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G = C23×C5⋊2C8  order 320 = 26·5

Direct product of C23 and C5⋊2C8

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

Aliases: C23×C5⋊2C8, C20.73C24, C24.6Dic5, C5⋊4(C23×C8), (C22×C10)⋊8C8, C10⋊4(C22×C8), (C23×C4).14D5, C4.72(C23×D5), (C23×C10).16C4, (C22×C20).62C4, C10.61(C23×C4), (C23×C20).20C2, C2.1(C23×Dic5), (C2×C20).883C23, C20.238(C22×C4), (C22×C4).471D10, C4.37(C22×Dic5), (C22×C4).21Dic5, C23.43(C2×Dic5), (C22×C20).568C22, C22.27(C22×Dic5), (C2×C10)⋊15(C2×C8), (C2×C20).494(C2×C4), (C2×C4).105(C2×Dic5), (C2×C4).826(C22×D5), (C2×C10).305(C22×C4), (C22×C10).207(C2×C4), SmallGroup(320,1452)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — C23×C5⋊2C8
Chief series
C1 — C5 — C10 — C20 — C5⋊2C8 — C2×C5⋊2C8 — C22×C5⋊2C8 — C23×C5⋊2C8
Lower central
C5 — C23×C5⋊2C8

Subgroups: 542 in 338 conjugacy classes, 287 normal (11 characteristic)
C1, C2, C2 [×14], C4, C4 [×7], C22 [×35], C5, C8 [×8], C2×C4 [×28], C23 [×15], C10, C10 [×14], C2×C8 [×28], C22×C4 [×14], C24, C20, C20 [×7], C2×C10 [×35], C22×C8 [×14], C23×C4, C5⋊2C8 [×8], C2×C20 [×28], C22×C10 [×15], C23×C8, C2×C5⋊2C8 [×28], C22×C20 [×14], C23×C10, C22×C5⋊2C8 [×14], C23×C20, C23×C5⋊2C8

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], D5, C2×C8 [×28], C22×C4 [×14], C24, Dic5 [×8], D10 [×7], C22×C8 [×14], C23×C4, C5⋊2C8 [×8], C2×Dic5 [×28], C22×D5 [×7], C23×C8, C2×C5⋊2C8 [×28], C22×Dic5 [×14], C23×D5, C22×C5⋊2C8 [×14], C23×Dic5, C23×C5⋊2C8

