metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C20).287D4, C10.26(C4⋊Q8), (C2×Dic5).5Q8, C22.46(Q8×D5), C2.8(C20⋊2D4), (C2×Dic5).67D4, (C22×C4).41D10, C22.245(D4×D5), C10.89(C4⋊D4), C10.45(C22⋊Q8), C2.21(D10⋊Q8), C2.6(Dic5⋊Q8), (C22×C20).29C22, C10.19(C42.C2), C23.375(C22×D5), C22.103(C4○D20), C22.99(D4⋊2D5), (C22×C10).344C23, C5⋊3(C23.81C23), C10.10C42.18C2, C2.11(Dic5.Q8), C10.62(C22.D4), (C22×Dic5).53C22, C2.12(C23.23D10), (C2×C4⋊C4).17D5, (C10×C4⋊C4).31C2, (C2×C10).79(C2×Q8), (C2×C10).448(C2×D4), (C2×C4).36(C5⋊D4), C22.135(C2×C5⋊D4), (C2×C10).154(C4○D4), (C2×C10.D4).16C2, SmallGroup(320,607)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C20).287D4
G = < a,b,c,d | a2=b20=c4=1, d2=b10, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=ab9, dcd-1=ab10c-1 >
Subgroups: 486 in 150 conjugacy classes, 61 normal (25 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2.C42, C2×C4⋊C4, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.81C23, C10.D4, C5×C4⋊C4, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C10.10C42, C10.10C42, C2×C10.D4, C2×C10.D4, C10×C4⋊C4, (C2×C20).287D4
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C5⋊D4, C22×D5, C23.81C23, C4○D20, D4×D5, D4⋊2D5, Q8×D5, C2×C5⋊D4, Dic5.Q8, D10⋊Q8, C23.23D10, C20⋊2D4, Dic5⋊Q8, (C2×C20).287D4
►Regular action on 320 points
Generators in S320
(1 153)(2 154)(3 155)(4 156)(5 157)(6 158)(7 159)(8 160)(9 141)(10 142)(11 143)(12 144)(13 145)(14 146)(15 147)(16 148)(17 149)(18 150)(19 151)(20 152)(21 239)(22 240)(23 221)(24 222)(25 223)(26 224)(27 225)(28 226)(29 227)(30 228)(31 229)(32 230)(33 231)(34 232)(35 233)(36 234)(37 235)(38 236)(39 237)(40 238)(41 124)(42 125)(43 126)(44 127)(45 128)(46 129)(47 130)(48 131)(49 132)(50 133)(51 134)(52 135)(53 136)(54 137)(55 138)(56 139)(57 140)(58 121)(59 122)(60 123)(61 200)(62 181)(63 182)(64 183)(65 184)(66 185)(67 186)(68 187)(69 188)(70 189)(71 190)(72 191)(73 192)(74 193)(75 194)(76 195)(77 196)(78 197)(79 198)(80 199)(81 162)(82 163)(83 164)(84 165)(85 166)(86 167)(87 168)(88 169)(89 170)(90 171)(91 172)(92 173)(93 174)(94 175)(95 176)(96 177)(97 178)(98 179)(99 180)(100 161)(101 302)(102 303)(103 304)(104 305)(105 306)(106 307)(107 308)(108 309)(109 310)(110 311)(111 312)(112 313)(113 314)(114 315)(115 316)(116 317)(117 318)(118 319)(119 320)(120 301)(201 245)(202 246)(203 247)(204 248)(205 249)(206 250)(207 251)(208 252)(209 253)(210 254)(211 255)(212 256)(213 257)(214 258)(215 259)(216 260)(217 241)(218 242)(219 243)(220 244)(261 286)(262 287)(263 288)(264 289)(265 290)(266 291)(267 292)(268 293)(269 294)(270 295)(271 296)(272 297)(273 298)(274 299)(275 300)(276 281)(277 282)(278 283)(279 284)(280 285)
(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)
