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## G = C2×C4×C5⋊2C8order 320 = 26·5

### Direct product of C2×C4 and C5⋊2C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C2×C4×C5⋊2C8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — C22×C5⋊2C8 — C2×C4×C5⋊2C8
 Lower central C5 — C2×C4×C5⋊2C8
 Upper central C1 — C2×C42

Generators and relations for C2×C4×C52C8
G = < a,b,c,d | a2=b4=c5=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 270 in 162 conjugacy classes, 135 normal (17 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C22×C4, C22×C4, C20, C2×C10, C2×C10, C4×C8, C2×C42, C22×C8, C52C8, C2×C20, C2×C20, C22×C10, C2×C4×C8, C2×C52C8, C4×C20, C22×C20, C22×C20, C4×C52C8, C22×C52C8, C2×C4×C20, C2×C4×C52C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C42, C2×C8, C22×C4, Dic5, D10, C4×C8, C2×C42, C22×C8, C52C8, C4×D5, C2×Dic5, C22×D5, C2×C4×C8, C2×C52C8, C4×Dic5, C2×C4×D5, C22×Dic5, C4×C52C8, C22×C52C8, C2×C4×Dic5, C2×C4×C52C8

Smallest permutation representation of C2×C4×C52C8
Regular action on 320 points
Generators in S320
(1 84)(2 85)(3 86)(4 87)(5 88)(6 81)(7 82)(8 83)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 105)(17 207)(18 208)(19 201)(20 202)(21 203)(22 204)(23 205)(24 206)(25 197)(26 198)(27 199)(28 200)(29 193)(30 194)(31 195)(32 196)(33 224)(34 217)(35 218)(36 219)(37 220)(38 221)(39 222)(40 223)(41 144)(42 137)(43 138)(44 139)(45 140)(46 141)(47 142)(48 143)(49 178)(50 179)(51 180)(52 181)(53 182)(54 183)(55 184)(56 177)(57 155)(58 156)(59 157)(60 158)(61 159)(62 160)(63 153)(64 154)(65 163)(66 164)(67 165)(68 166)(69 167)(70 168)(71 161)(72 162)(73 304)(74 297)(75 298)(76 299)(77 300)(78 301)(79 302)(80 303)(89 288)(90 281)(91 282)(92 283)(93 284)(94 285)(95 286)(96 287)(97 319)(98 320)(99 313)(100 314)(101 315)(102 316)(103 317)(104 318)(113 292)(114 293)(115 294)(116 295)(117 296)(118 289)(119 290)(120 291)(121 257)(122 258)(123 259)(124 260)(125 261)(126 262)(127 263)(128 264)(129 265)(130 266)(131 267)(132 268)(133 269)(134 270)(135 271)(136 272)(145 256)(146 249)(147 250)(148 251)(149 252)(150 253)(151 254)(152 255)(169 237)(170 238)(171 239)(172 240)(173 233)(174 234)(175 235)(176 236)(185 246)(186 247)(187 248)(188 241)(189 242)(190 243)(191 244)(192 245)(209 277)(210 278)(211 279)(212 280)(213 273)(214 274)(215 275)(216 276)(225 312)(226 305)(227 306)(228 307)(229 308)(230 309)(231 310)(232 311)
(1 14 189 153)(2 15 190 154)(3 16 191 155)(4 9 192 156)(5 10 185 157)(6 11 186 158)(7 12 187 159)(8 13 188 160)(17 150 167 39)(18 151 168 40)(19 152 161 33)(20 145 162 34)(21 146 163 35)(22 147 164 36)(23 148 165 37)(24 149 166 38)(25 276 142 176)(26 277 143 169)(27 278 144 170)(28 279 137 171)(29 280 138 172)(30 273 139 173)(31 274 140 174)(32 275 141 175)(41 238 199 210)(42 239 200 211)(43 240 193 212)(44 233 194 213)(45 234 195 214)(46 235 196 215)(47 236 197 216)(48 237 198 209)(49 257 116 95)(50 258 117 96)(51 259 118 89)(52 260 119 90)(53 261 120 91)(54 262 113 92)(55 263 114 93)(56 264 115 94)(57 86 