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=d-1 >

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

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

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

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

(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)| (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,81)(33,307)(34,308)(35,309)(36,310)(37,311)(38,312)(39,305)(40,306)(41,281)(42,282)(43,283)(44,284)(45,285)(46,286)(47,287)(48,288)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,292)(66,293)(67,294)(68,295)(69,296)(70,289)(71,290)(72,291)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,97)(80,98)(89,219)(90,220)(91,221)(92,222)(93,223)(94,224)(95,217)(96,218)(105,186)(106,187)(107,188)(108,189)(109,190)(110,191)(111,192)(112,185)(121,252)(122,253)(123,254)(124,255)(125,256)(126,249)(127,250)(128,251)(129,139)(130,140)(131,141)(132,142)(133,143)(134,144)(135,137)(136,138)(145,242)(146,243)(147,244)(148,245)(149,246)(150,247)(151,248)(152,241)(153,227)(154,228)(155,229)(156,230)(157,231)(158,232)(159,225)(160,226)(161,303)(162,304)(163,297)(164,298)(165,299)(166,300)(167,301)(168,302)(169,317)(170,318)(171,319)(172,320)(173,313)(174,314)(175,315)(176,316)(177,261)(178,262)(179,263)(180,264)(181,257)(182,258)(183,259)(184,260)(193,206)(194,207)(195,208)(196,201)(197,202)(198,203)(199,204)(200,205)(209,235)(210,236)(211,237)(212,238)(213,239)(214,240)(215,233)(216,234)(265,279)(266,280)(267,273)(268,274)(269,275)(270,276)(271,277)(272,278), (1,137)(2,138)(3,139)(4,140)(5,141)(6,142)(7,143)(8,144)(9,299)(10,300)(11,301)(12,302)(13,303)(14,304)(15,297)(16,298)(17,273)(18,274)(19,275)(20,276)(21,277)(22,278)(23,279)(24,280)(25,132)(26,133)(27,134)(28,135)(29,136)(30,129)(31,130)(32,131)(33,197)(34,198)(35,199)(36,200)(37,193)(38,194)(39,195)(40,196)(41,177)(42,178)(43,179)(44,180)(45,181)(46,182)(47,183)(48,184)(49,165)(50,166)(51,167)(52,168)(53,161)(54,162)(55,163)(56,164)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,65)(73,173)(74,174)(75,175)(76,176)(77,169)(78,170)(79,171)(80,172)(81,266)(82,267)(83,268)(84,269)(85,270)(86,271)(87,272)(88,265)(89,150)(90,151)(91,152)(92,145)(93,146)(94,147)(95,148)(96,149)(97,319)(98,320)(99,313)(100,314)(101,315)(102,316)(103,317)(104,318)(105,212)(106,213)(107,214)(108,215)(109,216)(110,209)(111,210)(112,211)(113,293)(114,294)(115,295)(116,296)(117,289)(118,290)(119,291)(120,292)(121,226)(122,227)(123,228)(124,229)(125,230)(126,231)(127,232)(128,225)(153,253)(154,254)(155,255)(156,256)(157,249)(158,250)(159,251)(160,252)(185,237)(186,238)(187,239)(188,240)(189,233)(190,234)(191,235)(192,236)(201,306)(202,307)(203,308)(204,309)(205,310)(206,311)(207,312)(208,305)(217,245)(218,246)(219,247)(220,248)(221,241)(222,242)(223,243)(224,244)(257,285)(258,286)(259,287)(260,288)(261,281)(262,282)(263,283)(264,284), (1,171)(2,172)(3,173)(4,174)(5,175)(6,176)(7,169)(8,170)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,202)(18,203)(19,204)(20,205)(21,206)(22,207)(23,208)(24,201)(25,316)(26,317)(27,318)(28,319)(29,320)(30,313)(31,314)(32,315)(33,267)(34,268)(35,269)(36,270)(37,271)(38,272)(39,265)(40,266)(49,287)(50,288)(51,281)(52,282)(53,283)(54,284)(55,285)(56,286)(57,153)(58,154)(59,155)(60,156)(61,157)(62,158)(63,159)(64,160)(65,252)(66,253)(67,254)(68,255)(69,256)(70,249)(71,250)(72,251)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,137)(80,138)(81,196)(82,197)(83,198)(84,199)(85,200)(86,193)(87,194)(88,195)(