direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C2×C20, C2.5(C23×C20), C4.17(C22×C20), (C2×C42).17C10, C10.78(C23×C4), C42.86(C2×C10), C22.17(Q8×C10), C10.56(C22×Q8), (C2×C20).708C23, (C4×C20).370C22, C20.221(C22×C4), (C2×C10).336C24, C22.9(C23×C10), (C22×Q8).10C10, C22.26(C22×C20), C23.68(C22×C10), (Q8×C10).282C22, (C22×C10).468C23, (C22×C20).594C22, C2.2(Q8×C2×C10), (C2×C4×C20).40C2, C2.3(C10×C4○D4), (C2×C4⋊C4).22C10, (C10×C4⋊C4).51C2, (Q8×C2×C10).20C2, C4⋊C4.80(C2×C10), (C2×C4).52(C2×C20), (C2×C20).446(C2×C4), C10.222(C2×C4○D4), (C2×Q8).70(C2×C10), (C2×C10).115(C2×Q8), C22.28(C5×C4○D4), (C5×C4⋊C4).405C22, (C2×C4).55(C22×C10), (C22×C4).98(C2×C10), (C2×C10).228(C4○D4), (C2×C10).347(C22×C4), SmallGroup(320,1518)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C2×C20
G = < a,b,c,d | a2=b20=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 322 in 298 conjugacy classes, 274 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C10, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C42, C2×C4⋊C4, C4×Q8, C22×Q8, C2×C20, C2×C20, C5×Q8, C22×C10, C2×C4×Q8, C4×C20, C5×C4⋊C4, C22×C20, C22×C20, Q8×C10, C2×C4×C20, C10×C4⋊C4, Q8×C20, Q8×C2×C10, Q8×C2×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, Q8, C23, C10, C22×C4, C2×Q8, C4○D4, C24, C20, C2×C10, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, C2×C20, C5×Q8, C22×C10, C2×C4×Q8, C22×C20, Q8×C10, C5×C4○D4, C23×C10, Q8×C20, C23×C20, Q8×C2×C10, C10×C4○D4, Q8×C2×C20
►Regular action on 320 points
Generators in S320
(1 304)(2 305)(3 306)(4 307)(5 308)(6 309)(7 310)(8 311)(9 312)(10 313)(11 314)(12 315)(13 316)(14 317)(15 318)(16 319)(17 320)(18 301)(19 302)(20 303)(21 290)(22 291)(23 292)(24 293)(25 294)(26 295)(27 296)(28 297)(29 298)(30 299)(31 300)(32 281)(33 282)(34 283)(35 284)(36 285)(37 286)(38 287)(39 288)(40 289)(41 248)(42 249)(43 250)(44 251)(45 252)(46 253)(47 254)(48 255)(49 256)(50 257)(51 258)(52 259)(53 260)(54 241)(55 242)(56 243)(57 244)(58 245)(59 246)(60 247)(61 173)(62 174)(63 175)(64 176)(65 177)(66 178)(67 179)(68 180)(69 161)(70 162)(71 163)(72 164)(73 165)(74 166)(75 167)(76 168)(77 169)(78 170)(79 171)(80 172)(81 152)(82 153)(83 154)(84 155)(85 156)(86 157)(87 158)(88 159)(89 160)(90 141)(91 142)(92 143)(93 144)(94 145)(95 146)(96 147)(97 148)(98 149)(99 150)(100 151)(101 229)(102 230)(103 231)(104 232)(105 233)(106 234)(107 235)(108 236)(109 237)(110 238)(111 239)(112 240)(113 221)(114 222)(115 223)(116 224)(117 225)(118 226)(119 227)(120 228)(121 183)(122 184)(123 185)(124 186)(125 187)(126 188)(127 189)(128 190)(129 191)(130 192)(131 193)(132 194)(133 195)(134 196)(135 197)(136 198)(137 199)(138 200)(139 181)(140 182)(201 271)(202 272)(203 273)(204 274)(205 275)(206 276)(207 277)(208 278)(209 279)(210 280)(211 261)(212 262)(213 263)(214 264)(215 265)(216 266)(217 267)(218 268)(219 269)(220 270)
(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)
(1 280 246 72)(2 261 247 73)(3 262 248 74)(4 263 249 75)(5 264 250 76)(6 265 251 77)(7 266 252 78)(8 267 253 79)(9 268 254 80)(10 269 255 61)(11 270 256 62)(12 271 257 63)(13 272 258 64)(14 273 259 65)(15 274 260 66)(16 275 241 67)(17 276 242 68)(18 277 243 69)(19 278 244 70)(20 279 245 71)(21 138 116 87)(22 139 117 88)(23 140 118 89)(24 121 119 90)(25 122 120 91)(26 123 101 92)(27 124 102 93)(28 125 103 94)(29 126 104 95)(30 127 105 96)(31 128 106 97)(32 129 107 98)(33 130 108 99)(34 131 109 100)(35 132 110 81)(36 133 111 82)(37 134 112 83)(38 135 113 84)(39 136 114 