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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), C10⋊4(C22×C8), (C22×C10)⋊8C8, C4.72(C23×D5), (C23×C4).14D5, C10.61(C23×C4), (C22×C20).62C4, (C23×C10).16C4, (C23×C20).20C2, C2.1(C23×Dic5), C20.238(C22×C4), (C2×C20).883C23, (C22×C4).471D10, C4.37(C22×Dic5), C23.43(C2×Dic5), (C22×C4).21Dic5, (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), (C22×C10).207(C2×C4), (C2×C10).305(C22×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
Upper central
C1 — C23×C4

Generators and relations for C23×C5⋊2C8
 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 >

Subgroups: 542 in 338 conjugacy classes, 287 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C23, C10, C10, C2×C8, C22×C4, C24, C20, C20, C2×C10, C22×C8, C23×C4, C5⋊2C8, C2×C20, C22×C10, C23×C8, C2×C5⋊2C8, C22×C20, C23×C10, C22×C5⋊2C8, C23×C20, C23×C5⋊2C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C2×C8, C22×C4, C24, Dic5, D10, C22×C8, C23×C4, C5⋊2C8, C2×Dic5, C22×D5, C23×C8, C2×C5⋊2C8, C22×Dic5, C23×D5, C22×C5⋊2C8, C23×Dic5, C23×C5⋊2C8

Smallest permutation representation of C23×C5⋊2C8
►Regular action on 320 points
Generators in S320
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 247)(10 248)(11 241)(12 242)(13 243)(14 244)(15 245)(16 246)(17 95)(18 96)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(33 111)(34 112)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 99)(42 100)(43 101)(44 102)(45 103)(46 104)(47 97)(48 98)(49 117)(50 118)(51 119)(52 120)(53 113)(54 114)(55 115)(56 116)(57 311)(58 312)(59 305)(60 306)(61 307)(62 308)(63 309)(64 310)(65 295)(66 296)(67 289)(68 290)(69 291)(70 292)(71 293)(72 294)(73 299)(74 300)(75 301)(76 302)(77 303)(78 304)(79 297)(80 298)(81 219)(82 220)(83 221)(84 222)(85 223)(86 224)(87 217)(88 218)(121 255)(122 256)(123 249)(124 250)(125 251)(126 252)(127 253)(128 254)(129 259)(130 260)(131 261)(132 262)(133 263)(134 264)(135 257)(136 258)(137 179)(138 180)(139 181)(140 182)(141 183)(142 184)(143 177)(144 178)(145 275)(146 276)(147 277)(148 278)(149 279)(150 280)(151 273)(152 274)(153 227)(154 228)(155 229)(156 230)(157 231)(158 232)(159 225)(160 226)(161 205)(162 206)(163 207)(164 208)(165 201)(166 202)(167 203)(168 204)(169 317)(170 318)(171 319)(172 320)(173 313)(174 314)(175 315)(176 316)(185 283)(186 284)(187 285)(188 286)(189 287)(190 288)(191 281)(192 282)(193 209)(194 210)(195 211)(196 212)(197 213)(198 214)(199 215)(200 216)(233 269)(234 270)(235 271)(236 272)(237 265)(238 266)(239 267)(240 268)

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

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

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

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

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

Matrix representation of C23×C5⋊2C8 ►in 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] >;
 

C23×C5⋊2C8 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
 

׿
×
⋊
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𝔽
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