(1 49 288 21)(2 131 289 238)(3 47 290 39)(4 129 291 236)(5 45 292 37)(6 127 293 234)(7 43 294 35)(8 125 295 232)(9 41 296 33)(10 123 297 230)(11 59 298 31)(12 121 299 228)(13 57 300 29)(14 139 281 226)(15 55 282 27)(16 137 283 224)(17 53 284 25)(18 135 285 222)(19 51 286 23)(20 133 287 240)(22 152 50 262)(24 150 52 280)(26 148 54 278)(28 146 56 276)(30 144 58 274)(32 142 60 272)(34 160 42 270)(36 158 44 268)(38 156 46 266)(40 154 48 264)(61 208 307 93)(62 251 308 173)(63 206 309 91)(64 249 310 171)(65 204 311 89)(66 247 312 169)(67 202 313 87)(68 245 314 167)(69 220 315 85)(70 243 316 165)(71 218 317 83)(72 241 318 163)(73 216 319 81)(74 259 320 161)(75 214 301 99)(76 257 302 179)(77 212 303 97)(78 255 304 177)(79 210 305 95)(80 253 306 175)(82 191 217 117)(84 189 219 115)(86 187 201 113)(88 185 203 111)(90 183 205 109)(92 181 207 107)(94 199 209 105)(96 197 211 103)(98 195 213 101)(100 193 215 119)(102 178 196 256)(104 176 198 254)(106 174 200 252)(108 172 182 250)(110 170 184 248)(112 168 186 246)(114 166 188 244)(116 164 190 242)(118 162 192 260)(120 180 194 258)(122 273 229 143)(124 271 231 141)(126 269 233 159)(128 267 235 157)(130 265 237 155)(132 263 239 153)(134 261 221 151)(136 279 223 149)(138 277 225 147)(140 275 227 145)
(1 316 11 306)(2 104 12 114)(3 314 13 304)(4 102 14 112)(5 312 15 302)(6 120 16 110)(7 310 17 320)(8 118 18 108)(9 308 19 318)(10 116 20 106)(21 94 31 84)(22 164 32 174)(23 92 33 82)(24 162 34 172)(25 90 35 100)(26 180 36 170)(27 88 37 98)(28 178 38 168)(29 86 39 96)(30 176 40 166)(41 217 51 207)(42 250 52 260)(43 215 53 205)(44 248 54 258)(45 213 55 203)(46 246 56 256)(47 211 57 201)(48 244 58 254)(49 209 59 219)(50 242 60 252)(61 272 71 262)(62 286 72 296)(63 270 73 280)(64 284 74 294)(65 268 75 278)(66 282 76 292)(67 266 77 276)(68 300 78 290)(69 264 79 274)(70 298 80 288)(81 232 91 222)(83 230 93 240)(85 228 95 238)(87 226 97 236)(89 224 99 234)(101 157 111 147)(103 155 113 145)(105 153 115 143)(107 151 117 141)(109 149 119 159)(121 210 131 220)(122 243 132 253)(123 208 133 218)(124 241 134 251)(125 206 135 216)(126 259 136 249)(127 204 137 214)(128 257 138 247)(129 202 139 212)(130 255 140 245)(142 317 152 307)(144 315 154 305)(146 313 156 303)(148 311 158 301)(150 309 160 319)(161 223 171 233)(163 221 173 231)(165 239 175 229)(167 237 177 227)(169 235 179 225)(181 261 191 271)(182 295 192 285)(183 279 193 269)(184 293 194 283)(185 277 195 267)(186 291 196 281)(187 275 197 265)(188 289 198 299)(189 273 199 263)(190 287 200 297)
G:=sub<Sym(320)| (1,153)(2,154)(3,155)(4,156)(5,157)(6,158)(7,159)(8,160)(9,141)(10,142)(11,143)(12,144)(13,145)(14,146)(15,147)(16,148)(17,149)(18,150)(19,151)(20,152)(21,239)(22,240)(23,221)(24,222)(25,223)(26,224)(27,225)(28,226)(29,227)(30,228)(31,229)(32,230)(33,231)(34,232)(35,233)(36,234)(37,235)(38,236)(39,237)(40,238)(41,124)(42,125)(43,126)(44,127)(45,128)(46,129)(47,130)(48,131)(49,132)(50,133)(51,134)(52,135)(53,136)(54,137)(55,138)(56,139)(57,140)(58,121)(59,122)(60,123)(61,200)(62,181)(63,182)(64,183)(65,184)(66,185)(67,186)(68,187)(69,188)(70,189)(71,190)(72,191)(73,192)(74,193)(75,194)(76,195)(77,196)(78,197)(79,198)(80,199)(81,162)(82,163)(83,164)(84,165)(85,166)(86,167)(87,168)(88,169)(89,170)(90,171)(91,172)(92,173)(93,174)(94,175)(95,176)(96,177)(97,178)(98,179)(99,180)(100,161)(101,302)(102,303)(103,304)(104,305)(105,306)(106,307)(107,308)(108,309)(109,310)(110,311)(111,312)(112,313)(113,314)(114,315)(115,316)(116,317)(117,318)(118,319)(119,320)(120,301)(201,245)(202,246)(203,247)(204,248)(205,249)(206,250)(207,251)(208,252)