metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C20).288D4, (C2×Dic5).6Q8, C22.47(Q8×D5), (C22×C4).99D10, C2.6(D10⋊3Q8), C10.71(C22⋊Q8), C2.7(C20.17D4), C10.40(C4.4D4), C10.21(C42.C2), C23.376(C22×D5), C10.26(C42⋊2C2), C22.104(C4○D20), (C22×C20).392C22, (C22×C10).346C23, C5⋊4(C23.83C23), C22.48(Q8⋊2D5), C22.100(D4⋊2D5), C10.10C42.19C2, C2.12(Dic5.Q8), C10.63(C22.D4), (C22×Dic5).54C22, C2.13(C23.23D10), (C2×C4⋊C4).19D5, (C10×C4⋊C4).32C2, (C2×C10).81(C2×Q8), (C2×C10).449(C2×D4), (C2×C4).37(C5⋊D4), C2.12(C4⋊C4⋊D5), C22.136(C2×C5⋊D4), (C2×C10).155(C4○D4), (C2×C10.D4).33C2, SmallGroup(320,609)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C20).288D4
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=ac-1 >
Subgroups: 438 in 134 conjugacy classes, 57 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, C23.83C23, C10.D4, C5×C4⋊C4, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C10.10C42, C10.10C42, C2×C10.D4, C10×C4⋊C4, (C2×C20).288D4
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C5⋊D4, C22×D5, C23.83C23, C4○D20, D4⋊2D5, Q8×D5, Q8⋊2D5, C2×C5⋊D4, Dic5.Q8, C4⋊C4⋊D5, C23.23D10, C20.17D4, D10⋊3Q8, (C2×C20).288D4
►Regular action on 320 points
Generators in S320
(1 227)(2 228)(3 229)(4 230)(5 231)(6 232)(7 233)(8 234)(9 235)(10 236)(11 237)(12 238)(13 239)(14 240)(15 221)(16 222)(17 223)(18 224)(19 225)(20 226)(21 99)(22 100)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)(41 157)(42 158)(43 159)(44 160)(45 141)(46 142)(47 143)(48 144)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 151)(56 152)(57 153)(58 154)(59 155)(60 156)(61 120)(62 101)(63 102)(64 103)(65 104)(66 105)(67 106)(68 107)(69 108)(70 109)(71 110)(72 111)(73 112)(74 113)(75 114)(76 115)(77 116)(78 117)(79 118)(80 119)(121 186)(122 187)(123 188)(124 189)(125 190)(126 191)(127 192)(128 193)(129 194)(130 195)(131 196)(132 197)(133 198)(134 199)(135 200)(136 181)(137 182)(138 183)(139 184)(140 185)(161 298)(162 299)(163 300)(164 281)(165 282)(166 283)(167 284)(168 285)(169 286)(170 287)(171 288)(172 289)(173 290)(174 291)(175 292)(176 293)(177 294)(178 295)(179 296)(180 297)(201 274)(202 275)(203 276)(204 277)(205 278)(206 279)(207 280)(208 261)(209 262)(210 263)(211 264)(212 265)(213 266)(214 267)(215 268)(216 269)(217 270)(218 271)(219 272)(220 273)(241 316)(242 317)(243 318)(244 319)(245 320)(246 301)(247 302)(248 303)(249 304)(250 305)(251 306)(252 307)(253 308)(254 309)(255 310)(256 311)(257 312)(258 313)(259 314)(260 315)
(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)
(1 158 84 313)(2 41 85 257)(3 156 86 311)(4 59 87 255)(5 154 88 309)(6 57 89 253)(7 152 90 307)(8 55 91 251)(9 150 92 305)(10 53 93 249)(11 148 94 303)(12 51 95 247)(13 146 96 301)(14 49 97 245)(15 144 98 319)(16 47 99 243)(17 142 100 317)(18 45 81 241)(19 160 82 315)(20 43 83 259)(21 318 222 143)(22 242 223 46)(23 316 224 141)(24 260 225 44)(25 314 226 159)(26 258 227 42)(27 312 228 157)(28 256 229 60)(29 310 230 155)(30 254 231 58)(31 308 232 153)(32 252 233 56)(33 306 234 151)(34 250 235 54)(35 304 236 149)(36 248 237 52)(37 