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