,276),(205,277),(206,278),(207,279),(208,280),(209,223),(210,224),(211,217),(212,218),(213,219),(214,220),(215,221),(216,222),(225,315),(226,316),(227,317),(228,318),(229,319),(230,320),(231,313),(232,314),(241,291),(242,292),(243,293),(244,294),(245,295),(246,296),(247,289),(248,290)], [(1,49,157,35,161),(2,36,50,162,158),(3,163,37,159,51),(4,160,164,52,38),(5,53,153,39,165),(6,40,54,166,154),(7,167,33,155,55),(8,156,168,56,34),(9,231,213,205,29),(10,206,232,30,214),(11,31,207,215,225),(12,216,32,226,208),(13,227,209,201,25),(14,202,228,26,210),(15,27,203,211,229),(16,212,28,230,204),(17,317,223,273,115),(18,274,318,116,224),(19,117,275,217,319),(20,218,118,320,276),(21,313,219,277,119),(22,278,314,120,220),(23,113,279,221,315),(24,222,114,316,280),(41,69,137,183,249),(42,184,70,250,138),(43,251,177,139,71),(44,140,252,72,178),(45,65,141,179,253),(46,180,66,254,142),(47,255,181,143,67),(48,144,256,68,182),(57,83,91,175,285),(58,176,84,286,92),(59,287,169,93,85),(60,94,288,86,170),(61,87,95,171,281),(62,172,88,282,96),(63,283,173,89,81),(64,90,284,82,174),(73,267,261,189,151),(74,190,268,152,262),(75,145,191,263,269),(76,264,146,270,192),(77,271,257,185,147),(78,186,272,148,258),(79,149,187,259,265),(80,260,150,266,188),(97,233,289,199,125),(98,200,234,126,290),(99,127,193,291,235),(100,292,128,236,194),(101,237,293,195,121),(102,196,238,122,294),(103,123,197,295,239),(104,296,124,240,198),(105,247,303,309,131),(106,310,248,132,304),(107,133,311,297,241),(108,298,134,242,312),(109,243,299,305,135),(110,306,244,136,300),(111,129,307,301,245),(112,302,130,246,308)], [(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 ⊆ GL₇(𝔽₄₁)

40	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	40	0	0	0	0
0	0	0	40	0	0	0
0	0	0	0	40	0	0
0	0	0	0	0	40	0
0	0	0	0	0	0	40

1	0	0	0	0	0	0
0	40	0	0	0	0	0
0	0	40	0	0	0	0
0	0	0	40	0	0	0
0	0	0	0	40	0	0
0	0	0	0	0	40	0
0	0	0	0	0	0	40

40	0	0	0	0	0	0
0	40	0	0	0	0	0
0	0	40	0	0	0	0
0	0	0	40	0	0	0
0	0	0	0	40	0	0
0	0	0	0	0	40	0
0	0	0	0	0	0	40

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	0	0	0	40
0	0	0	1	0	0	40
0	0	0	0	1	0	40
0	0	0	0	0	1	40

40	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	24	4	1	29
0	0	0	25	33	25	12
0	0	0	8	16	29	13
0	0	0	12	17	17	37

G:=sub<GL(7,GF(41))| [40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,40,40,40,40],[40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,24,25,8,12,0,0,0,4,33,16,17,0,0,0,1,25,29,17,0,0,0,29,12,13,37] >;

80 conjugacy classes

class	1	2A	···	2O	4A	···	4P	5	8A	···	8AF	10A	···	10O
order	1	2	···	2	4	···	4	5	8	···	8	10	···	10
size	1	1	···	1	5	···	5	4	5	···	5	4	···	4

80 irreducible representations

dim	1	1	1	1	1	1	4	4	4
type	+	+	+				+	-	+
image	C₁	C₂	C₂	C₄	C₄	C₈	F₅	C₅⋊C₈	C₂×F₅
kernel	C₂³×C₅⋊C₈	C₂²×C₅⋊C₈	C₂³×Dic₅	C₂²×Dic₅	C₂³×C₁₀	C₂²×C₁₀	C₂⁴	C₂³	C₂³
# reps	1	14	1	14	2	32	1	8	7

In GAP, Magma, Sage, TeX

C_2^3\times C_5\rtimes C_8

% in TeX

G:=Group("C2^3xC5:C8");

// GroupNames label

G:=SmallGroup(320,1605);

// by ID

G=gap.SmallGroup(320,1605);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,102,6278,818]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^5=e^8=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,e*d*e^-1=d^3>;

// generators/relations

G = C₂³×C₅⋊C₈ order 320 = 2⁶·5

Direct product of C₂³ and C₅⋊C₈

40	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	24	4	1	29
0	0	0	25	33	25	12
0	0	0	8	16	29	13
0	0	0	12	17	17	37

40	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	24	4	1	29
0	0	0	25	33	25	12
0	0	0	8	16	29	13
0	0	0	12	17	17	37

G = C23×C5⋊C8 order 320 = 26·5

Direct product of C23 and C5⋊C8

G = C₂³×C₅⋊C₈ order 320 = 2⁶·5

Direct product of C₂³ and C₅⋊C₈

40	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	24	4	1	29
0	0	0	25	33	25	12
0	0	0	8	16	29	13
0	0	0	12	17	17	37