direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20.6Q8, C42.273D10, C20.77(C2×Q8), (C2×C20).57Q8, (C2×C42).20D5, C10.3(C22×Q8), (C2×C10).13C24, C10⋊1(C42.C2), C4.43(C2×Dic10), (C2×C4).52Dic10, (C2×C20).779C23, (C4×C20).313C22, (C22×C4).435D10, (C2×Dic5).2C23, C2.5(C22×Dic10), C22.60(C23×D5), C22.66(C4○D20), C4⋊Dic5.287C22, C22.35(C2×Dic10), C23.311(C22×D5), (C22×C10).375C23, (C22×C20).501C22, C10.D4.94C22, (C22×Dic5).73C22, (C2×C4×C20).14C2, C5⋊1(C2×C42.C2), C2.7(C2×C4○D20), C10.2(C2×C4○D4), (C2×C10).47(C2×Q8), (C2×C4⋊Dic5).26C2, (C2×C10).94(C4○D4), (C2×C4).647(C22×D5), (C2×C10.D4).18C2, SmallGroup(320,1141)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C20.6Q8
G = < a,b,c,d | a2=b20=c4=1, d2=b10c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b10c-1 >
Subgroups: 606 in 226 conjugacy classes, 127 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C2×C42, C2×C4⋊C4, C42.C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C2×C42.C2, C10.D4, C4⋊Dic5, C4×C20, C22×Dic5, C22×C20, C22×C20, C20.6Q8, C2×C10.D4, C2×C4⋊Dic5, C2×C4×C20, C2×C20.6Q8
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, C24, D10, C42.C2, C22×Q8, C2×C4○D4, Dic10, C22×D5, C2×C42.C2, C2×Dic10, C4○D20, C23×D5, C20.6Q8, C22×Dic10, C2×C4○D20, C2×C20.6Q8
►Regular action on 320 points
Generators in S320
(1 187)(2 188)(3 189)(4 190)(5 191)(6 192)(7 193)(8 194)(9 195)(10 196)(11 197)(12 198)(13 199)(14 200)(15 181)(16 182)(17 183)(18 184)(19 185)(20 186)(21 292)(22 293)(23 294)(24 295)(25 296)(26 297)(27 298)(28 299)(29 300)(30 281)(31 282)(32 283)(33 284)(34 285)(35 286)(36 287)(37 288)(38 289)(39 290)(40 291)(41 135)(42 136)(43 137)(44 138)(45 139)(46 140)(47 121)(48 122)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 129)(56 130)(57 131)(58 132)(59 133)(60 134)(61 152)(62 153)(63 154)(64 155)(65 156)(66 157)(67 158)(68 159)(69 160)(70 141)(71 142)(72 143)(73 144)(74 145)(75 146)(76 147)(77 148)(78 149)(79 150)(80 151)(81 244)(82 245)(83 246)(84 247)(85 248)(86 249)(87 250)(88 251)(89 252)(90 253)(91 254)(92 255)(93 256)(94 257)(95 258)(96 259)(97 260)(98 241)(99 242)(100 243)(101 219)(102 220)(103 201)(104 202)(105 203)(106 204)(107 205)(108 206)(109 207)(110 208)(111 209)(112 210)(113 211)(114 212)(115 213)(116 214)(117 215)(118 216)(119 217)(120 218)(161 314)(162 315)(163 316)(164 317)(165 318)(166 319)(167 320)(168 301)(169 302)(170 303)(171 304)(172 305)(173 306)(174 307)(175 308)(176 309)(177 310)(178 311)(179 312)(180 313)(221 271)(222 272)(223 273)(224 274)(225 275)(226 276)(227 277)(228 278)(229 279)(230 280)(231 261)(232 262)(233 263)(234 264)(235 265)(236 266)(237 267)(238 268)(239 269)(240 270)
(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)