302 238 147)(38 246 239 50)(39 320 240 145)(40 244 221 48)(61 273 163 187)(62 219 164 121)(63 271 165 185)(64 217 166 139)(65 269 167 183)(66 215 168 137)(67 267 169 181)(68 213 170 135)(69 265 171 199)(70 211 172 133)(71 263 173 197)(72 209 174 131)(73 261 175 195)(74 207 176 129)(75 279 177 193)(76 205 178 127)(77 277 179 191)(78 203 180 125)(79 275 161 189)(80 201 162 123)(101 272 281 186)(102 218 282 140)(103 270 283 184)(104 216 284 138)(105 268 285 182)(106 214 286 136)(107 266 287 200)(108 212 288 134)(109 264 289 198)(110 210 290 132)(111 262 291 196)(112 208 292 130)(113 280 293 194)(114 206 294 128)(115 278 295 192)(116 204 296 126)(117 276 297 190)(118 202 298 124)(119 274 299 188)(120 220 300 122)
(1 116 11 106)(2 66 12 76)(3 114 13 104)(4 64 14 74)(5 112 15 102)(6 62 16 72)(7 110 17 120)(8 80 18 70)(9 108 19 118)(10 78 20 68)(21 291 31 281)(22 163 32 173)(23 289 33 299)(24 161 34 171)(25 287 35 297)(26 179 36 169)(27 285 37 295)(28 177 38 167)(29 283 39 293)(30 175 40 165)(41 182 51 192)(42 126 52 136)(43 200 53 190)(44 124 54 134)(45 198 55 188)(46 122 56 132)(47 196 57 186)(48 140 58 130)(49 194 59 184)(50 138 60 128)(61 233 71 223)(63 231 73 221)(65 229 75 239)(67 227 77 237)(69 225 79 235)(81 172 91 162)(82 298 92 288)(83 170 93 180)(84 296 94 286)(85 168 95 178)(86 294 96 284)(87 166 97 176)(88 292 98 282)(89 164 99 174)(90 290 100 300)(101 222 111 232)(103 240 113 230)(105 238 115 228)(107 236 117 226)(109 234 119 224)(121 143 131 153)(123 141 133 151)(125 159 135 149)(127 157 137 147)(129 155 139 145)(142 187 152 197)(144 185 154 195)(146 183 156 193)(148 181 158 191)(150 199 160 189)(201 316 211 306)(202 250 212 260)(203 314 213 304)(204 248 214 258)(205 312 215 302)(206 246 216 256)(207 310 217 320)(208 244 218 254)(209 308 219 318)(210 242 220 252)(241 264 251 274)(243 262 253 272)(245 280 255 270)(247 278 257 268)(249 276 259 266)(261 319 271 309)(263 317 273 307)(265 315 275 305)(267 313 277 303)(269 311 279 301)
G:=sub<Sym(320)| (1,227)(2,228)(3,229)(4,230)(5,231)(6,232)(7,233)(8,234)(9,235)(10,236)(11,237)(12,238)(13,239)(14,240)(15,221)(16,222)(17,223)(18,224)(19,225)(20,226)(21,99)(22,100)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,157)(42,158)(43,159)(44,160)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,153)(58,154)(59,155)(60,156)(61,120)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(121,186)(122,187)(123,188)(124,189)(125,190)(126,191)(127,192)(128,193)(129,194)(130,195)(131,196)(132,197)(133,198)(134,199)(135,200)(136,181)(137,182)(138,183)(139,184)(140,185)(161,298)(162,299)(163,300)(164,281)(165,282)(166,283)(167,284)(168,285)(169,286)(170,287)(171,288)(172,289)(173,290)(174,291)(175,292)(176,293)(177,294)(178,295)(179,296)(180,297)(201,274)(202,275)(203,276)(204,277)(205,278)(206,279)(207,280)(208,261)(209,262)(210,263)(211,264)(212,265)(213,266)(214,267)(215,268)(216,269)(217,270)(218,271)(219,272)(220,273)(241,316)(242,317)(243,318)(244,319)(245,320)(246,301)(247,302)(248,303)(249,304)(250,305)(251,306)(252,307)(253,308)(254,309)(255