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## G = C2×C42.D5order 320 = 26·5

### Direct product of C2 and C42.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C42.D5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — C22×C5⋊2C8 — C2×C42.D5
 Lower central C5 — C10 — C2×C42.D5
 Upper central C1 — C22×C4 — C2×C42

Generators and relations for C2×C42.D5
G = < a,b,c,d,e | a2=b4=c4=d5=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bc2, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 270 in 146 conjugacy classes, 103 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C22×C4, C22×C4, C20, C20, C2×C10, C2×C10, C8⋊C4, C2×C42, C22×C8, C52C8, C2×C20, C2×C20, C2×C20, C22×C10, C2×C8⋊C4, C2×C52C8, C4×C20, C22×C20, C22×C20, C42.D5, C22×C52C8, C2×C4×C20, C2×C42.D5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C42, M4(2), C22×C4, Dic5, D10, C8⋊C4, C2×C42, C2×M4(2), C4×D5, C2×Dic5, C22×D5, C2×C8⋊C4, C4.Dic5, C4×Dic5, C2×C4×D5, C22×Dic5, C42.D5, C2×C4.Dic5, C2×C4×Dic5, C2×C42.D5

Smallest permutation representation of C2×C42.D5
Regular action on 320 points
Generators in S320
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 235)(10 236)(11 237)(12 238)(13 239)(14 240)(15 233)(16 234)(25 44)(26 45)(27 46)(28 47)(29 48)(30 41)(31 42)(32 43)(33 190)(34 191)(35 192)(36 185)(37 186)(38 187)(39 188)(40 189)(49 296)(50 289)(51 290)(52 291)(53 292)(54 293)(55 294)(56 295)(57 320)(58 313)(59 314)(60 315)(61 316)(62 317)(63 318)(64 319)(65 285)(66 286)(67 287)(68 288)(69 281)(70 282)(71 283)(72 284)(73 225)(74 226)(75 227)(76 228)(77 229)(78 230)(79 231)(80 232)(81 172)(82 173)(83 174)(84 175)(85 176)(86 169)(87 170)(88 171)(89 223)(90 224)(91 217)(92 218)(93 219)(94 220)(95 221)(96 222)(97 178)(98 179)(99 180)(100 181)(101 182)(102 183)(103 184)(104 177)(105 306)(106 307)(107 308)(108 309)(109 310)(110 311)(111 312)(112 305)(113 165)(114 166)(115 167)(116 168)(117 161)(118 162)(119 163)(120 164)(121 280)(122 273)(123 274)(124 275)(125 276)(126 277)(127 278)(128 279)(129 153)(130 154)(131 155)(132 156)(133 157)(134 158)(135 159)(136 160)(137 242)(138 243)(139 244)(140 245)(141 246)(142 247)(143 248)(144 241)(145 258)(146 259)(147 260)(148 261)(149 262)(150 263)(151 264)(152 257)(193 206)(194 207)(195 208)(196 201)(197 202)(198 203)(199 204)(200 205)(209 302)(210 303)(211 304)(212 297)(213 298)(214 299)(215 300)(216 301)(249 268)(250 269)(251 270)(252 271)(253 272)(254 265)(255 266)(256 267)
(1 118 197 257)(2 115 198 262)(3 120 199 259)(4 117 200 264)(5 114 193 261)(6 119 194 258)(7 116 195 263)(8 113 196 260)(9 271 97 296)(10 268 98 293)(11 265 99 290)(12 270 100 295)(13 267 101 292)(14 272 102 289)(15 269 103 294)(16 266 104 291)(17 164 204 146)(18 161 205 151)(19 166 206 148)(20 163 207 145)(21 168 208 150)(22 165 201 147)(23 162 202 152)(24 167 203 149)(25 132 310 169)(26 129 311 174)(27 134 312 171)(28 131 305 176)(29 136 306 173)(30 133 307 170)(31 130 308 175)(32 135 309 172)(33 242 212 66)(34 247 213 71)(35 244 214 68)(36 241 215 65)(37 246 216 70)(38 243 209 67)(39 248 210 72)(40 245 211 69)(41 157 106 87)(42 154 107 84)(43 159 108 81)(44 156 109 86)(45 153 110 83)(46 158 111 88)(47 155 112 85)(48 160 105 82)(49 235 252 178)(50 240 253 183)(51 237 254 180)(52 234 255 177)(53 239 256 182)(54 236 249 179)(55 233 250 184)(56 238 251 181)(57 232 276 219)(58 229 277 224)(59 226 278 221)(60 231 279 218)(61 