89,235)(90,236)(91,237)(92,238)(93,239)(94,240)(95,233)(96,234)(97,135)(98,136)(99,129)(100,130)(101,131)(102,132)(103,133)(104,134)(105,242)(106,243)(107,244)(108,245)(109,246)(110,247)(111,248)(112,241)(113,227)(114,228)(115,229)(116,230)(117,231)(118,232)(119,225)(120,226)(121,292)(122,293)(123,294)(124,295)(125,296)(126,289)(127,290)(128,291)(145,186)(146,187)(147,188)(148,189)(149,190)(150,191)(151,192)(152,185)(161,263)(162,264)(163,257)(164,258)(165,259)(166,260)(167,261)(168,262)(177,301)(178,302)(179,303)(180,304)(181,297)(182,298)(183,299)(184,300)(209,219)(210,220)(211,221)(212,222)(213,223)(214,224)(215,217)(216,218)(273,307)(274,308)(275,309)(276,310)(277,311)(278,312)(279,305)(280,306), (1,199,157,49,233)(2,234,50,158,200)(3,193,159,51,235)(4,236,52,160,194)(5,195,153,53,237)(6,238,54,154,196)(7,197,155,55,239)(8,240,56,156,198)(9,215,28,204,231)(10,232,205,29,216)(11,209,30,206,225)(12,226,207,31,210)(13,211,32,208,227)(14,228,201,25,212)(15,213,26,202,229)(16,230,203,27,214)(17,115,45,223,317)(18,318,224,46,116)(19,117,47,217,319)(20,320,218,48,118)(21,119,41,219,313)(22,314,220,42,120)(23,113,43,221,315)(24,316,222,44,114)(33,255,163,187,143)(34,144,188,164,256)(35,249,165,189,137)(36,138,190,166,250)(37,251,167,191,139)(38,140,192,168,252)(39,253,161,185,141)(40,142,186,162,254)(57,283,91,175,88)(58,81,176,92,284)(59,285,93,169,82)(60,83,170,94,286)(61,287,95,171,84)(62,85,172,96,288)(63,281,89,173,86)(64,87,174,90,282)(65,272,74,151,262)(66,263,152,75,265)(67,266,76,145,264)(68,257,146,77,267)(69,268,78,147,258)(70,259,148,79,269)(71,270,80,149,260)(72,261,150,73,271)(97,275,289,183,245)(98,246,184,290,276)(99,277,291,177,247)(100,248,178,292,278)(101,279,293,179,241)(102,242,180,294,280)(103,273,295,181,243)(104,244,182,296,274)(105,304,123,306,132)(106,133,307,124,297)(107,298,125,308,134)(108,135,309,126,299)(109,300,127,310,136)(110,129,311,128,301)(111,302,121,312,130)(112,131,305,122,303), (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( (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,81)(33,307)(34,308)(35,309)(36,310)(37,311)(38,312)(39,305)(40,306)(41,281)(42,282)(43,283)(44,284)(45,285)(46,286)(47,287)(48,288)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,292)(66,293)(67,294)(68,295)(69,296)(70,289)(71,290)(72,291)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,97)(80,98)(89,219)(90,220)(91,221)(92,222)(93,223)(94,224)(95,217)(96,218)(105,186)(106,187)(107,188)(108,189)(109,190)(110,191)(111,192)(112,185)(121,252)(122,253)(123,254)(124,255)(125,256)(126,249)(127,250)(128,251)(129,139)(130,140)(131,141)(132,142)(133,143)(134,144)(135,137)(136,138)(145,242)(146,243)(147,244)(148,245)(149,246)(150,247)(151,248)(152,241)(153,227)(154,228)(155,229)(156,230)(157,231)(158,232)(159,225)(160,226)(161,303)(162,304)(163,297)(164,298)(165,299)(166,300)(167,301)(168,302)(169,317)(170,318)(171,319)(172,320)(173,313)(174,314)(175,315)(176,316)(177,261)(178,262)(179,263)(180,264)(181,257)(182,258)(183,259)(184,260)(193,206)(194,207)(195,208)(196,201)(197,202)(198,203)(199,204)(200,205)(209,235)(210,236)(211,237)(212,238)(213,239)(214,240)(215,233)(216,234)(265,279)(266,280)(267,273)(268,274)(269,275)(270,276)(271,277)(272,278), (1,137)(2,138)(3,139)(4,140)(5,141)(6,142)(7,143)(8,144)(9,299)(10,300)(11,301)(12,302)(13,303)(14,304)(15,297)(16,298)(17,273)(18,274)(19,275)(20,276)(21,277)(22,278)(23,279)(24,280)(25,132)(26,133)(27,134)(28,135)(29,136)(30,129)(31,130)(32,131)(33,197)(34,198)(35,199)(36,200)(37,193)(38