,310)(256,311)(257,312)(258,313)(259,314)(260,315), (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), (1,158,84,313)(2,41,85,257)(3,156,86,311)(4,59,87,255)(5,154,88,309)(6,57,89,253)(7,152,90,307)(8,55,91,251)(9,150,92,305)(10,53,93,249)(11,148,94,303)(12,51,95,247)(13,146,96,301)(14,49,97,245)(15,144,98,319)(16,47,99,243)(17,142,100,317)(18,45,81,241)(19,160,82,315)(20,43,83,259)(21,318,222,143)(22,242,223,46)(23,316,224,141)(24,260,225,44)(25,314,226,159)(26,258,227,42)(27,312,228,157)(28,256,229,60)(29,310,230,155)(30,254,231,58)(31,308,232,153)(32,252,233,56)(33,306,234,151)(34,250,235,54)(35,304,236,149)(36,248,237,52)(37,302,238,147)(38,246,239,50)(39,320,240,145)(40,244,221,48)(61,273,163,187)(62,219,164,121)(63,271,165,185)(64,217,166,139)(65,269,167,183)(66,215,168,137)(67,267,169,181)(68,213,170,135)(69,265,171,199)(70,211,172,133)(71,263,173,197)(72,209,174,131)(73,261,175,195)(74,207,176,129)(75,279,177,193)(76,205,178,127)(77,277,179,191)(78,203,180,125)(79,275,161,189)(80,201,162,123)(101,272,281,186)(102,218,282,140)(103,270,283,184)(104,216,284,138)(105,268,285,182)(106,214,286,136)(107,266,287,200)(108,212,288,134)(109,264,289,198)(110,210,290,132)(111,262,291,196)(112,208,292,130)(113,280,293,194)(114,206,294,128)(115,278,295,192)(116,204,296,126)(117,276,297,190)(118,202,298,124)(119,274,299,188)(120,220,300,122), (1,116,11,106)(2,66,12,76)(3,114,13,104)(4,64,14,74)(5,112,15,102)(6,62,16,72)(7,110,17,120)(8,80,18,70)(9,108,19,118)(10,78,20,68)(21,291,31,281)(22,163,32,173)(23,289,33,299)(24,161,34,171)(25,287,35,297)(26,179,36,169)(27,285,37,295)(28,177,38,167)(29,283,39,293)(30,175,40,165)(41,182,51,192)(42,126,52,136)(43,200,53,190)(44,124,54,134)(45,198,55,188)(46,122,56,132)(47,196,57,186)(48,140,58,130)(49,194,59,184)(50,138,60,128)(61,233,71,223)(63,231,73,221)(65,229,75,239)(67,227,77,237)(69,225,79,235)(81,172,91,162)(82,298,92,288)(83,170,93,180)(84,296,94,286)(85,168,95,178)(86,294,96,284)(87,166,97,176)(88,292,98,282)(89,164,99,174)(90,290,100,300)(101,222,111,232)(103,240,113,230)(105,238,115,228)(107,236,117,226)(109,234,119,224)(121,143,131,153)(123,141,133,151)(125,159,135,149)(127,157,137,147)(129,155,139,145)(142,187,152,197)(144,185,154,195)(146,183,156,193)(148,181,158,191)(150,199,160,189)(201,316,211,306)(202,250,212,260)(203,314,213,304)(204,248,214,258)(205,312,215,302)(206,246,216,256)(207,310,217,320)(208,244,218,254)(209,308,219,318)(210,242,220,252)(241,264,251,274)(243,262,253,272)(245,280,255,270)(247,278,257,268)(249,276,259,266)(261,319,271,309)(263,317,273,307)(265,315,275,305)(267,313,277,303)(269,311,279,301)>;