228 280 223)(62 225 273 220)(63 230 274 217)(64 227 275 222)(73 122 94 317)(74 127 95 314)(75 124 96 319)(76 121 89 316)(77 126 90 313)(78 123 91 318)(79 128 92 315)(80 125 93 320)(137 297 286 190)(138 302 287 187)(139 299 288 192)(140 304 281 189)(141 301 282 186)(142 298 283 191)(143 303 284 188)(144 300 285 185)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)(129 131 133 135)(130 132 134 136)(137 139 141 143)(138 140 142 144)(145 147 149 151)(146 148 150 152)(153 155 157 159)(154 156 158 160)(161 163 165 167)(162 164 166 168)(169 171 173 175)(170 172 174 176)(177 179 181 183)(178 180 182 184)(185 187 189 191)(186 188 190 192)(193 195 197 199)(194 196 198 200)(201 203 205 207)(202 204 206 208)(209 211 213 215)(210 212 214 216)(217 219 221 223)(218 220 222 224)(225 227 229 231)(226 228 230 232)(233 235 237 239)(234 236 238 240)(241 243 245 247)(242 244 246 248)(249 251 253 255)(250 252 254 256)(257 259 261 263)(258 260 262 264)(265 267 269 271)(266 268 270 272)(273 275 277 279)(274 276 278 280)(281 283 285 287)(282 284 286 288)(289 291 293 295)(290 292 294 296)(297 299 301 303)(298 300 302 304)(305 307 309 311)(306 308 310 312)(313 315 317 319)(314 316 318 320)
(1 100 45 92 186)(2 187 93 46 101)(3 102 47 94 188)(4 189 95 48 103)(5 104 41 96 190)(6 191 89 42 97)(7 98 43 90 192)(8 185 91 44 99)(9 194 298 76 107)(10 108 77 299 195)(11 196 300 78 109)(12 110 79 301 197)(13 198 302 80 111)(14 112 73 303 199)(15 200 304 74 105)(16 106 75 297 193)(17 183 28 220 39)(18 40 221 29 184)(19 177 30 222 33)(20 34 223 31 178)(21 179 32 224 35)(22 36 217 25 180)(23 181 26 218 37)(24 38 219 27 182)(49 163 247 61 130)(50 131 62 248 164)(51 165 241 63 132)(52 133 64 242 166)(53 167 243 57 134)(54 135 58 244 168)(55 161 245 59 136)(56 129 60 246 162)(65 274 169 254 147)(66 148 255 170 275)(67 276 171 256 149)(68 150 249 172 277)(69 278 173 250 151)(70 152 251 174 279)(71 280 175 252 145)(72 146 253 176 273)(81 126 288 263 268)(82 269 264 281 127)(83 128 282 257 270)(84 271 258 283 121)(85 122 284 259 272)(86 265 260 285 123)(87 124 286 261 266)(88 267 262 287 125)(113 144 318 156 290)(114 291 157 319 137)(115 138 320 158 292)(116 293 159 313 139)(117 140 314 160 294)(118 295 153 315 141)(119 142 316 154 296)(120 289 155 317 143)(201 215 230 310 237)(202 238 311 231 216)(203 209 232 312 239)(204 240 305 225 210)(205 211 226 306 233)(206 234 307 227 212)(207 213 228 308 235)(208 236 309 229 214)
(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)| (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,235)(10,236)(11,237)(12,238)(13,239)(14,240)(15,233)(16,234)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43)(33,190)(34,191)(35,192)(36,185)(37,186)(38,187)(39,188)(40,189)(49,296)(50,289)(51,290)(52,291)(53,292)(54,293)(55,294)(56,295)(57,320)(58,313)(59,314)(60,315)(61,316)(62,317)(63,318)(64,319)(65,285)(66,286)(67,287)(68,288)(69,281)(70,282)(71,283)(72,284)(73,225)(74,226)(75,227)(76,228)(77,229)(78,230)(79,231)(80,232)(81,172)(82,173)(83,174)(84,175)(85,176)(86,169)(87,170)(88,171)(89,223)(90,224)(91,217)(92,218)(93,219)(94,220)(95,221)(96,222)(97,178)(98,179)(99,180)(100,181)(101,182)(102,183)(103,184)(104,177)(105,306)(106,307)(107,308)(108,309)(109,310)(110,311)(111,312)(112,305)(113,165)(114,166)(115,167)(116,168)(117,161)(118,162)(119,163)(120,164)(121,280)(122,273)(123,274)(124,275)(125,276)(126,277)(127,278)(128,279