,194)(39,195)(40,196)(41,177)(42,178)(43,179)(44,180)(45,181)(46,182)(47,183)(48,184)(49,165)(50,166)(51,167)(52,168)(53,161)(54,162)(55,163)(56,164)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,65)(73,173)(74,174)(75,175)(76,176)(77,169)(78,170)(79,171)(80,172)(81,266)(82,267)(83,268)(84,269)(85,270)(86,271)(87,272)(88,265)(89,150)(90,151)(91,152)(92,145)(93,146)(94,147)(95,148)(96,149)(97,319)(98,320)(99,313)(100,314)(101,315)(102,316)(103,317)(104,318)(105,212)(106,213)(107,214)(108,215)(109,216)(110,209)(111,210)(112,211)(113,293)(114,294)(115,295)(116,296)(117,289)(118,290)(119,291)(120,292)(121,226)(122,227)(123,228)(124,229)(125,230)(126,231)(127,232)(128,225)(153,253)(154,254)(155,255)(156,256)(157,249)(158,250)(159,251)(160,252)(185,237)(186,238)(187,239)(188,240)(189,233)(190,234)(191,235)(192,236)(201,306)(202,307)(203,308)(204,309)(205,310)(206,311)(207,312)(208,305)(217,245)(218,246)(219,247)(220,248)(221,241)(222,242)(223,243)(224,244)(257,285)(258,286)(259,287)(260,288)(261,281)(262,282)(263,283)(264,284), (1,171)(2,172)(3,173)(4,174)(5,175)(6,176)(7,169)(8,170)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,202)(18,203)(19,204)(20,205)(21,206)(22,207)(23,208)(24,201)(25,316)(26,317)(27,318)(28,319)(29,320)(30,313)(31,314)(32,315)(33,267)(34,268)(35,269)(36,270)(37,271)(38,272)(39,265)(40,266)(49,287)(50,288)(51,281)(52,282)(53,283)(54,284)(55,285)(56,286)(57,153)(58,154)(59,155)(60,156)(61,157)(62,158)(63,159)(64,160)(65,252)(66,253)(67,254)(68,255)(69,256)(70,249)(71,250)(72,251)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,137)(80,138)(81,196)(82,197)(83,198)(84,199)(85,200)(86,193)(87,194)(88,195)(89,235)(90,236)(91,237)(92,238)(93,239)(94,240)(95,233)(96,234)(97,135)(98,136)(99,129)(100,130)(101,131)(102,132)(103,133)(104,134)(105,242)(106,243)(107,244)(108,245)(109,246)(110,247)(111,248)(112,241)(113,227)(114,228)(115,229)(116,230)(117,231)(118,232)(119,225)(120,226)(121,292)(122,293)(123,294)(124,295)(125,296)(126,289)(127,290)(128,291)(145,186)(146,187)(147,188)(148,189)(149,190)(150,191)(151,192)(152,185)(161,263)(162,264)(163,257)(164,258)(165,259)(166,260)(167,261)(168,262)(177,301)(178,302)(179,303)(180,304)(181,297)(182,298)(183,299)(184,300)(209,219)(210,220)(211,221)(212,222)(213,223)(214,224)(215,217)(216,218)(273,307)(274,308)(275,309)(276,310)(277,311)(278,312)(279,305)(280,306), (1,199,157,49,233)(2,234,50,158,200)(3,193,159,51,235)(4,236,52,160,194)(5,195,153,53,237)(6,238,54,154,196)(7,197,155,55,239)(8,240,56,156,198)(9,215,28,204,231)(10,232,205,29,216)(11,209,30,206,225)(12,226,207,31,210)(13,211,32,208,227)(14,228,201,25,212)(15,213,26,202,229)(16,230,203,27,214)(17,115,45,223,317)(18,318,224,46,116)(19,117,47,217,319)(20,320,218,48,118)(21,119,41,219,313)(22,314,220,42,120)(23,113,43,221,315)(24,316,222,44,114)(33,255,163,187,143)(34,144,188,164,256)(35,249,165,189,137)(36,138,190,166,250)(37,251,167,191,139)(38,140,192,168,252)(39,253,161,185,141)(40,142,186,162,254)(57,283,91,175,88)(58,81,176,92,284)(59,285,93,169,82)(60,83,170,94,286)(61,287,95,171,84)(62,85,172,96,288)(63,281,89,173,86)(64,87,174,90,282)(65,272,74,151,262)(66,263,152,75,265)(67,266,76,145,264)(68,257,146,77,267)(69,268,78,147,258)(70,259,148,79,269)(71,270,80,149,260)(72,261,150,73,271)(97,275,289,183,245)(98,246,184,290,276)(99,277,291,177,247)(100,248,178,292,278)(101,279,293,179,241)(102,242,180,294,280)(103,273,295,181,243)(104,244,182,296,274)(105,304,123,306,132)(106,133,307,124,297)(107,298,125,308,134)(108,135