(209,253)(210,254)(211,255)(212,256)(213,257)(214,258)(215,259)(216,260)(217,241)(218,242)(219,243)(220,244)(261,286)(262,287)(263,288)(264,289)(265,290)(266,291)(267,292)(268,293)(269,294)(270,295)(271,296)(272,297)(273,298)(274,299)(275,300)(276,281)(277,282)(278,283)(279,284)(280,285), (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), (1,49,288,21)(2,131,289,238)(3,47,290,39)(4,129,291,236)(5,45,292,37)(6,127,293,234)(7,43,294,35)(8,125,295,232)(9,41,296,33)(10,123,297,230)(11,59,298,31)(12,121,299,228)(13,57,300,29)(14,139,281,226)(15,55,282,27)(16,137,283,224)(17,53,284,25)(18,135,285,222)(19,51,286,23)(20,133,287,240)(22,152,50,262)(24,150,52,280)(26,148,54,278)(28,146,56,276)(30,144,58,274)(32,142,60,272)(34,160,42,270)(36,158,44,268)(38,156,46,266)(40,154,48,264)(61,208,307,93)(62,251,308,173)(63,206,309,91)(64,249,310,171)(65,204,311,89)(66,247,312,169)(67,202,313,87)(68,245,314,167)(69,220,315,85)(70,243,316,165)(71,218,317,83)(72,241,318,163)(73,216,319,81)(74,259,320,161)(75,214,301,99)(76,257,302,179)(77,212,303,97)(78,255,304,177)(79,210,305,95)(80,253,306,175)(82,191,217,117)(84,189,219,115)(86,187,201,113)(88,185,203,111)(90,183,205,109)(92,181,207,107)(94,199,209,105)(96,197,211,103)(98,195,213,101)(100,193,215,119)(102,178,196,256)(104,176,198,254)(106,174,200,252)(108,172,182,250)(110,170,184,248)(112,168,186,246)(114,166,188,244)(116,164,190,242)(118,162,192,260)(120,180,194,258)(122,273,229,143)(124,271,231,141)(126,269,233,159)(128,267,235,157)(130,265,237,155)(132,263,239,153)(134,261,221,151)(136,279,223,149)(138,277,225,147)(140,275,227,145), (1,316,11,306)(2,104,12,114)(3,314,13,304)(4,102,14,112)(5,312,15,302)(6,120,16,110)(7,310,17,320)(8,118,18,108)(9,308,19,318)(10,116,20,106)(21,94,31,84)(22,164,32,174)(23,92,33,82)(24,162,34,172)(25,90,35,100)(26,180,36,170)(27,88,37,98)(28,178,38,168)(29,86,39,96)(30,176,40,166)(41,217,51,207)(42,250,52,260)(43,215,53,205)(44,248,54,258)(45,213,55,203)(46,246,56,256)(47,211,57,201)(48,244,58,254)(49,209,59,219)(50,242,60,252)(61,272,71,262)(62,286,72,296)(63,270,73,280)(64,284,74,294)(65,268,75,278)(66,282,76,292)(67,266,77,276)(68,300,78,290)(69,264,79,274)(70,298,80,288)(81,232,91,222)(83,230,93,240)(85,228,95,238)(87,226,97,236)(89,224,99,234)(101,157,111,147)(103,155,113,145)(105,153,115,143)(107,151,117,141)(109,149,119,159)(121,210,131,220)(122,243,132,253)(123,208,133,218)(124,241,134,251)(125,206,135,216)(126,259,136,249)(127,204,137,214)(128,257,138,247)(129,202,139,212)(130,255,140,245)(142,317,152,307)(144,315,154,305)(146,313,156,303)(148,311,158,301)(150,309,160,319)(161,223,171,233)(163,221,173,231)(165,239,175,229)(167,237,177,227)(169,235,179,225)(181,261,191,271)(182,295,192,285)(183,279,193,269)(184,293,194,283)(185,277,195,267)(186,291,196,281)(187,275,197,265)(188,289,198,299)(189,273,199,263)(190,287,200,297)>;