105 244)(58 87 106 245)(59 88 107 246)(60 81 108 247)(61 82 109 248)(62 83 110 241)(63 84 111 242)(64 85 112 243)(65 218 203 249)(66 219 204 250)(67 220 205 251)(68 221 206 252)(69 222 207 253)(70 223 208 254)(71 224 201 255)(72 217 202 256)(73 267 228 101)(74 268 229 102)(75 269 230 103)(76 270 231 104)(77 271 232 97)(78 272 225 98)(79 265 226 99)(80 266 227 100)(121 295 286 178)(122 296 287 179)(123 289 288 180)(124 290 281 181)(125 291 282 182)(126 292 283 183)(127 293 284 184)(128 294 285 177)(129 305 313 302)(130 306 314 303)(131 307 315 304)(132 308 316 297)(133 309 317 298)(134 310 318 299)(135 311 319 300)(136 312 320 301)
(1 146 141 291 298)(2 299 292 142 147)(3 148 143 293 300)(4 301 294 144 149)(5 150 137 295 302)(6 303 296 138 151)(7 152 139 289 304)(8 297 290 140 145)(9 136 285 170 166)(10 167 171 286 129)(11 130 287 172 168)(12 161 173 288 131)(13 132 281 174 162)(14 163 175 282 133)(15 134 283 176 164)(16 165 169 284 135)(17 279 121 313 157)(18 158 314 122 280)(19 273 123 315 159)(20 160 316 124 274)(21 275 125 317 153)(22 154 318 126 276)(23 277 127 319 155)(24 156 320 128 278)(25 36 190 310 183)(26 184 311 191 37)(27 38 192 312 177)(28 178 305 185 39)(29 40 186 306 179)(30 180 307 187 33)(31 34 188 308 181)(32 182 309 189 35)(41 252 87 78 115)(42 116 79 88 253)(43 254 81 80 117)(44 118 73 82 255)(45 256 83 74 119)(46 120 75 84 249)(47 250 85 76 113)(48 114 77 86 251)(49 226 246 222 200)(50 193 223 247 227)(51 228 248 224 194)(52 195 217 241 229)(53 230 242 218 196)(54 197 219 243 231)(55 232 244 220 198)(56 199 221 245 225)(57 205 209 263 97)(58 98 264 210 206)(59 207 211 257 99)(60 100 258 212 208)(61 201 213 259 101)(62 102 260 214 202)(63 203 215 261 103)(64 104 262 216 204)(65 235 91 269 111)(66 112 270 92 236)(67 237 93 271 105)(68 106 272 94 238)(69 239 95 265 107)(70 108 266 96 240)(71 233 89 267 109)(72 110 268 90 234)
(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)| (1,84)(2,85)(3,86)(4,87)(5,88)(6,81)(7,82)(8,83)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,105)(17,207)(18,208)(19,201)(20,202)(21,203)(22,204)(23,205)(24,206)(25,197)(26,198)(27,199)(28,200)(29,193)(30,194)(31,195)(32,196)(33,224)(34,217)(35,218)(36,219)(37,220)(38,221)(39,222)(40,223)(41,144)(42,137)(43,138)(44,139)(45,140)(46,141)(47,142)(48,143)(49,178)(50,179)(51,180)(52,181)(53,182)(54,183)(55,184)(56,177)(57,155)(58,156)(59,157)(60,158)(61,159)(62,160)(63,153)(64,154)(65,163)(66,164)(67,165)(68,166)(69,167)(70,168)(71,161)(72,162)(73,304)(74,297)(75,298)(76,299)(77,300)(78,301)(79,302)(80,303)(89,288)(90,281)(91,282)(92,283)(93,284)(94,285)(95,286)(96,287)(97,319)(98,320)(99,313)(100,314)(101,315)(102,316)(103,317)(104,318)(113,292)(114,293)(115,294)(116,295)(117,296)(118,289)(119,290)(120,291)(121,257)(122,258)(123,259)(124,260)(125,261)(126,262)(127,263)(128,264)(129,265)(130,266)(131,267)(132,268)(133,269)(134,270)(135,271)(136,272)(145,256)(146,249)(147,250)(148,251)(149,252)(150,253)(151,254)(152,255)(169,237)(170,238)(171,239)(172,