,309,126,299)(109,300,127,310,136)(110,129,311,128,301)(111,302,121,312,130)(112,131,305,122,303), (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([(1,28),(2,29),(3,30),(4,31),(5,32),(6,25),(7,26),(8,27),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,82),(18,83),(19,84),(20,85),(21,86),(22,87),(23,88),(24,81),(33,307),(34,308),(35,309),(36,310),(37,311),(38,312),(39,305),(40,306),(41,281),(42,282),(43,283),(44,284),(45,285),(46,286),(47,287),(48,288),(57,113),(58,114),(59,115),(60,116),(61,117),(62,118),(63,119),(64,120),(65,292),(66,293),(67,294),(68,295),(69,296),(70,289),(71,290),(72,291),(73,99),(74,100),(75,101),(76,102),(77,103),(78,104),(79,97),(80,98),(89,219),(90,220),(91,221),(92,222),(93,223),(94,224),(95,217),(96,218),(105,186),(106,187),(107,188),(108,189),(109,190),(110,191),(111,192),(112,185),(121,252),(122,253),(123,254),(124,255),(125,256),(126,249),(127,250),(128,251),(129,139),(130,140),(131,141),(132,142),(133,143),(134,144),(135,137),(136,138),(145,242),(146,243),(147,244),(148,245),(149,246),(150,247),(151,248),(152,241),(153,227),(154,228),(155,229),(156,230),(157,231),(158,232),(159,225),(160,226),(161,303),(162,304),(163,297),(164,298),(165,299),(166,300),(167,301),(168,302),(169,317),(170,318),(171,319),(172,320),(173,313),(174,314),(175,315),(176,316),(177,261),(178,262),(179,263),(180,264),(181,257),(182,258),(183,259),(184,260),(193,206),(194,207),(195,208),(196,201),(197,202),(198,203),(199,204),(200,205),(209,235),(210,236),(211,237),(212,238),(213,239),(214,240),(215,233),(216,234),(265,279),(266,280),(267,273),(268,274),(269,275),(270,276),(271,277),(272,278)], [(1,137),(2,138),(3,139),(4,140),(5,141),(6,142),(7,143),(8,144),(9,299),(10,300),(11,301),(12,302),(13,303),(14,304),(15,297),(16,298),(17,273),(18,274),(19,275),(20,276),(21,277),(22,278),(23,279),(24,280),(25,132),(26,133),(27,134),(28,135),(29,136),(30,129),(31,130),(32,131),(33,197),(34,198),(35,199),(36,200),(37,193),(38,194),(39,195),(40,196),(41,177),(42,178),(43,179),(44,180),(45,181),(46,182),(47,183),(48,184),(49,165),(50,166),(51,167),(52,168),(53,161),(54,162),(55,163),(56,164),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72),(64,65),(73,173),(74,174),(75,175),(76,176),(77,169),(78,170),(79,171),(80,172),(81,266),(82,267),(83,268),(84,269),(85,270),(86,271),(87,272),(88,265),(89,150),(90,151),(91,152),(92,145),(93,146),(94,147),(95,148),(96,149),(97,319),(98,320),(99,313),(100,314),(101,315),(102,316),(103,317),(104,318),(105,212),(106,213),(107,214),(108,215),(109,216),(110,209),(111,210),(112,211),(113,293),(114,294),(115,295),(116,296),(117,289),(118,290),(119,291),(120,292),(121,226),(122,227),(123,228),(124,229),(125,230),(126,231),(127,232),(128,225),(153,253),(154,254),(155,255),(156,256),(157,249),(158,250),(159,251),(160,252),(185,237),(186,238),(187,239),(188,240),(189,233),(190,234),(191,235),(192,236),(201,306),(202,307),(203,308),(204,309),(205,310),(206,311),(207,312),(208,305),(217,245),(218,246),(219,247),(220,248),(221,241),(222,242),(223,243),(224,244),(257,285),(258,286),(259,287),(260,288),(261,281),(262,282),(263,283),(264,284)], [(1,171),(2,172),(3,173),(4,174),(5,175),(6,176),(7,169),(8,170),(9,47),(10,48),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,202),(18,203),(19,204),(20,205),(21,206),(22,207),(23,208),(24,201),(25,316),(26,317),(27,318),(28,319),(29,320),(30,313),(31,314),(32,315),(33,267),(34,268),(35,269),(36,270),(37,271),(38,272),(39,265),(40,266),(49,287),(50,288),(51,281),(52,282),(