G:=Group( (1,227)(2,228)(3,229)(4,230)(5,231)(6,232)(7,233)(8,234)(9,235)(10,236)(11,237)(12,238)(13,239)(14,240)(15,221)(16,222)(17,223)(18,224)(19,225)(20,226)(21,99)(22,100)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,157)(42,158)(43,159)(44,160)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,153)(58,154)(59,155)(60,156)(61,120)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(121,186)(122,187)(123,188)(124,189)(125,190)(126,191)(127,192)(128,193)(129,194)(130,195)(131,196)(132,197)(133,198)(134,199)(135,200)(136,181)(137,182)(138,183)(139,184)(140,185)(161,298)(162,299)(163,300)(164,281)(165,282)(166,283)(167,284)(168,285)(169,286)(170,287)(171,288)(172,289)(173,290)(174,291)(175,292)(176,293)(177,294)(178,295)(179,296)(180,297)(201,274)(202,275)(203,276)(204,277)(205,278)(206,279)(207,280)(208,261)(209,262)(210,263)(211,264)(212,265)(213,266)(214,267)(215,268)(216,269)(217,270)(218,271)(219,272)(220,273)(241,316)(242,317)(243,318)(244,319)(245,320)(246,301)(247,302)(248,303)(249,304)(250,305)(251,306)(252,307)(253,308)(254,309)(255,310)(256,311)(257,312)(258,313)(259,314)(260,315), (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), (1,158,84,313)(2,41,85,257)(3,156,86,311)(4,59,87,255)(5,154,88,309)(6,57,89,253)(7,152,90,307)(8,55,91,251)(9,150,92,305)(10,53,93,249)(11,148,94,303)(12,51,95,247)(13,146,96,301)(14,49,97,245)(15,144,98,319)(16,47,99,243)(17,142,100,317)(18,45,81,241)(19,160,82,315)(20,43,83,259)(21,318,222,143)(22,242,223,46)(23,316,224,141)(24,260,225,44)(25,314,226,159)(26,258,227,42)(27,312,228,157)(28,256,229,60)(29,310,230,155)(30,254,231,58)(31,308,232,153)(32,252,233,56)(33,306,234,151)(34,250,235,54)(35,304,236,149)(36,248,237,52)(37,302,238,147)(38,246,239,50)(39,320,240,145)(40,244,221,48)(61,273,163,187)(62,219,164,121)(63,271,165,185)(64,217,166,139)(65,269,167,183)(66,215,168,137)(67,267,169,181)(68,213,170,135)(69,265,171,199)(70,211,172,133)(71,263,173,197)(72,209,174,131)(73,261,175,195)(74,207,176,129)(75,279,177,193)(76,205,178,127)(77,277,179,191)(78,203,180,125)(79,275,161,189)(80,201,162,123)(101,272,281,186)(102,218,282,140)(103,270,283,184)(104,216,284,138)(105,268,285,182)(106,214,286,136)(107,266,287,200)(108,212,288,134)(109,264,289,198)(110,210,290,132)(111,262,291,196)(112,208,292,130)(113,280,293,194)(114,206,294,128)(115,278,295,192)(116,204,296,126)(117,276,297,190)(118,202,298,124)(119,274,299,188)(120,220,300,122), (1,116,11,106)(2,66,12,76)(3,114,13,104)(4,64,14,74)(5,112,15,102)(6,62,16,72)(7,110,17,120)(8,80,18,70)(9,108,19,118)(10,78,20,68)(21,291,31,281)(22,163,32,173)(23,289,33,299)(24,161,34,171)(25,287,35,297)(26,179,36,169)(27,285,37,295)(28,177,38,167)(29,283,39,293)(30,175,40,165)(41,182,51,192)(42,126,52,136)(43,200,53,190)(44,124,54,134)(45,198,55,188)(46,122,56,132)(47,196,57,186)(48,140,58,130)(49,194,59,184)(50,138,60,128)(61,233,71,223)(63,231,73,221)(65,229,75,239)(67,227,77,237)(69,225,79,235)(81,172,91,162)(82,298,92,288)(83,170,93,180)(84,296,94,286)(85,168,95,178)(86,294,96,284)(87,166,97,176)(88,292,98,282)(89,164,99,174)(90,290,100,300)(101,222,111,232)(103,240,113,230)(105,238,115,228)(107,236,117,226)(109,234,119,224)(121,143,131,153)(123,141,133,151)(125,159,135,149)(127,157,137,147)(129,155,139,145)(142,187,152,197)(144,185,154,195)(146,183,156,193)(148,181,158,191)(150,199,160,189)(201,316,211,306)(202,250,212,260)(203,314,213,304)(204,248,214,258)(205,312,215,302)(206,246,216,256)(207,310,217,320)(208,244,218,254)(209,308,219,318)(210,242,220,252)(241,264,251,274)(243,262,253,272)(245,280,255,270)(247,278,257,268)(249,276,259,266)(261,319,271,309)(263,317,273,307)(265,315,275,305)(267,313,277,303)(269,311,279,301) );