240)(173,233)(174,234)(175,235)(176,236)(185,246)(186,247)(187,248)(188,241)(189,242)(190,243)(191,244)(192,245)(209,277)(210,278)(211,279)(212,280)(213,273)(214,274)(215,275)(216,276)(225,312)(226,305)(227,306)(228,307)(229,308)(230,309)(231,310)(232,311), (1,14,189,153)(2,15,190,154)(3,16,191,155)(4,9,192,156)(5,10,185,157)(6,11,186,158)(7,12,187,159)(8,13,188,160)(17,150,167,39)(18,151,168,40)(19,152,161,33)(20,145,162,34)(21,146,163,35)(22,147,164,36)(23,148,165,37)(24,149,166,38)(25,276,142,176)(26,277,143,169)(27,278,144,170)(28,279,137,171)(29,280,138,172)(30,273,139,173)(31,274,140,174)(32,275,141,175)(41,238,199,210)(42,239,200,211)(43,240,193,212)(44,233,194,213)(45,234,195,214)(46,235,196,215)(47,236,197,216)(48,237,198,209)(49,257,116,95)(50,258,117,96)(51,259,118,89)(52,260,119,90)(53,261,120,91)(54,262,113,92)(55,263,114,93)(56,264,115,94)(57,86,105,244)(58,87,106,245)(59,88,107,246)(60,81,108,247)(61,82,109,248)(62,83,110,241)(63,84,111,242)(64,85,112,243)(65,218,203,249)(66,219,204,250)(67,220,205,251)(68,221,206,252)(69,222,207,253)(70,223,208,254)(71,224,201,255)(72,217,202,256)(73,267,228,101)(74,268,229,102)(75,269,230,103)(76,270,231,104)(77,271,232,97)(78,272,225,98)(79,265,226,99)(80,266,227,100)(121,295,286,178)(122,296,287,179)(123,289,288,180)(124,290,281,181)(125,291,282,182)(126,292,283,183)(127,293,284,184)(128,294,285,177)(129,305,313,302)(130,306,314,303)(131,307,315,304)(132,308,316,297)(133,309,317,298)(134,310,318,299)(135,311,319,300)(136,312,320,301), (1,146,141,291,298)(2,299,292,142,147)(3,148,143,293,300)(4,301,294,144,149)(5,150,137,295,302)(6,303,296,138,151)(7,152,139,289,304)(8,297,290,140,145)(9,136,285,170,166)(10,167,171,286,129)(11,130,287,172,168)(12,161,173,288,131)(13,132,281,174,162)(14,163,175,282,133)(15,134,283,176,164)(16,165,169,284,135)(17,279,121,313,157)(18,158,314,122,280)(19,273,123,315,159)(20,160,316,124,274)(21,275,125,317,153)(22,154,318,126,276)(23,277,127,319,155)(24,156,320,128,278)(25,36,190,310,183)(26,184,311,191,37)(27,38,192,312,177)(28,178,305,185,39)(29,40,186,306,179)(30,180,307,187,33)(31,34,188,308,181)(32,182,309,189,35)(41,252,87,78,115)(42,116,79,88,253)(43,254,81,80,117)(44,118,73,82,255)(45,256,83,74,119)(46,120,75,84,249)(47,250,85,76,113)(48,114,77,86,251)(49,226,246,222,200)(50,193,223,247,227)(51,228,248,224,194)(52,195,217,241,229)(53,230,242,218,196)(54,197,219,243,231)(55,232,244,220,198)(56,199,221,245,225)(57,205,209,263,97)(58,98,264,210,206)(59,207,211,257,99)(60,100,258,212,208)(61,201,213,259,101)(62,102,260,214,202)(63,203,215,261,103)(64,104,262,216,204)(65,235,91,269,111)(66,112,270,92,236)(67,237,93,271,105)(68,106,272,94,238)(69,239,95,265,107)(70,108,266,96,240)(71,233,89,267,109)(72,110,268,90,234), (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( (1,84)(2,85)(3,86)(4,87)(5,88)(6,81)(7,82)(8,83)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,105)(17,207)(18,208)(19,201)(20,202)(21,203)(22,204)(23,205)(24,206)(25,197)(26,198)(27,199)(28,200