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## G = C23×C5⋊C8order 320 = 26·5

### Direct product of C23 and C5⋊C8

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

Generators and relations for C23×C5⋊C8
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 >

Subgroups: 746 in 338 conjugacy classes, 236 normal (9 characteristic)
C1, C2, C2, C4, C22, C5, C8, C2×C4, C23, C10, C10, C2×C8, C22×C4, C24, Dic5, Dic5, C2×C10, C22×C8, C23×C4, C5⋊C8, C2×Dic5, C22×C10, C23×C8, C2×C5⋊C8, C22×Dic5, C23×C10, C22×C5⋊C8, C23×Dic5, C23×C5⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C24, F5, C22×C8, C23×C4, C5⋊C8, C2×F5, C23×C8, C2×C5⋊C8, C22×F5, C22×C5⋊C8, C23×F5, C23×C5⋊C8

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

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

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

C23×C5⋊C8 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

׿
×
𝔽