G=PermutationGroup([[(1,227),(2,228),(3,229),(4,230),(5,231),(6,232),(7,233),(8,234),(9,235),(10,236),(11,237),(12,238),(13,239),(14,240),(15,221),(16,222),(17,223),(18,224),(19,225),(20,226),(21,99),(22,100),(23,81),(24,82),(25,83),(26,84),(27,85),(28,86),(29,87),(30,88),(31,89),(32,90),(33,91),(34,92),(35,93),(36,94),(37,95),(38,96),(39,97),(40,98),(41,157),(42,158),(43,159),(44,160),(45,141),(46,142),(47,143),(48,144),(49,145),(50,146),(51,147),(52,148),(53,149),(54,150),(55,151),(56,152),(57,153),(58,154),(59,155),(60,156),(61,120),(62,101),(63,102),(64,103),(65,104),(66,105),(67,106),(68,107),(69,108),(70,109),(71,110),(72,111),(73,112),(74,113),(75,114),(76,115),(77,116),(78,117),(79,118),(80,119),(121,186),(122,187),(123,188),(124,189),(125,190),(126,191),(127,192),(128,193),(129,194),(130,195),(131,196),(132,197),(133,198),(134,199),(135,200),(136,181),(137,182),(138,183),(139,184),(140,185),(161,298),(162,299),(163,300),(164,281),(165,282),(166,283),(167,284),(168,285),(169,286),(170,287),(171,288),(172,289),(173,290),(174,291),(175,292),(176,293),(177,294),(178,295),(179,296),(180,297),(201,274),(202,275),(203,276),(204,277),(205,278),(206,279),(207,280),(208,261),(209,262),(210,263),(211,264),(212,265),(213,266),(214,267),(215,268),(216,269),(217,270),(218,271),(219,272),(220,273),(241,316),(242,317),(243,318),(244,319),(245,320),(246,301),(247,302),(248,303),(249,304),(250,305),(251,306),(252,307),(253,308),(254,309),(255,310),(256,311),(257,312),(258,313),(259,314),(260,315)], [(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)], [(1,158,84,313),(2,41,85,257),(3,156,86,311),(4,59,87,255),(5,154,88,309),(6,57,89,253),(7,152,90,307),(8,55,91,251),(9,150,92,305),(10,53,93,249),(11,148,94,303),(12,51,95,247),(13,146,96,301),(14,49,97,245),(15,144,98,319),(16,47,99,243),(17,142,100,317),(18,45,81,241),(19,160,82,315),(20,43,83,259),(21,318,222,143),(22,242,223,46),(23,316,224,141),(24,260,225,44),(25,314,226,159),(26,258,227,42),(27,312,228,157),(28,256,229,60),(29,310,230,155),(30,254,231,58),(31,308,232,153),(32,252,233,56),(33,306,234,151),(34,250,235,54),(35,304,236,149),(36,248,237,52),(37,302,238,147),(38,246,239,50),(39,320,240,145),(40,244,221,48),(61,273,163,187),(62,219,164,121),(63,271,165,185),(64,217,166,139),(65,269,167,183),(66,215,168,137),(67,267,169,181),(68,213,170,135),(69,265,171,199),(70,211,172,133),(71,263,173,197),(72,209,174,131),(73,261,175,195),(74,207,176,129),(75,279,177,193),(76,205,178,127),(77,277,179,191),(78,203,180,125),(79,275,161,189),(80,201,162,123),(101,272,281,186),(102,218,282,140),(103,270,283,184),(104,216,284,138),(105,268,285,182),(106,214,286,136),(107,266,287,200),(108,212,288,134),(109,264,289,198),(110,210,290,132),(111,262,291,196),(112,208,292,130),(113,280,293,194),(114,206,294,128),(115,278,295,192),(116,204,296,126),(117,276,297