G:=Group( (1,153)(2,154)(3,155)(4,156)(5,157)(6,158)(7,159)(8,160)(9,141)(10,142)(11,143)(12,144)(13,145)(14,146)(15,147)(16,148)(17,149)(18,150)(19,151)(20,152)(21,239)(22,240)(23,221)(24,222)(25,223)(26,224)(27,225)(28,226)(29,227)(30,228)(31,229)(32,230)(33,231)(34,232)(35,233)(36,234)(37,235)(38,236)(39,237)(40,238)(41,124)(42,125)(43,126)(44,127)(45,128)(46,129)(47,130)(48,131)(49,132)(50,133)(51,134)(52,135)(53,136)(54,137)(55,138)(56,139)(57,140)(58,121)(59,122)(60,123)(61,200)(62,181)(63,182)(64,183)(65,184)(66,185)(67,186)(68,187)(69,188)(70,189)(71,190)(72,191)(73,192)(74,193)(75,194)(76,195)(77,196)(78,197)(79,198)(80,199)(81,162)(82,163)(83,164)(84,165)(85,166)(86,167)(87,168)(88,169)(89,170)(90,171)(91,172)(92,173)(93,174)(94,175)(95,176)(96,177)(97,178)(98,179)(99,180)(100,161)(101,302)(102,303)(103,304)(104,305)(105,306)(106,307)(107,308)(108,309)(109,310)(110,311)(111,312)(112,313)(113,314)(114,315)(115,316)(116,317)(117,318)(118,319)(119,320)(120,301)(201,245)(202,246)(203,247)(204,248)(205,249)(206,250)(207,251)(208,252)(209,253)(210,254)(211,255)(212,256)(213,257)(214,258)(215,259)(216,260)(217,241)(218,242)(219,243)(220,244)(261,286)(262,287)(263,288)(264,289)(265,290)(266,291)(267,292)(268,293)(269,294)(270,295)(271,296)(272,297)(273,298)(274,299)(275,300)(276,281)(277,282)(278,283)(279,284)(280,285), (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), (1,49,288,21)(2,131,289,238)(3,47,290,39)(4,129,291,236)(5,45,292,37)(6,127,293,234)(7,43,294,35)(8,125,295,232)(9,41,296,33)(10,123,297,230)(11,59,298,31)(12,121,299,228)(13,57,300,29)(14,139,281,226)(15,55,282,27)(16,137,283,224)(17,53,284,25)(18,135,285,222)(19,51,286,23)(20,133,287,240)(22,152,50,262)(24,150,52,280)(26,148,54,278)(28,146,56,276)(30,144,58,274)(32,142,60,272)(34,160,42,270)(36,158,44,268)(38,156,46,266)(40,154,48,264)(61,208,307,93)(62,251,308,173)(63,206,309,91)(64,249,310,171)(65,204,311,89)(66,247,312,169)(67,202,313,87)(68,245,314,167)(69,220,315,85)(70,243,316,165)(71,218,317,83)(72,241,318,163)(73,216,319,81)(74,259,320,161)(75,214,301,99)(76,257,302,179)(77,212,303,97)(78,255,304,177)(79,210,305,95)(80,253,306,175)(82,191,217,117)(84,189,219,115)(86,187,201,113)(88,185,203,111)(90,183,205,109)(92,181,207,107)(94,199,209,105)(96,197,211,103)(98,195,213,101)(100,193,215,119)(102,178,196,256)(104,176,198,254)(106,174,200,252)(108,172,182,250)(110,170,184,248)(112,168,186,246)(114,166,188,244)(116,164,190,242)(118,162,192,260)(120,180,194,258)(122,273,229,143)(124,271,231,141)(126,269,233,159)(128,267,235,157)(130,265,237,155)(132,263,239,153)(134,261,221,151)(136,279,223,149)(138,277,225,147)(140,275,227,145), (1,316,11,306)(2,104,12,114)(3,314,13,304)(4,102,14,112)(5,312,15,302)(6,120,16,110)(7,310,17,320)(8,118,18,108)(9,308,19,318)(10,116,20,106)(21,94,31,84)(22,164,32,174)(23,92,33,82)(24,162,34,172)(25,90,35,100)(26,180,36,170)(27,88,37,98)(28,178,38,168)(29,86,39,96)(30,176,40,166)(41,217,51,207)(42,250,52,260)(43,215,53,205)(44,248,54,258)(45,213,55,203)(46,246,56,256)(47,211,57,201)(48,244,58,254)(49,209,59,219)(50,242,60,252)(61,272,71,262)(62,286,72,296)(63,270,73,280)(64,284,74,294)(65,268,75,278)(66,282,76,292)(67,266,77,276)(68,300,78,290)(69,264,79,274)(70,298,80,288)(81,232,91,222)(83,230,93,240)(85,228,95,238)(87,226,97,236)(89,224,99,234)(101,157,111,147)(103,155,113,145)(105,153,115,143)(107,151,117,141)(109,149,119,159)(121,210,131,220)(122,243,132,253)(123,208,133,218)(124,241,134,251)(125,206,135,216)(126,259,136,249)(127,204,137,214)(128,257,138,247)(129,202,139,212)(130,255,140,245)(142,317,152,307)(144,315,154,305)(146,313,156,303)(148,311,158,301)(150,309,160,319)(161,223,171,233)(163,221,173,231)(165,239,175,229)(167,237,177,227)(169,235,179,225)(181,261,191,271)(182,295,192,285)(183,279,193,269)(184,293,194,283)(185,277,195,267)(186,291,196,281)(187,275,197,265)(188,289,198,299)(189,273,199,263)(190,287,200,297) );