)(29,193)(30,194)(31,195)(32,196)(33,224)(34,217)(35,218)(36,219)(37,220)(38,221)(39,222)(40,223)(41,144)(42,137)(43,138)(44,139)(45,140)(46,141)(47,142)(48,143)(49,178)(50,179)(51,180)(52,181)(53,182)(54,183)(55,184)(56,177)(57,155)(58,156)(59,157)(60,158)(61,159)(62,160)(63,153)(64,154)(65,163)(66,164)(67,165)(68,166)(69,167)(70,168)(71,161)(72,162)(73,304)(74,297)(75,298)(76,299)(77,300)(78,301)(79,302)(80,303)(89,288)(90,281)(91,282)(92,283)(93,284)(94,285)(95,286)(96,287)(97,319)(98,320)(99,313)(100,314)(101,315)(102,316)(103,317)(104,318)(113,292)(114,293)(115,294)(116,295)(117,296)(118,289)(119,290)(120,291)(121,257)(122,258)(123,259)(124,260)(125,261)(126,262)(127,263)(128,264)(129,265)(130,266)(131,267)(132,268)(133,269)(134,270)(135,271)(136,272)(145,256)(146,249)(147,250)(148,251)(149,252)(150,253)(151,254)(152,255)(169,237)(170,238)(171,239)(172,240)(173,233)(174,234)(175,235)(176,236)(185,246)(186,247)(187,248)(188,241)(189,242)(190,243)(191,244)(192,245)(209,277)(210,278)(211,279)(212,280)(213,273)(214,274)(215,275)(216,276)(225,312)(226,305)(227,306)(228,307)(229,308)(230,309)(231,310)(232,311), (1,14,189,153)(2,15,190,154)(3,16,191,155)(4,9,192,156)(5,10,185,157)(6,11,186,158)(7,12,187,159)(8,13,188,160)(17,150,167,39)(18,151,168,40)(19,152,161,33)(20,145,162,34)(21,146,163,35)(22,147,164,36)(23,148,165,37)(24,149,166,38)(25,276,142,176)(26,277,143,169)(27,278,144,170)(28,279,137,171)(29,280,138,172)(30,273,139,173)(31,274,140,174)(32,275,141,175)(41,238,199,210)(42,239,200,211)(43,240,193,212)(44,233,194,213)(45,234,195,214)(46,235,196,215)(47,236,197,216)(48,237,198,209)(49,257,116,95)(50,258,117,96)(51,259,118,89)(52,260,119,90)(53,261,120,91)(54,262,113,92)(55,263,114,93)(56,264,115,94)(57,86,105,244)(58,87,106,245)(59,88,107,246)(60,81,108,247)(61,82,109,248)(62,83,110,241)(63,84,111,242)(64,85,112,243)(65,218,203,249)(66,219,204,250)(67,220,205,251)(68,221,206,252)(69,222,207,253)(70,223,208,254)(71,224,201,255)(72,217,202,256)(73,267,228,101)(74,268,229,102)(75,269,230,103)(76,270,231,104)(77,271,232,97)(78,272,225,98)(79,265,226,99)(80,266,227,100)(121,295,286,178)(122,296,287,179)(123,289,288,180)(124,290,281,181)(125,291,282,182)(126,292,283,183)(127,293,284,184)(128,294,285,177)(129,305,313,302)(130,306,314,303)(131,307,315,304)(132,308,316,297)(133,309,317,298)(134,310,318,299)(135,311,319,300)(136,312,320,301), (1,146,141,291,298)(2,299,292,142,147)(3,148,143,293,300)(4,301,294,144,149)(5,150,137,295,302)(6,303,296,138,151)(7,152,139,289,304)(8,297,290,140,145)(9,136,285,170,166)(10,167,171,286,129)(11,130,287,172,168)(12,161,173,288,131)(13,132,281,174,162)(14,163,175,282,133)(15,134,283,176,164)(16,165,169,284,135)(17,279,121,313,157)(18,158,314,122,280)(19,273,123,315,159)(20,160,316,124,274)(21,275,125,317,153)(22,154,318,126,276)(23,277,127,319,155)(24,156,320,128,278)(25,36,190,310,183)(26,184,311,191,37)(27,38,192,312,177)(28,178,305,185,39)(29,40,186,306,179)(30,180,307,187,33)(31,34,188,308,181)(32,182,309,189,35