(1 295 311 87)(2 296 312 88)(3 297 313 89)(4 298 314 90)(5 299 315 91)(6 300 316 92)(7 281 317 93)(8 282 318 94)(9 283 319 95)(10 284 320 96)(11 285 301 97)(12 286 302 98)(13 287 303 99)(14 288 304 100)(15 289 305 81)(16 290 306 82)(17 291 307 83)(18 292 308 84)(19 293 309 85)(20 294 310 86)(21 175 247 184)(22 176 248 185)(23 177 249 186)(24 178 250 187)(25 179 251 188)(26 180 252 189)(27 161 253 190)(28 162 254 191)(29 163 255 192)(30 164 256 193)(31 165 257 194)(32 166 258 195)(33 167 259 196)(34 168 260 197)(35 169 241 198)(36 170 242 199)(37 171 243 200)(38 172 244 181)(39 173 245 182)(40 174 246 183)(41 150 201 239)(42 151 202 240)(43 152 203 221)(44 153 204 222)(45 154 205 223)(46 155 206 224)(47 156 207 225)(48 157 208 226)(49 158 209 227)(50 159 210 228)(51 160 211 229)(52 141 212 230)(53 142 213 231)(54 143 214 232)(55 144 215 233)(56 145 216 234)(57 146 217 235)(58 147 218 236)(59 148 219 237)(60 149 220 238)(61 105 271 137)(62 106 272 138)(63 107 273 139)(64 108 274 140)(65 109 275 121)(66 110 276 122)(67 111 277 123)(68 112 278 124)(69 113 279 125)(70 114 280 126)(71 115 261 127)(72 116 262 128)(73 117 263 129)(74 118 264 130)(75 119 265 131)(76 120 266 132)(77 101 267 133)(78 102 268 134)(79 103 269 135)(80 104 270 136)
(1 145 301 224)(2 144 302 223)(3 143 303 222)(4 142 304 221)(5 141 305 240)(6 160 306 239)(7 159 307 238)(8 158 308 237)(9 157 309 236)(10 156 310 235)(11 155 311 234)(12 154 312 233)(13 153 313 232)(14 152 314 231)(15 151 315 230)(16 150 316 229)(17 149 317 228)(18 148 318 227)(19 147 319 226)(20 146 320 225)(21 123 257 101)(22 122 258 120)(23 121 259 119)(24 140 260 118)(25 139 241 117)(26 138 242 116)(27 137 243 115)(28 136 244 114)(29 135 245 113)(30 134 246 112)(31 133 247 111)(32 132 248 110)(33 131 249 109)(34 130 250 108)(35 129 251 107)(36 128 252 106)(37 127 253 105)(38 126 254 104)(39 125 255 103)(40 124 256 102)(41 82 211 300)(42 81 212 299)(43 100 213 298)(44 99 214 297)(45 98 215 296)(46 97 216 295)(47 96 217 294)(48 95 218 293)(49 94 219 292)(50 93 220 291)(51 92 201 290)(52 91 202 289)(53 90 203 288)(54 89 204 287)(55 88 205 286)(56 87 206 285)(57 86 207 284)(58 85 208 283)(59 84 209 282)(60 83 210 281)(61 161 261 200)(62 180 262 199)(63 179 263 198)(64 178 264 197)(65 177 265 196)(66 176 266 195)(67 175 267 194)(68 174 268 193)(69 173 269 192)(70 172 270 191)(71 171 271 190)(72 170 272 189)(73 169 273 188)(74 168 274 187)(75 167 275 186)(76 166 276 185)(77 165 277 184)(78 164 278 183)(79 163 279 182)(80 162 280 181)
G:=sub<Sym(320)| (1,187)(2,188)(3,189)(4,190)(5,191)(6,192)(7,193)(8,194)(9,195)(10,196)(11,197)(12,198)(13,199)(14,200)(15,181)(16,182)(17,183)(18,184)(19,185)(20,186)(21,292)(22,293)(23,294)(24,295)(25,296)(26,297)(27,298)(28,299)(29,300)(30,281)(31,282)(32,283)(33,284)(34,285)(35,286)(36,287)(37,288)(38,289)(39,290)(40,291)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,152)(62,153)(63,154)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151)(81,244)(82,245)(83,246)(84,247)(85,248)(86,249)(87,250)(88,251)(89,252)(90,253)(91,254)(92,255)(93,256)(94,257)(95,258)(96,259)(97,260)(98,241)(99,242)(100,243)(101,219)(102,220)(103,201)(104,202)(105,203)(106,204)(107,205)(108,206)(109,207)(110,208)(111,209)(112,210)(113,211)(114,212)(115,213)(116,214)(117,215)(118,216)(119,217)(120,218)(161,314)(162,315)(163,316)(164,317)(165,318)(166,319)(167,320)(168,301)(169,302)(170,303)(171,304)(172,305)(173,306)(174,307)(175,308)(176,309)(177,310)(178,311