)(129,153)(130,154)(131,155)(132,156)(133,157)(134,158)(135,159)(136,160)(137,242)(138,243)(139,244)(140,245)(141,246)(142,247)(143,248)(144,241)(145,258)(146,259)(147,260)(148,261)(149,262)(150,263)(151,264)(152,257)(193,206)(194,207)(195,208)(196,201)(197,202)(198,203)(199,204)(200,205)(209,302)(210,303)(211,304)(212,297)(213,298)(214,299)(215,300)(216,301)(249,268)(250,269)(251,270)(252,271)(253,272)(254,265)(255,266)(256,267), (1,118,197,257)(2,115,198,262)(3,120,199,259)(4,117,200,264)(5,114,193,261)(6,119,194,258)(7,116,195,263)(8,113,196,260)(9,271,97,296)(10,268,98,293)(11,265,99,290)(12,270,100,295)(13,267,101,292)(14,272,102,289)(15,269,103,294)(16,266,104,291)(17,164,204,146)(18,161,205,151)(19,166,206,148)(20,163,207,145)(21,168,208,150)(22,165,201,147)(23,162,202,152)(24,167,203,149)(25,132,310,169)(26,129,311,174)(27,134,312,171)(28,131,305,176)(29,136,306,173)(30,133,307,170)(31,130,308,175)(32,135,309,172)(33,242,212,66)(34,247,213,71)(35,244,214,68)(36,241,215,65)(37,246,216,70)(38,243,209,67)(39,248,210,72)(40,245,211,69)(41,157,106,87)(42,154,107,84)(43,159,108,81)(44,156,109,86)(45,153,110,83)(46,158,111,88)(47,155,112,85)(48,160,105,82)(49,235,252,178)(50,240,253,183)(51,237,254,180)(52,234,255,177)(53,239,256,182)(54,236,249,179)(55,233,250,184)(56,238,251,181)(57,232,276,219)(58,229,277,224)(59,226,278,221)(60,231,279,218)(61,228,280,223)(62,225,273,220)(63,230,274,217)(64,227,275,222)(73,122,94,317)(74,127,95,314)(75,124,96,319)(76,121,89,316)(77,126,90,313)(78,123,91,318)(79,128,92,315)(80,125,93,320)(137,297,286,190)(138,302,287,187)(139,299,288,192)(140,304,281,189)(141,301,282,186)(142,298,283,191)(143,303,284,188)(144,300,285,185), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160)(161,163,165,167)(162,164,166,168)(169,171,173,175)(170,172,174,176)(177,179,181,183)(178,180,182,184)(185,187,189,191)(186,188,190,192)(193,195,197,199)(194,196,198,200)(201,203,205,207)(202,204,206,208)(209,211,213,215)(210,212,214,216)(217,219,221,223)(218,220,222,224)(225,227,229,231)(226,228,230,232)(233,235,237,239)(234,236,238,240)(241,243,245,247)(242,244,246,248)(249,251,253,255)(250,252,254,256)(257,259,261,263)(258,260,262,264)(265,267,269,271)(266,268,270,272)(273,275,277,279)(274,276,278,280)(281,283,285,287)(282,284,286,288)(289,291,293,295)(290,292,294,296)(297,299,301,303)(298,300,302,304)(305,307,309,311)(306,308,310,312)(313,315,317,319)(314,316,318,320), (1,100,45,92,186)(2,187,93,46,101)(3,102,47,94,188)(4,189,95,48,103)(5,104,41,96,190)(6,191,89,42,97)(7,98,43,90,192)(8,185,91,44,99)(9,194,298,76,107)(10,108,77,299,195)(11,196,300,78,109)(12,110,79,301,197)(13,198,302,80,111)(14,112,73,303,199)(15,200,304,74,105)(16,106,75,297,193)(17,183,28,220,39)(18,40,221,29,184)(19,177,30,222,33)(20,34,223,31,178)(21,179,32,224,35)(22,36,217,25,180)(23,181,26,218,37)(24,38,219,27,182)(49,163,247,61,130)(50,131,62,248,164)(51,165,241,63,132)(52,133,64,242,166)(53,167,243,57,134)(54,135,58,244,168)(55,161,245,59,136)(56,129,60,246,162)(65,274,169,254,147)(66,148,255,170,275)(67,276,171,256,149)(68,150,249,172,277)(69,278,173,250,151)(70,152,251,174,279)(71,280,175,252,145)(72,146,253,176,273)(81,126,288,263,268)(82,269,264,281,127)(83,128,282,257,270)(84,271,258,283,121)(85,122,284,259,272)(86,265,260,285,123)(87,124,286,261,266)(88,267,262,287,125)(113,144,318,156,290)(114,291,157,319,137)(115,138,320,