G=PermutationGroup([[(1,153),(2,154),(3,155),(4,156),(5,157),(6,158),(7,159),(8,160),(9,141),(10,142),(11,143),(12,144),(13,145),(14,146),(15,147),(16,148),(17,149),(18,150),(19,151),(20,152),(21,239),(22,240),(23,221),(24,222),(25,223),(26,224),(27,225),(28,226),(29,227),(30,228),(31,229),(32,230),(33,231),(34,232),(35,233),(36,234),(37,235),(38,236),(39,237),(40,238),(41,124),(42,125),(43,126),(44,127),(45,128),(46,129),(47,130),(48,131),(49,132),(50,133),(51,134),(52,135),(53,136),(54,137),(55,138),(56,139),(57,140),(58,121),(59,122),(60,123),(61,200),(62,181),(63,182),(64,183),(65,184),(66,185),(67,186),(68,187),(69,188),(70,189),(71,190),(72,191),(73,192),(74,193),(75,194),(76,195),(77,196),(78,197),(79,198),(80,199),(81,162),(82,163),(83,164),(84,165),(85,166),(86,167),(87,168),(88,169),(89,170),(90,171),(91,172),(92,173),(93,174),(94,175),(95,176),(96,177),(97,178),(98,179),(99,180),(100,161),(101,302),(102,303),(103,304),(104,305),(105,306),(106,307),(107,308),(108,309),(109,310),(110,311),(111,312),(112,313),(113,314),(114,315),(115,316),(116,317),(117,318),(118,319),(119,320),(120,301),(201,245),(202,246),(203,247),(204,248),(205,249),(206,250),(207,251),(208,252),(209,253),(210,254),(211,255),(212,256),(213,257),(214,258),(215,259),(216,260),(217,241),(218,242),(219,243),(220,244),(261,286),(262,287),(263,288),(264,289),(265,290),(266,291),(267,292),(268,293),(269,294),(270,295),(271,296),(272,297),(273,298),(274,299),(275,300),(276,281),(277,282),(278,283),(279,284),(280,285)], [(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)], [(1,49,288,21),(2,131,289,238),(3,47,290,39),(4,129,291,236),(5,45,292,37),(6,127,293,234),(7,43,294,35),(8,125,295,232),(9,41,296,33),(10,123,297,230),(11,59,298,31),(12,121,299,228),(13,57,300,29),(14,139,281,226),(15,55,282,27),(16,137,283,224),(17,53,284,25),(18,135,285,222),(19,51,286,23),(20,133,287,240),(22,152,50,262),(24,150,52,280),(26,148,54,278),(28,146,56,276),(30,144,58,274),(32,142,60,272),(34,160,42,270),(36,158,44,268),(38,156,46,266),(40,154,48,264),(61,208,307,93),(62,251,308,173),(63,206,309,91),(64,249,310,171),(65,204,311,89),(66,247,312,169),(67,202,313,87),(68,245,314,167),(69,220,315,85),(70,243,316,165),(71,218,317,83),(72,241,318,163),(73,216,319,81),(74,259,320,161),(75,214,301,99),(76,257,302,179),(77,212,303,97),(78,255,304,177),(79,210,305,95),(80,253,306,175),(82,191,217,117),(84,189,219,115),(86,187,201,113),(88,185,203,111),(90,183,205,109),(92,181,207,107),(94,199,209,105),(96,197,211,103),(98,195,213,101),(100,193,215,119),(102,178,196,256),(104,176,198,254),(106,174,200,252),(108,172,182,250),(110,170,184,248),(112,168,186,246),(114,166,188,244),(116