)(179,312)(180,313)(221,271)(222,272)(223,273)(224,274)(225,275)(226,276)(227,277)(228,278)(229,279)(230,280)(231,261)(232,262)(233,263)(234,264)(235,265)(236,266)(237,267)(238,268)(239,269)(240,270), (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), (1,295,311,87)(2,296,312,88)(3,297,313,89)(4,298,314,90)(5,299,315,91)(6,300,316,92)(7,281,317,93)(8,282,318,94)(9,283,319,95)(10,284,320,96)(11,285,301,97)(12,286,302,98)(13,287,303,99)(14,288,304,100)(15,289,305,81)(16,290,306,82)(17,291,307,83)(18,292,308,84)(19,293,309,85)(20,294,310,86)(21,175,247,184)(22,176,248,185)(23,177,249,186)(24,178,250,187)(25,179,251,188)(26,180,252,189)(27,161,253,190)(28,162,254,191)(29,163,255,192)(30,164,256,193)(31,165,257,194)(32,166,258,195)(33,167,259,196)(34,168,260,197)(35,169,241,198)(36,170,242,199)(37,171,243,200)(38,172,244,181)(39,173,245,182)(40,174,246,183)(41,150,201,239)(42,151,202,240)(43,152,203,221)(44,153,204,222)(45,154,205,223)(46,155,206,224)(47,156,207,225)(48,157,208,226)(49,158,209,227)(50,159,210,228)(51,160,211,229)(52,141,212,230)(53,142,213,231)(54,143,214,232)(55,144,215,233)(56,145,216,234)(57,146,217,235)(58,147,218,236)(59,148,219,237)(60,149,220,238)(61,105,271,137)(62,106,272,138)(63,107,273,139)(64,108,274,140)(65,109,275,121)(66,110,276,122)(67,111,277,123)(68,112,278,124)(69,113,279,125)(70,114,280,126)(71,115,261,127)(72,116,262,128)(73,117,263,129)(74,118,264,130)(75,119,265,131)(76,120,266,132)(77,101,267,133)(78,102,268,134)(79,103,269,135)(80,104,270,136), (1,145,301,224)(2,144,302,223)(3,143,303,222)(4,142,304,221)(5,141,305,240)(6,160,306,239)(7,159,307,238)(8,158,308,237)(9,157,309,236)(10,156,310,235)(11,155,311,234)(12,154,312,233)(13,153,313,232)(14,152,314,231)(15,151,315,230)(16,150,316,229)(17,149,317,228)(18,148,318,227)(19,147,319,226)(20,146,320,225)(21,123,257,101)(22,122,258,120)(23,121,259,119)(24,140,260,118)(25,139,241,117)(26,138,242,116)(27,137,243,115)(28,136,244,114)(29,135,245,113)(30,134,246,112)(31,133,247,111)(32,132,248,110)(33,131,249,109)(34,130,250,108)(35,129,251,107)(36,128,252,106)(37,127,253,105)(38,126,254,104)(39,125,255,103)(40,124,256,102)(41,82,211,300)(42,81,212,299)(43,100,213,298)(44,99,214,297)(45,98,215,296)(46,97,216,295)(47,96,217,294)(48,95,218,293)(49,94,219,292)(50,93,220,291)(51,92,201,290)(52,91,202,289)(53,90,203,288)(54,89,204,287)(55,88,205,286)(56,87,206,285)(57,86,207,284)(58,85,208,283)(59,84,209,282)(60,83,210,281)(61,161,261,200)(62,180,262,199)(63,179,263,198)(64,178,264,197)(65,177,265,196)(66,176,266,195)(67,175,267,194)(68,174,268,193)(69,173,269,192)(70,172,270,191)(71,171,271,190)(72,170,272,189)(73,169,273,188)(74,168,274,187)(75,167,275,186)(76,166,276,185)(77,165,277,184)(78,164,278,183)(79,163,279,182)(80,162,280,181)>;