,190),(118,202,298,124),(119,274,299,188),(120,220,300,122)], [(1,116,11,106),(2,66,12,76),(3,114,13,104),(4,64,14,74),(5,112,15,102),(6,62,16,72),(7,110,17,120),(8,80,18,70),(9,108,19,118),(10,78,20,68),(21,291,31,281),(22,163,32,173),(23,289,33,299),(24,161,34,171),(25,287,35,297),(26,179,36,169),(27,285,37,295),(28,177,38,167),(29,283,39,293),(30,175,40,165),(41,182,51,192),(42,126,52,136),(43,200,53,190),(44,124,54,134),(45,198,55,188),(46,122,56,132),(47,196,57,186),(48,140,58,130),(49,194,59,184),(50,138,60,128),(61,233,71,223),(63,231,73,221),(65,229,75,239),(67,227,77,237),(69,225,79,235),(81,172,91,162),(82,298,92,288),(83,170,93,180),(84,296,94,286),(85,168,95,178),(86,294,96,284),(87,166,97,176),(88,292,98,282),(89,164,99,174),(90,290,100,300),(101,222,111,232),(103,240,113,230),(105,238,115,228),(107,236,117,226),(109,234,119,224),(121,143,131,153),(123,141,133,151),(125,159,135,149),(127,157,137,147),(129,155,139,145),(142,187,152,197),(144,185,154,195),(146,183,156,193),(148,181,158,191),(150,199,160,189),(201,316,211,306),(202,250,212,260),(203,314,213,304),(204,248,214,258),(205,312,215,302),(206,246,216,256),(207,310,217,320),(208,244,218,254),(209,308,219,318),(210,242,220,252),(241,264,251,274),(243,262,253,272),(245,280,255,270),(247,278,257,268),(249,276,259,266),(261,319,271,309),(263,317,273,307),(265,315,275,305),(267,313,277,303),(269,311,279,301)]])
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 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | Q8 | D4 | D5 | C4○D4 | D10 | C5⋊D4 | C4○D20 | D4⋊2D5 | Q8×D5 | Q8⋊2D5 |
| kernel | (C2×C20).288D4 | C10.10C42 | C2×C10.D4 | C10×C4⋊C4 | C2×Dic5 | C2×C20 | C2×C4⋊C4 | C2×C10 | C22×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 5 | 1 | 1 | 2 | 2 | 2 | 10 | 6 | 8 | 16 | 4 | 2 | 2 |
Matrix representation of (C2×C20).288D4 ►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 |
| 3 | 24 | 0 | 0 | 0 | 0 |
| 17 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 3 |
| 0 | 0 | 0 | 0 | 10 | 25 |
| 25 | 29 | 0 | 0 | 0 | 0 |
| 18 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 32 | 0 | 0 |
| 0 | 0 | 22 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 20 | 15 | 0 | 0 | 0 | 0 |
| 39 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 30 | 0 | 0 |
| 0 | 0 | 14 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 37 |
| 0 | 0 | 0 | 0 | 0 | 9 |
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],[3,17,0,0,0,0,24,40,0,0,0,0,0,0,7,40,0,0,0,0,1,0,0,0,0,0,0,0,16,10,0,0,0,0,3,25],[25,18,0,0,0,0,29,16,0,0,0,0,0,0,19,22,0,0,0,0,32,22,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[20,39,0,0,0,0,15,21,0,0,0,0,0,0,14,14,0,0,0,0,30,27,0,0,0,0,0,0,32,0,0,0,0,0,37,9] >;
(C2×C20).288D4 in GAP, Magma, Sage, TeX
(C_2\times C_{20})._{288}D_4 % in TeX
G:=Group("(C2xC20).288D4"); // GroupNames label
G:=SmallGroup(320,609);
// by ID
G=gap.SmallGroup(320,609);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,232,254,387,268,12550]);
// 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*c^-1>;
// generators/relations