158,292)(116,293,159,313,139)(117,140,314,160,294)(118,295,153,315,141)(119,142,316,154,296)(120,289,155,317,143)(201,215,230,310,237)(202,238,311,231,216)(203,209,232,312,239)(204,240,305,225,210)(205,211,226,306,233)(206,234,307,227,212)(207,213,228,308,235)(208,236,309,229,214), (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( (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,235)(10,236)(11,237)(12,238)(13,239)(14,240)(15,233)(16,234)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43)(33,190)(34,191)(35,192)(36,185)(37,186)(38,187)(39,188)(40,189)(49,296)(50,289)(51,290)(52,291)(53,292)(54,293)(55,294)(56,295)(57,320)(58,313)(59,314)(60,315)(61,316)(62,317)(63,318)(64,319)(65,285)(66,286)(67,287)(68,288)(69,281)(70,282)(71,283)(72,284)(73,225)(74,226)(75,227)(76,228)(77,229)(78,230)(79,231)(80,232)(81,172)(82,173)(83,174)(84,175)(85,176)(86,169)(87,170)(88,171)(89,223)(90,224)(91,217)(92,218)(93,219)(94,220)(95,221)(96,222)(97,178)(98,179)(99,180)(100,181)(101,182)(102,183)(103,184)(104,177)(105,306)(106,307)(107,308)(108,309)(109,310)(110,311)(111,312)(112,305)(113,165)(114,166)(115,167)(116,168)(117,161)(118,162)(119,163)(120,164)(121,280)(122,273)(123,274)(124,275)(125,276)(126,277)(127,278)(128,279)(129,153)(130,154)(131,155)(132,156)(133,157)(134,158)(135,159)(136,160)(137,242)(138,243)(139,244)(140,245)(141,246)(142,247)(143,248)(144,241)(145,258)(146,259)(147,260)(148,261)(149,262)(150,263)(151,264)(152,257)(193,206)(194,207)(195,208)(196,201)(197,202)(198,203)(199,204)(200,205)(209,302)(210,303)(211,304)(212,297)(213,298)(214,299)(215,300)(216,301)(249,268)(250,269)(251,270)(252,271)(253,272)(254,265)(255,266)(256,267), (1,118,197,257)(2,115,198,262)(3,120,199,259)(4,117,200,264)(5,114,193,261)(6,119,194,258)(7,116,195,263)(8,113,196,260)(9,271,97,296)(10,268,98,293)(11,265,99,290)(12,270,100,295)(13,267,101,292)(14,272,102,289)(15,269,103,294)(16,266,104,291)(17,164,204,146)(18,161,205,151)(19,166,206,148)(20,163,207,145)(21,168,208,150)(22,165,201,147)(23,162,202,152)(24,167,203,149)(25,132,310,169)(26,129,311,174)(27,134,312,171)(28,131,305,176)(29,136,306,173)(30,133,307,170)(31,130,308,175)(32,135,309,172)(33,242,212,66)(34,247,213,71)(35,244,214,68)(36,241,215,65)(37,246,216,70)(38,243,209,67)(39,248,210,72)(40,245,211,69)(41,157,106,87)(42,154,107,84)(43,159,108,81)(44,156,109,86)(45,153,110,83)(46,158,111,88)(47,155,112,85)(48,160,105,82)(49,235,252,178)(50,240,253,183)(51,237,254,180)(52,234,255,177)(53,239,256,182)(54,236,249,179)(55,233,250,184)(56,238,251,181)(57,232,276,219)(58,229,277,224)(59,226,278,221)(60,231,279,218)(61,228,280,223)(62,225,273,220)(63,230,274,217)(64,227,275,222)(73,122,94,317)(74,127,95,314)(75,124,96,319)(76,121,89,316)(77,126,90,313)(78,123,91,318)(79,128,92,315)(80,125,93,320)(137,297,286,190)(138,302,287,187)(139,299,288,192)(140,304,281,189)(141,301,282,186)(142,298,283,191)(143,303,284,188)(144,300,285,185), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160)(161,163,165,167)(162,164,166,168)(169,171,173,175)(170,172,174,176)(177,179,181,183)(178,180,182,184)(185,187,189,191)(186,188,190,192)(193,195,197,199)(194,196,198,200)(201,203,205,207)(202,204,206,208)(209,211,213,215)(210,212,214,216)(217,219,221,223)(218,220,222,224)(225,227,229,231)(226,228,230,232)(233,235,237,239)(234,236,238,240)(241,243,245,247)(242,244,246,248)(249,251,253,255)(250