G:=Group( (1,187)(2,188)(3,189)(4,190)(5,191)(6,192)(7,193)(8,194)(9,195)(10,196)(11,197)(12,198)(13,199)(14,200)(15,181)(16,182)(17,183)(18,184)(19,185)(20,186)(21,292)(22,293)(23,294)(24,295)(25,296)(26,297)(27,298)(28,299)(29,300)(30,281)(31,282)(32,283)(33,284)(34,285)(35,286)(36,287)(37,288)(38,289)(39,290)(40,291)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,152)(62,153)(63,154)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151)(81,244)(82,245)(83,246)(84,247)(85,248)(86,249)(87,250)(88,251)(89,252)(90,253)(91,254)(92,255)(93,256)(94,257)(95,258)(96,259)(97,260)(98,241)(99,242)(100,243)(101,219)(102,220)(103,201)(104,202)(105,203)(106,204)(107,205)(108,206)(109,207)(110,208)(111,209)(112,210)(113,211)(114,212)(115,213)(116,214)(117,215)(118,216)(119,217)(120,218)(161,314)(162,315)(163,316)(164,317)(165,318)(166,319)(167,320)(168,301)(169,302)(170,303)(171,304)(172,305)(173,306)(174,307)(175,308)(176,309)(177,310)(178,311)(179,312)(180,313)(221,271)(222,272)(223,273)(224,274)(225,275)(226,276)(227,277)(228,278)(229,279)(230,280)(231,261)(232,262)(233,263)(234,264)(235,265)(236,266)(237,267)(238,268)(239,269)(240,270), (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), (1,295,311,87)(2,296,312,88)(3,297,313,89)(4,298,314,90)(5,299,315,91)(6,300,316,92)(7,281,317,93)(8,282,318,94)(9,283,319,95)(10,284,320,96)(11,285,301,97)(12,286,302,98)(13,287,303,99)(14,288,304,100)(15,289,305,81)(16,290,306,82)(17,291,307,83)(18,292,308,84)(19,293,309,85)(20,294,310,86)(21,175,247,184)(22,176,248,185)(23,177,249,186)(24,178,250,187)(25,179,251,188)(26,180,252,189)(27,161,253,190)(28,162,254,191)(29,163,255,192)(30,164,256,193)(31,165,257,194)(32,166,258,195)(33,167,259,196)(34,168,260,197)(35,169,241,198)(36,170,242,199)(37,171,243,200)(38,172,244,181)(39,173,245,182)(40,174,246,183)(41,150,201,239)(42,151,202,240)(43,152,203,221)(44,153,204,222)(45,154,205,223)(46,155,206,224)(47,156,207,225)(48,157,208,226)(49,158,209,227)(50,159,210,228)(51,160,211,229)(52,141,212,230)(53,142,213,231)(54,143,214,232)(55,144,215,233)(56,145,216,234)(57,146,217,235)(58,147,218,236)(59,148,219,237)(60,149,220,238)(61,105,271,137)(62,106,272,138)(63,107,273,139)(64,108,274,140)(65,109,275,121)(66,110,276,122)(67,111,277,123)(68,112,278,124)(69,113,279,125)(70,114,280,126)(71,115,261,127)(72,116,262,128)(73,117,263,129)(74,118,264,130)(75,119,265,131)(76,120,266,132)(77,101,267,133)(78,102,268,134)(79,103,269,135)(80,104,270,136), (1,145,301,224)(2,144,302,223)(3,143,303,222)(4,142,304,221)(5,141,305,240)(6,160,306,239)(7,159,307,238)(8,158,308,237)(9,157,309,236)(10,156,310,235)(11,155,311,234)(12,154,312,233)(13,153,313,232)(14,152,314,231)(15,151,315,230)(16,150,316,229)(17,149,317,228)(18,148,318,227)(19,147,319,226)(20,146,320,225)(21,123,257,101)(22,122,258,120)(23,121,259,119)(24,140,260,118)(25,139,241,117)(26,138,242,116)(27,137,243,115)(28,136,244,114)(29,135,245,113)(30,134,246,112)(31,133,247,111)(32,132,248,110)(33,131,249,109)(34,130,250,108)(35,129,251,107)(36,128,252,106)(37,127,253,105)(38,126,254,104)(39,125,255,103)(40,124,256,102)(41,82,211,300)(42,81,212,299)(43,100,213,298)(44,99,214,297)(45,98,215,296)(46,97,216,295)(47,96,217,294)(48,95,218,293)(49,94,219,292)(50,93,220,291)(51,92,201,290)(52,91,202,289)(53,90,203,288)(54,89,204,287)(55,88,205,286)(56,87,206,285)(57,86,207,284)(58,85,208,283)(59,84,209,282)(60,83,210,281)(61,161,261,200)(62,180,262,199)(63,179,263,198)(64,178,264,197)(65,177,265,196)(66,176,266,195)(67,175,267,194)(68,174,268,193)(69,173,269,192)(70,172,270,191)(71,171,271,190)(72,170,272,189)(73,169,273,188)(74,168,274,187)(75,167,275,186)(76,166,276,185)(77,165,277,184)(78,164,278,183)(79,163,279,182)(80,162,280,181) );