,164,190,242),(118,162,192,260),(120,180,194,258),(122,273,229,143),(124,271,231,141),(126,269,233,159),(128,267,235,157),(130,265,237,155),(132,263,239,153),(134,261,221,151),(136,279,223,149),(138,277,225,147),(140,275,227,145)], [(1,316,11,306),(2,104,12,114),(3,314,13,304),(4,102,14,112),(5,312,15,302),(6,120,16,110),(7,310,17,320),(8,118,18,108),(9,308,19,318),(10,116,20,106),(21,94,31,84),(22,164,32,174),(23,92,33,82),(24,162,34,172),(25,90,35,100),(26,180,36,170),(27,88,37,98),(28,178,38,168),(29,86,39,96),(30,176,40,166),(41,217,51,207),(42,250,52,260),(43,215,53,205),(44,248,54,258),(45,213,55,203),(46,246,56,256),(47,211,57,201),(48,244,58,254),(49,209,59,219),(50,242,60,252),(61,272,71,262),(62,286,72,296),(63,270,73,280),(64,284,74,294),(65,268,75,278),(66,282,76,292),(67,266,77,276),(68,300,78,290),(69,264,79,274),(70,298,80,288),(81,232,91,222),(83,230,93,240),(85,228,95,238),(87,226,97,236),(89,224,99,234),(101,157,111,147),(103,155,113,145),(105,153,115,143),(107,151,117,141),(109,149,119,159),(121,210,131,220),(122,243,132,253),(123,208,133,218),(124,241,134,251),(125,206,135,216),(126,259,136,249),(127,204,137,214),(128,257,138,247),(129,202,139,212),(130,255,140,245),(142,317,152,307),(144,315,154,305),(146,313,156,303),(148,311,158,301),(150,309,160,319),(161,223,171,233),(163,221,173,231),(165,239,175,229),(167,237,177,227),(169,235,179,225),(181,261,191,271),(182,295,192,285),(183,279,193,269),(184,293,194,283),(185,277,195,267),(186,291,196,281),(187,275,197,265),(188,289,198,299),(189,273,199,263),(190,287,200,297)]])
62 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4F | 4G | ··· | 4N | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | D4 | Q8 | D4 | D5 | C4○D4 | D10 | C5⋊D4 | C4○D20 | D4×D5 | D4⋊2D5 | Q8×D5 |
| kernel | (C2×C20).287D4 | C10.10C42 | C2×C10.D4 | C10×C4⋊C4 | C2×Dic5 | C2×Dic5 | C2×C20 | C2×C4⋊C4 | C2×C10 | C22×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 3 | 3 | 1 | 2 | 4 | 2 | 2 | 6 | 6 | 8 | 16 | 2 | 2 | 4 |
Matrix representation of (C2×C20).287D4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 23 |
| 0 | 0 | 0 | 0 | 18 | 35 |
| 0 | 19 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 16 | 0 | 0 |
| 0 | 0 | 14 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 16 |
| 0 | 0 | 0 | 0 | 2 | 16 |
| 30 | 22 | 0 | 0 | 0 | 0 |
| 28 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 38 | 0 | 0 |
| 0 | 0 | 20 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 0 | 0 | 1 | 35 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,13,0,0,0,0,22,0,0,0,0,0,0,0,0,40,0,0,0,0,1,7,0,0,0,0,0,0,20,18,0,0,0,0,23,35],[0,28,0,0,0,0,19,0,0,0,0,0,0,0,29,14,0,0,0,0,16,12,0,0,0,0,0,0,25,2,0,0,0,0,16,16],[30,28,0,0,0,0,22,11,0,0,0,0,0,0,15,20,0,0,0,0,38,26,0,0,0,0,0,0,6,1,0,0,0,0,6,35] >;
(C2×C20).287D4 in GAP, Magma, Sage, TeX
(C_2\times C_{20})._{287}D_4 % in TeX
G:=Group("(C2xC20).287D4"); // GroupNames label
G:=SmallGroup(320,607);
// by ID
G=gap.SmallGroup(320,607);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,232,254,387,100,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=a*b^9,d*c*d^-1=a*b^10*c^-1>;
// generators/relations