,252,254,256)(257,259,261,263)(258,260,262,264)(265,267,269,271)(266,268,270,272)(273,275,277,279)(274,276,278,280)(281,283,285,287)(282,284,286,288)(289,291,293,295)(290,292,294,296)(297,299,301,303)(298,300,302,304)(305,307,309,311)(306,308,310,312)(313,315,317,319)(314,316,318,320), (1,100,45,92,186)(2,187,93,46,101)(3,102,47,94,188)(4,189,95,48,103)(5,104,41,96,190)(6,191,89,42,97)(7,98,43,90,192)(8,185,91,44,99)(9,194,298,76,107)(10,108,77,299,195)(11,196,300,78,109)(12,110,79,301,197)(13,198,302,80,111)(14,112,73,303,199)(15,200,304,74,105)(16,106,75,297,193)(17,183,28,220,39)(18,40,221,29,184)(19,177,30,222,33)(20,34,223,31,178)(21,179,32,224,35)(22,36,217,25,180)(23,181,26,218,37)(24,38,219,27,182)(49,163,247,61,130)(50,131,62,248,164)(51,165,241,63,132)(52,133,64,242,166)(53,167,243,57,134)(54,135,58,244,168)(55,161,245,59,136)(56,129,60,246,162)(65,274,169,254,147)(66,148,255,170,275)(67,276,171,256,149)(68,150,249,172,277)(69,278,173,250,151)(70,152,251,174,279)(71,280,175,252,145)(72,146,253,176,273)(81,126,288,263,268)(82,269,264,281,127)(83,128,282,257,270)(84,271,258,283,121)(85,122,284,259,272)(86,265,260,285,123)(87,124,286,261,266)(88,267,262,287,125)(113,144,318,156,290)(114,291,157,319,137)(115,138,320,158,292)(116,293,159,313,139)(117,140,314,160,294)(118,295,153,315,141)(119,142,316,154,296)(120,289,155,317,143)(201,215,230,310,237)(202,238,311,231,216)(203,209,232,312,239)(204,240,305,225,210)(205,211,226,306,233)(206,234,307,227,212)(207,213,228,308,235)(208,236,309,229,214), (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([[(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,235),(10,236),(11,237),(12,238),(13,239),(14,240),(15,233),(16,234),(25,44),(26,45),(27,46),(28,47),(29,48),(30,41),(31,42),(32,43),(33,190),(34,191),(35,192),(36,185),(37,186),(38,187),(39,188),(40,189),(49,296),(50,289),(51,290),(52,291),(53,292),(54,293),(55,294),(56,295),(57,320),(58,313),(59,314),(60,315),(61,316),(62,317),(63,318),(64,319),(65,285),(66,286),(67,287),(68,288),(69,281),(70,282),(71,283),(72,284),(73,225),(74,226),(75,227),(76,228),(77,229),(78,230),(79,231),(80,232),(81,172),(82,173),(83,174),(84,175),(85,176),(86,169),(87,170),(88,171),(89,223),(90,224),(91,217),(92,218),(93,219),(94,220),(95,221),(96,222),(97,178),(98,179),(99,180),(100,181),(101,182),(102,183),(103,184),(104,177),(105,306),(106,307),(107,308),(108,309),(109,310),(110,311),(111,312),(112,305),(113,165),(114,166),(115,167),(116,168),(117,161),(118,162),(119,163),(120,164),(121,280),(122,273),(123,274),(124,275),(125,276),(126,277),(127,278),(128,279),(129,153),(130,154),(131,155),(132,156),(133,157),(134,158),(135,159),(136,160),(137,242),(138,243),(139,244),(140,245),(141,246),(142,247),(143,248),(144,241),(145,258),(146,259),(147,260),(148,261),(149,262),(150,263),(151,264),(152,257),(193,206),(194,207),(195,208),(196,201),(197,202),(198,203),(199,204),(200,205),(209,302),(210,303),(211,304),(212,297),(213,298),(214,299),(215,300),(216,301),(249,268),(250,269),(251,270),(252,271),(253,272),(254,265),(255,266),(256,267)], [(1,118,197,257),(2,115,198,262),(3,120,199,259),(4,117,200,264),(5,114,193,261),(6,119,194,258),(7,116,195,263),(8,113,196,260),(9,271,97,296),(10,268,98,293),(11,265,99,290),(12,270,100,295),(13,267,101,292),(14,272,102,289),(15,269,103,294),(16,266,104,291),(17,164,204,146),(18,161,205,151),(19,166,206,148