G=PermutationGroup([[(1,187),(2,188),(3,189),(4,190),(5,191),(6,192),(7,193),(8,194),(9,195),(10,196),(11,197),(12,198),(13,199),(14,200),(15,181),(16,182),(17,183),(18,184),(19,185),(20,186),(21,292),(22,293),(23,294),(24,295),(25,296),(26,297),(27,298),(28,299),(29,300),(30,281),(31,282),(32,283),(33,284),(34,285),(35,286),(36,287),(37,288),(38,289),(39,290),(40,291),(41,135),(42,136),(43,137),(44,138),(45,139),(46,140),(47,121),(48,122),(49,123),(50,124),(51,125),(52,126),(53,127),(54,128),(55,129),(56,130),(57,131),(58,132),(59,133),(60,134),(61,152),(62,153),(63,154),(64,155),(65,156),(66,157),(67,158),(68,159),(69,160),(70,141),(71,142),(72,143),(73,144),(74,145),(75,146),(76,147),(77,148),(78,149),(79,150),(80,151),(81,244),(82,245),(83,246),(84,247),(85,248),(86,249),(87,250),(88,251),(89,252),(90,253),(91,254),(92,255),(93,256),(94,257),(95,258),(96,259),(97,260),(98,241),(99,242),(100,243),(101,219),(102,220),(103,201),(104,202),(105,203),(106,204),(107,205),(108,206),(109,207),(110,208),(111,209),(112,210),(113,211),(114,212),(115,213),(116,214),(117,215),(118,216),(119,217),(120,218),(161,314),(162,315),(163,316),(164,317),(165,318),(166,319),(167,320),(168,301),(169,302),(170,303),(171,304),(172,305),(173,306),(174,307),(175,308),(176,309),(177,310),(178,311),(179,312),(180,313),(221,271),(222,272),(223,273),(224,274),(225,275),(226,276),(227,277),(228,278),(229,279),(230,280),(231,261),(232,262),(233,263),(234,264),(235,265),(236,266),(237,267),(238,268),(239,269),(240,270)], [(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)], [(1,295,311,87),(2,296,312,88),(3,297,313,89),(4,298,314,90),(5,299,315,91),(6,300,316,92),(7,281,317,93),(8,282,318,94),(9,283,319,95),(10,284,320,96),(11,285,301,97),(12,286,302,98),(13,287,303,99),(14,288,304,100),(15,289,305,81),(16,290,306,82),(17,291,307,83),(18,292,308,84),(19,293,309,85),(20,294,310,86),(21,175,247,184),(22,176,248,185),(23,177,249,186),(24,178,250,187),(25,179,251,188),(26,180,252,189),(27,161,253,190),(28,162,254,191),(29,163,255,192),(30,164,256,193),(31,165,257,194),(32,166,258,195),(33,167,259,196),(34,168,260,197),(35,169,241,198),(36,170,242,199),(37,171,243,200),(38,172,244,181),(39,173,245,182),(40,174,246,183),(41,150,201,239),(42,151,202,240),(43,152,203,221),(44,153,204,222),(45,154,205,223),(46,155,206,224),(47,156,207,225),(48,157,208,226),(49,158,209,227),(50,159,210,228),(51,160,211,229),(52,141,212,230),(53,142,213,231),(54,143,214,232),(55,144,215,233),(56,145,216,234),(57,146,217,235),(58,147,218,236),(59,148,219,237),(60,149,220,238),(61,105,271,137),(62,106,272,138),(63,107,273,139),(64,108,274,140),(65,109,275,121),(66,110,276,122),(67,111,277,123),(68,112,278,124),(69,113,279,125),(70,114,280,126),(71,115,261,127),(72,116