)(41,252,87,78,115)(42,116,79,88,253)(43,254,81,80,117)(44,118,73,82,255)(45,256,83,74,119)(46,120,75,84,249)(47,250,85,76,113)(48,114,77,86,251)(49,226,246,222,200)(50,193,223,247,227)(51,228,248,224,194)(52,195,217,241,229)(53,230,242,218,196)(54,197,219,243,231)(55,232,244,220,198)(56,199,221,245,225)(57,205,209,263,97)(58,98,264,210,206)(59,207,211,257,99)(60,100,258,212,208)(61,201,213,259,101)(62,102,260,214,202)(63,203,215,261,103)(64,104,262,216,204)(65,235,91,269,111)(66,112,270,92,236)(67,237,93,271,105)(68,106,272,94,238)(69,239,95,265,107)(70,108,266,96,240)(71,233,89,267,109)(72,110,268,90,234), (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([[(1,84),(2,85),(3,86),(4,87),(5,88),(6,81),(7,82),(8,83),(9,106),(10,107),(11,108),(12,109),(13,110),(14,111),(15,112),(16,105),(17,207),(18,208),(19,201),(20,202),(21,203),(22,204),(23,205),(24,206),(25,197),(26,198),(27,199),(28,200),(29,193),(30,194),(31,195),(32,196),(33,224),(34,217),(35,218),(36,219),(37,220),(38,221),(39,222),(40,223),(41,144),(42,137),(43,138),(44,139),(45,140),(46,141),(47,142),(48,143),(49,178),(50,179),(51,180),(52,181),(53,182),(54,183),(55,184),(56,177),(57,155),(58,156),(59,157),(60,158),(61,159),(62,160),(63,153),(64,154),(65,163),(66,164),(67,165),(68,166),(69,167),(70,168),(71,161),(72,162),(73,304),(74,297),(75,298),(76,299),(77,300),(78,301),(79,302),(80,303),(89,288),(90,281),(91,282),(92,283),(93,284),(94,285),(95,286),(96,287),(97,319),(98,320),(99,313),(100,314),(101,315),(102,316),(103,317),(104,318),(113,292),(114,293),(115,294),(116,295),(117,296),(118,289),(119,290),(120,291),(121,257),(122,258),(123,259),(124,260),(125,261),(126,262),(127,263),(128,264),(129,265),(130,266),(131,267),(132,268),(133,269),(134,270),(135,271),(136,272),(145,256),(146,249),(147,250),(148,251),(149,252),(150,253),(151,254),(152,255),(169,237),(170,238),(171,239),(172,240),(173,233),(174,234),(175,235),(176,236),(185,246),(186,247),(187,248),(188,241),(189,242),(190,243),(191,244),(192,245),(209,277),(210,278),(211,279),(212,280),(213,273),(214,274),(215,275),(216,276),(225,312),(226,305),(227,306),(228,307),(229,308),(230,309),(231,310),(232,311)], [(1,14,189,153),(2,15,190,154),(3,16,191,155),(4,9,192,156),(5,10,185,157),(6,11,186,158),(7,12,187,159),(8,13,188,160),(17,150,167,39),(18,151,168,40),(19,152,161,33),(20,145,162,34),(21,146,163,35),(22,147,164,36),(23,148,165,37),(24,149,166,38),(25,276,142,176),(26,277,143,169),(27,278,144,170),(28,279,137,171),(29,280,138,172),(30,273,139,173),(31,274,140,174),(32,275,141,175),(41,238,199,210),(42,239,200,211),(43,240,193,212),(44,233,194,213),(45,234,195,214),(46,235,196,215),(47,236,197,216),(48,237,198,209),(49,257,116,95),(50,258,117,96),(51,259,118,89),(52,260,119,90),(53,261,120,91),(54,262,113,92),(55,263,114,93),(56,264,115,94),(57,86,105,244),(58,87,106,245),(59,88,107,246),(60,81,108,247),(61,82,109,248),(62,83,110,241),(63,84,111,242),(64,85,112,243),(65,218,203,249),(66,219,204,250),(67,220,205,251),(68,221,206