85)(40 137 115 86)(41 166 306 212)(42 167 307 213)(43 168 308 214)(44 169 309 215)(45 170 310 216)(46 171 311 217)(47 172 312 218)(48 173 313 219)(49 174 314 220)(50 175 315 201)(51 176 316 202)(52 177 317 203)(53 178 318 204)(54 179 319 205)(55 180 320 206)(56 161 301 207)(57 162 302 208)(58 163 303 209)(59 164 304 210)(60 165 305 211)(141 293 183 227)(142 294 184 228)(143 295 185 229)(144 296 186 230)(145 297 187 231)(146 298 188 232)(147 299 189 233)(148 300 190 234)(149 281 191 235)(150 282 192 236)(151 283 193 237)(152 284 194 238)(153 285 195 239)(154 286 196 240)(155 287 197 221)(156 288 198 222)(157 289 199 223)(158 290 200 224)(159 291 181 225)(160 292 182 226)
(1 40 246 115)(2 21 247 116)(3 22 248 117)(4 23 249 118)(5 24 250 119)(6 25 251 120)(7 26 252 101)(8 27 253 102)(9 28 254 103)(10 29 255 104)(11 30 256 105)(12 31 257 106)(13 32 258 107)(14 33 259 108)(15 34 260 109)(16 35 241 110)(17 36 242 111)(18 37 243 112)(19 38 244 113)(20 39 245 114)(41 225 306 291)(42 226 307 292)(43 227 308 293)(44 228 309 294)(45 229 310 295)(46 230 311 296)(47 231 312 297)(48 232 313 298)(49 233 314 299)(50 234 315 300)(51 235 316 281)(52 236 317 282)(53 237 318 283)(54 238 319 284)(55 239 320 285)(56 240 301 286)(57 221 302 287)(58 222 303 288)(59 223 304 289)(60 224 305 290)(61 126 269 95)(62 127 270 96)(63 128 271 97)(64 129 272 98)(65 130 273 99)(66 131 274 100)(67 132 275 81)(68 133 276 82)(69 134 277 83)(70 135 278 84)(71 136 279 85)(72 137 280 86)(73 138 261 87)(74 139 262 88)(75 140 263 89)(76 121 264 90)(77 122 265 91)(78 123 266 92)(79 124 267 93)(80 125 268 94)(141 168 183 214)(142 169 184 215)(143 170 185 216)(144 171 186 217)(145 172 187 218)(146 173 188 219)(147 174 189 220)(148 175 190 201)(149 176 191 202)(150 177 192 203)(151 178 193 204)(152 179 194 205)(153 180 195 206)(154 161 196 207)(155 162 197 208)(156 163 198 209)(157 164 199 210)(158 165 200 211)(159 166 181 212)(160 167 182 213)
G:=sub<Sym(320)| (1,304)(2,305)(3,306)(4,307)(5,308)(6,309)(7,310)(8,311)(9,312)(10,313)(11,314)(12,315)(13,316)(14,317)(15,318)(16,319)(17,320)(18,301)(19,302)(20,303)(21,290)(22,291)(23,292)(24,293)(25,294)(26,295)(27,296)(28,297)(29,298)(30,299)(31,300)(32,281)(33,282)(34,283)(35,284)(36,285)(37,286)(38,287)(39,288)(40,289)(41,248)(42,249)(43,250)(44,251)(45,252)(46,253)(47,254)(48,255)(49,256)(50,257)(51,258)(52,259)(53,260)(54,241)(55,242)(56,243)(57,244)(58,245)(59,246)(60,247)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,161)(70,162)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,169)(78,170)(79,171)(80,172)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,141)(91,142)(92,143)(93,144)(94,145)(95,146)(96,147)(97,148)(98,149)(99,150)(100,151)(101,229)(102,230)(103,231)(104,232)(105,233)(106,234)(107,235)(108,236)(109,237)(110,238)(111,239)(112,240)(113,221)(114,222)(115,223)(116,224)(117,225)(118,226)(119,227)(120,228)(121,183)(122,184)(123,185)(124,186)(125,187)(126,188)(127,189)(128,190)(129,191)(130,192)(131,193)(132,194)(133,195)(134,196)(135,197)(136,198)(137,199)(138,200)(139,181)(140,182)(201,271)(202,272)(203,273)(204,274)(205,275)(206,276)(207,277)(208,278)(209,279)(210,280)(211,261)(212,262)(213,263)(214,264)(215,265)(216,266)(217,267)(218,268)(219,269)(220,270), (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), (1,280,246,72)(2,261,247,73)(3,262,248,74)(4,263,249,75)(5,264,250,76)(6,265,251,77)(7,266,252,78)(8,267,253,79)(9,268,254,80)(10,269,255,61)(11,270,256,62)(12,