),(20,163,207,145),(21,168,208,150),(22,165,201,147),(23,162,202,152),(24,167,203,149),(25,132,310,169),(26,129,311,174),(27,134,312,171),(28,131,305,176),(29,136,306,173),(30,133,307,170),(31,130,308,175),(32,135,309,172),(33,242,212,66),(34,247,213,71),(35,244,214,68),(36,241,215,65),(37,246,216,70),(38,243,209,67),(39,248,210,72),(40,245,211,69),(41,157,106,87),(42,154,107,84),(43,159,108,81),(44,156,109,86),(45,153,110,83),(46,158,111,88),(47,155,112,85),(48,160,105,82),(49,235,252,178),(50,240,253,183),(51,237,254,180),(52,234,255,177),(53,239,256,182),(54,236,249,179),(55,233,250,184),(56,238,251,181),(57,232,276,219),(58,229,277,224),(59,226,278,221),(60,231,279,218),(61,228,280,223),(62,225,273,220),(63,230,274,217),(64,227,275,222),(73,122,94,317),(74,127,95,314),(75,124,96,319),(76,121,89,316),(77,126,90,313),(78,123,91,318),(79,128,92,315),(80,125,93,320),(137,297,286,190),(138,302,287,187),(139,299,288,192),(140,304,281,189),(141,301,282,186),(142,298,283,191),(143,303,284,188),(144,300,285,185)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112),(113,115,117,119),(114,116,118,120),(121,123,125,127),(122,124,126,128),(129,131,133,135),(130,132,134,136),(137,139,141,143),(138,140,142,144),(145,147,149,151),(146,148,150,152),(153,155,157,159),(154,156,158,160),(161,163,165,167),(162,164,166,168),(169,171,173,175),(170,172,174,176),(177,179,181,183),(178,180,182,184),(185,187,189,191),(186,188,190,192),(193,195,197,199),(194,196,198,200),(201,203,205,207),(202,204,206,208),(209,211,213,215),(210,212,214,216),(217,219,221,223),(218,220,222,224),(225,227,229,231),(226,228,230,232),(233,235,237,239),(234,236,238,240),(241,243,245,247),(242,244,246,248),(249,251,253,255),(250,252,254,256),(257,259,261,263),(258,260,262,264),(265,267,269,271),(266,268,270,272),(273,275,277,279),(274,276,278,280),(281,283,285,287),(282,284,286,288),(289,291,293,295),(290,292,294,296),(297,299,301,303),(298,300,302,304),(305,307,309,311),(306,308,310,312),(313,315,317,319),(314,316,318,320)], [(1,100,45,92,186),(2,187,93,46,101),(3,102,47,94,188),(4,189,95,48,103),(5,104,41,96,190),(6,191,89,42,97),(7,98,43,90,192),(8,185,91,44,99),(9,194,298,76,107),(10,108,77,299,195),(11,196,300,78,109),(12,110,79,301,197),(13,198,302,80,111),(14,112,73,303,199),(15,200,304,74,105),(16,106,75,297,193),(17,183,28,220,39),(18,40,221,29,184),(19,177,30,222,33),(20,34,223,31,178),(21,179,32,224,35),(22,36,217,25,180),(23,181,26,218,37),(24,38,219,27,182),(49,163,247,61,130),(50,131,62,248,164),(51,165,241,63,132),(52,133,64,242,166),(53,167,243,57,134),(54,135,58,244,168),(55,161,245,59,136),(56,129,60,246,162),(65,274,169,254,147),(66,148,255,170,275),(67,276,171,256,149),(68,150,249,172,277),(69,278,173,250,151),(70,152,251,174,279),(71,280,175,252,145),(72,146,253,176,273),(81,126,288,263,268),(82,269,264,281,127),(83,128,282,257,270),(84,271,258,283,121),(85,122,284,259,272),(86,265,260,285,123),(87,124,286,261,266),(88,267,262,287,125),(113,144,318,156,290),(114,291,157,319,137),(115,138,320,158,292),(116,293,159,313,139),(117,140,314,160,294),(118,295,153,315,141),(119,142,316,154,296),(120,289,155,317,143),(201,215,230,310,237),(202,238,311,231,216),(203,209,232,312,239),(204,240,305,225,210),(205,211,226,306,233),(206,234,307,227,212),(207,213,228,308,235),(208,236,309,229,214)], [(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)]])