53,283),(54,284),(55,285),(56,286),(57,153),(58,154),(59,155),(60,156),(61,157),(62,158),(63,159),(64,160),(65,252),(66,253),(67,254),(68,255),(69,256),(70,249),(71,250),(72,251),(73,139),(74,140),(75,141),(76,142),(77,143),(78,144),(79,137),(80,138),(81,196),(82,197),(83,198),(84,199),(85,200),(86,193),(87,194),(88,195),(89,235),(90,236),(91,237),(92,238),(93,239),(94,240),(95,233),(96,234),(97,135),(98,136),(99,129),(100,130),(101,131),(102,132),(103,133),(104,134),(105,242),(106,243),(107,244),(108,245),(109,246),(110,247),(111,248),(112,241),(113,227),(114,228),(115,229),(116,230),(117,231),(118,232),(119,225),(120,226),(121,292),(122,293),(123,294),(124,295),(125,296),(126,289),(127,290),(128,291),(145,186),(146,187),(147,188),(148,189),(149,190),(150,191),(151,192),(152,185),(161,263),(162,264),(163,257),(164,258),(165,259),(166,260),(167,261),(168,262),(177,301),(178,302),(179,303),(180,304),(181,297),(182,298),(183,299),(184,300),(209,219),(210,220),(211,221),(212,222),(213,223),(214,224),(215,217),(216,218),(273,307),(274,308),(275,309),(276,310),(277,311),(278,312),(279,305),(280,306)], [(1,199,157,49,233),(2,234,50,158,200),(3,193,159,51,235),(4,236,52,160,194),(5,195,153,53,237),(6,238,54,154,196),(7,197,155,55,239),(8,240,56,156,198),(9,215,28,204,231),(10,232,205,29,216),(11,209,30,206,225),(12,226,207,31,210),(13,211,32,208,227),(14,228,201,25,212),(15,213,26,202,229),(16,230,203,27,214),(17,115,45,223,317),(18,318,224,46,116),(19,117,47,217,319),(20,320,218,48,118),(21,119,41,219,313),(22,314,220,42,120),(23,113,43,221,315),(24,316,222,44,114),(33,255,163,187,143),(34,144,188,164,256),(35,249,165,189,137),(36,138,190,166,250),(37,251,167,191,139),(38,140,192,168,252),(39,253,161,185,141),(40,142,186,162,254),(57,283,91,175,88),(58,81,176,92,284),(59,285,93,169,82),(60,83,170,94,286),(61,287,95,171,84),(62,85,172,96,288),(63,281,89,173,86),(64,87,174,90,282),(65,272,74,151,262),(66,263,152,75,265),(67,266,76,145,264),(68,257,146,77,267),(69,268,78,147,258),(70,259,148,79,269),(71,270,80,149,260),(72,261,150,73,271),(97,275,289,183,245),(98,246,184,290,276),(99,277,291,177,247),(100,248,178,292,278),(101,279,293,179,241),(102,242,180,294,280),(103,273,295,181,243),(104,244,182,296,274),(105,304,123,306,132),(106,133,307,124,297),(107,298,125,308,134),(108,135,309,126,299),(109,300,127,310,136),(110,129,311,128,301),(111,302,121,312,130),(112,131,305,122,303)], [(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
00100
000400
000040
,
10000
040000
004000
00010
00001
,
10000
01000
00100
000640
00010
,
140000
040000
004000
0002329
000318

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],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,1,0,0,0,40,0],[14,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,23,3,0,0,0,29,18] >;
 

128 conjugacy classes

class 1 2A···2O4A···4P5A5B8A···8AF10A···10AD20A···20AF
order12···24···4558···810···1020···20
size11···11···1225···52···22···2

128 irreducible representations

dim11111122222
type++++-+-
imageC1C2C2C4C4C8D5Dic5D10Dic5C5⋊2C8
kernelC23×C5⋊2C8C22×C5⋊2C8C23×C20C22×C20C23×C10C22×C10C23×C4C22×C4C22×C4C24C23
# reps11411423221414232

In GAP, Magma, Sage, TeX 

C_2^3\times C_5\rtimes_2C_8
 
% in TeX
 
G:=Group("C2^3xC5:2C8");
 
// GroupNames label
 
G:=SmallGroup(320,1452);
 
// by ID
 
G=gap.SmallGroup(320,1452);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,102,12550]);
 
// 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^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