,262,128),(73,117,263,129),(74,118,264,130),(75,119,265,131),(76,120,266,132),(77,101,267,133),(78,102,268,134),(79,103,269,135),(80,104,270,136)], [(1,145,301,224),(2,144,302,223),(3,143,303,222),(4,142,304,221),(5,141,305,240),(6,160,306,239),(7,159,307,238),(8,158,308,237),(9,157,309,236),(10,156,310,235),(11,155,311,234),(12,154,312,233),(13,153,313,232),(14,152,314,231),(15,151,315,230),(16,150,316,229),(17,149,317,228),(18,148,318,227),(19,147,319,226),(20,146,320,225),(21,123,257,101),(22,122,258,120),(23,121,259,119),(24,140,260,118),(25,139,241,117),(26,138,242,116),(27,137,243,115),(28,136,244,114),(29,135,245,113),(30,134,246,112),(31,133,247,111),(32,132,248,110),(33,131,249,109),(34,130,250,108),(35,129,251,107),(36,128,252,106),(37,127,253,105),(38,126,254,104),(39,125,255,103),(40,124,256,102),(41,82,211,300),(42,81,212,299),(43,100,213,298),(44,99,214,297),(45,98,215,296),(46,97,216,295),(47,96,217,294),(48,95,218,293),(49,94,219,292),(50,93,220,291),(51,92,201,290),(52,91,202,289),(53,90,203,288),(54,89,204,287),(55,88,205,286),(56,87,206,285),(57,86,207,284),(58,85,208,283),(59,84,209,282),(60,83,210,281),(61,161,261,200),(62,180,262,199),(63,179,263,198),(64,178,264,197),(65,177,265,196),(66,176,266,195),(67,175,267,194),(68,174,268,193),(69,173,269,192),(70,172,270,191),(71,171,271,190),(72,170,272,189),(73,169,273,188),(74,168,274,187),(75,167,275,186),(76,166,276,185),(77,165,277,184),(78,164,278,183),(79,163,279,182),(80,162,280,181)]])
92 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | Dic10 | C4○D20 |
| kernel | C2×C20.6Q8 | C20.6Q8 | C2×C10.D4 | C2×C4⋊Dic5 | C2×C4×C20 | C2×C20 | C2×C42 | C2×C10 | C42 | C22×C4 | C2×C4 | C22 |
| # reps | 1 | 8 | 4 | 2 | 1 | 4 | 2 | 8 | 8 | 6 | 16 | 32 |
Matrix representation of C2×C20.6Q8 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 30 | 0 | 0 | 0 | 0 |
| 27 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 |
| 0 | 0 | 11 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 0 | 0 | 0 | 0 | 30 | 14 |
| 18 | 40 | 0 | 0 | 0 | 0 |
| 36 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 1 | 0 | 0 |
| 0 | 0 | 40 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 39 | 25 | 0 | 0 | 0 | 0 |
| 8 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 12 | 0 | 0 |
| 0 | 0 | 23 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 38 |
| 0 | 0 | 0 | 0 | 20 | 15 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,27,0,0,0,0,30,16,0,0,0,0,0,0,32,11,0,0,0,0,30,27,0,0,0,0,0,0,9,30,0,0,0,0,11,14],[18,36,0,0,0,0,40,23,0,0,0,0,0,0,24,40,0,0,0,0,1,17,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[39,8,0,0,0,0,25,2,0,0,0,0,0,0,16,23,0,0,0,0,12,25,0,0,0,0,0,0,26,20,0,0,0,0,38,15] >;
C2×C20.6Q8 in GAP, Magma, Sage, TeX
C_2\times C_{20}._6Q_8 % in TeX
G:=Group("C2xC20.6Q8"); // GroupNames label
G:=SmallGroup(320,1141);
// by ID
G=gap.SmallGroup(320,1141);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,100,675,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=b^10*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^10*c^-1>;
// generators/relations