104 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4P 5A 5B 8A ··· 8P 10A ··· 10N 20A ··· 20AV order 1 2 ··· 2 4 ··· 4 4 ··· 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 ··· 1 1 ··· 1 2 ··· 2 2 2 10 ··· 10 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + - + - + image C1 C2 C2 C2 C4 C4 C4 D5 M4(2) Dic5 D10 Dic5 D10 C4×D5 C4.Dic5 kernel C2×C42.D5 C42.D5 C22×C5⋊2C8 C2×C4×C20 C2×C5⋊2C8 C4×C20 C22×C20 C2×C42 C2×C10 C42 C42 C22×C4 C22×C4 C2×C4 C22 # reps 1 4 2 1 16 4 4 2 8 4 4 4 2 16 32

Matrix representation of C2×C42.D5 in GL4(𝔽41) generated by

 40 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 32 0 0 0 0 1 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 32 0 0 0 0 32
,
 1 0 0 0 0 1 0 0 0 0 18 0 0 0 0 16
,
 1 0 0 0 0 40 0 0 0 0 0 16 0 0 2 0
G:=sub<GL(4,GF(41))| [40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,32,0,0,0,0,1,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,32],[1,0,0,0,0,1,0,0,0,0,18,0,0,0,0,16],[1,0,0,0,0,40,0,0,0,0,0,2,0,0,16,0] >;

C2×C42.D5 in GAP, Magma, Sage, TeX

C_2\times C_4^2.D_5
% in TeX

G:=Group("C2xC4^2.D5");
// GroupNames label

G:=SmallGroup(320,548);
// by ID

G=gap.SmallGroup(320,548);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,100,136,12550]);
// Polycyclic

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

׿
×
𝔽