,252),(69,222,207,253),(70,223,208,254),(71,224,201,255),(72,217,202,256),(73,267,228,101),(74,268,229,102),(75,269,230,103),(76,270,231,104),(77,271,232,97),(78,272,225,98),(79,265,226,99),(80,266,227,100),(121,295,286,178),(122,296,287,179),(123,289,288,180),(124,290,281,181),(125,291,282,182),(126,292,283,183),(127,293,284,184),(128,294,285,177),(129,305,313,302),(130,306,314,303),(131,307,315,304),(132,308,316,297),(133,309,317,298),(134,310,318,299),(135,311,319,300),(136,312,320,301)], [(1,146,141,291,298),(2,299,292,142,147),(3,148,143,293,300),(4,301,294,144,149),(5,150,137,295,302),(6,303,296,138,151),(7,152,139,289,304),(8,297,290,140,145),(9,136,285,170,166),(10,167,171,286,129),(11,130,287,172,168),(12,161,173,288,131),(13,132,281,174,162),(14,163,175,282,133),(15,134,283,176,164),(16,165,169,284,135),(17,279,121,313,157),(18,158,314,122,280),(19,273,123,315,159),(20,160,316,124,274),(21,275,125,317,153),(22,154,318,126,276),(23,277,127,319,155),(24,156,320,128,278),(25,36,190,310,183),(26,184,311,191,37),(27,38,192,312,177),(28,178,305,185,39),(29,40,186,306,179),(30,180,307,187,33),(31,34,188,308,181),(32,182,309,189,35),(41,252,87,78,115),(42,116,79,88,253),(43,254,81,80,117),(44,118,73,82,255),(45,256,83,74,119),(46,120,75,84,249),(47,250,85,76,113),(48,114,77,86,251),(49,226,246,222,200),(50,193,223,247,227),(51,228,248,224,194),(52,195,217,241,229),(53,230,242,218,196),(54,197,219,243,231),(55,232,244,220,198),(56,199,221,245,225),(57,205,209,263,97),(58,98,264,210,206),(59,207,211,257,99),(60,100,258,212,208),(61,201,213,259,101),(62,102,260,214,202),(63,203,215,261,103),(64,104,262,216,204),(65,235,91,269,111),(66,112,270,92,236),(67,237,93,271,105),(68,106,272,94,238),(69,239,95,265,107),(70,108,266,96,240),(71,233,89,267,109),(72,110,268,90,234)], [(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 ··· 2G 4A ··· 4X 5A 5B 8A ··· 8AF 10A ··· 10N 20A ··· 20AV order 1 2 ··· 2 4 ··· 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 ··· 1 1 ··· 1 2 2 5 ··· 5 2 ··· 2 2 ··· 2

128 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + - + - + image C1 C2 C2 C2 C4 C4 C4 C8 D5 Dic5 D10 Dic5 D10 C5⋊2C8 C4×D5 kernel C2×C4×C5⋊2C8 C4×C5⋊2C8 C22×C5⋊2C8 C2×C4×C20 C2×C5⋊2C8 C4×C20 C22×C20 C2×C20 C2×C42 C42 C42 C22×C4 C22×C4 C2×C4 C2×C4 # reps 1 4 2 1 16 4 4 32 2 4 4 4 2 32 16

Matrix representation of C2×C4×C52C8 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 34 40 0 0 1 0
,
 32 0 0 0 0 40 0 0 0 0 32 10 0 0 32 9
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,34,1,0,0,40,0],[32,0,0,0,0,40,0,0,0,0,32,32,0,0,10,9] >;

C2×C4×C52C8 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_5\rtimes_2C_8
% in TeX

G:=Group("C2xC4xC5:2C8");
// GroupNames label

G:=SmallGroup(320,547);
// by ID

G=gap.SmallGroup(320,547);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,100,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^5=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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