271,257,63)(13,272,258,64)(14,273,259,65)(15,274,260,66)(16,275,241,67)(17,276,242,68)(18,277,243,69)(19,278,244,70)(20,279,245,71)(21,138,116,87)(22,139,117,88)(23,140,118,89)(24,121,119,90)(25,122,120,91)(26,123,101,92)(27,124,102,93)(28,125,103,94)(29,126,104,95)(30,127,105,96)(31,128,106,97)(32,129,107,98)(33,130,108,99)(34,131,109,100)(35,132,110,81)(36,133,111,82)(37,134,112,83)(38,135,113,84)(39,136,114,85)(40,137,115,86)(41,166,306,212)(42,167,307,213)(43,168,308,214)(44,169,309,215)(45,170,310,216)(46,171,311,217)(47,172,312,218)(48,173,313,219)(49,174,314,220)(50,175,315,201)(51,176,316,202)(52,177,317,203)(53,178,318,204)(54,179,319,205)(55,180,320,206)(56,161,301,207)(57,162,302,208)(58,163,303,209)(59,164,304,210)(60,165,305,211)(141,293,183,227)(142,294,184,228)(143,295,185,229)(144,296,186,230)(145,297,187,231)(146,298,188,232)(147,299,189,233)(148,300,190,234)(149,281,191,235)(150,282,192,236)(151,283,193,237)(152,284,194,238)(153,285,195,239)(154,286,196,240)(155,287,197,221)(156,288,198,222)(157,289,199,223)(158,290,200,224)(159,291,181,225)(160,292,182,226), (1,40,246,115)(2,21,247,116)(3,22,248,117)(4,23,249,118)(5,24,250,119)(6,25,251,120)(7,26,252,101)(8,27,253,102)(9,28,254,103)(10,29,255,104)(11,30,256,105)(12,31,257,106)(13,32,258,107)(14,33,259,108)(15,34,260,109)(16,35,241,110)(17,36,242,111)(18,37,243,112)(19,38,244,113)(20,39,245,114)(41,225,306,291)(42,226,307,292)(43,227,308,293)(44,228,309,294)(45,229,310,295)(46,230,311,296)(47,231,312,297)(48,232,313,298)(49,233,314,299)(50,234,315,300)(51,235,316,281)(52,236,317,282)(53,237,318,283)(54,238,319,284)(55,239,320,285)(56,240,301,286)(57,221,302,287)(58,222,303,288)(59,223,304,289)(60,224,305,290)(61,126,269,95)(62,127,270,96)(63,128,271,97)(64,129,272,98)(65,130,273,99)(66,131,274,100)(67,132,275,81)(68,133,276,82)(69,134,277,83)(70,135,278,84)(71,136,279,85)(72,137,280,86)(73,138,261,87)(74,139,262,88)(75,140,263,89)(76,121,264,90)(77,122,265,91)(78,123,266,92)(79,124,267,93)(80,125,268,94)(141,168,183,214)(142,169,184,215)(143,170,185,216)(144,171,186,217)(145,172,187,218)(146,173,188,219)(147,174,189,220)(148,175,190,201)(149,176,191,202)(150,177,192,203)(151,178,193,204)(152,179,194,205)(153,180,195,206)(154,161,196,207)(155,162,197,208)(156,163,198,209)(157,164,199,210)(158,165,200,211)(159,166,181,212)(160,167,182,213)>;
G:=Group( (1,304)(2,305)(3,306)(4,307)(5,308)(6,309)(7,310)(8,311)(9,312)(10,313)(11,314)(12,315)(13,316)(14,317)(15,318)(16,319)(17,320)(18,301)(19,302)(20,303)(21,290)(22,291)(23,292)(24,293)(25,294)(26,295)(27,296)(28,297)(29,298)(30,299)(31,300)(32,281)(33,282)(34,283)(35,284)(36,285)(37,286)(38,287)(39,288)(40,289)(41,248)(42,249)(43,250)(44,251)(45,252)(46,253)(47,254)(48,255)(49,256)(50,257)(51,258)(52,259)(53,260)(54,241)(55,242)(56,243)(57,244)(58,245)(59,246)(60,247)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,161)(70,162)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,169)(78,170)(79,171)(80,172)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,141)(91,142)(92,143)(93,144)(94,145)(95,146)(96,147)(97,148)(98,149)(99,150)(100,151)(101,229)(102,230)(103,231)(104,232)(105,233)(106,234)(107,235)(108,236)(109,237)(110,238)(111,239)(112,240)(113,221)(114,222)(115,223)(116,224)(117,225)(118,226)(119,227)(120,228)(121,183)(122,184)(123,185)(124,186)(125,187)(126,188)(127,189)(128,190)(129,191)(130,192)(131,193)(132,194)(133,195)(134,196)(135,197)(136,198)(137,199)(138,200)(139,181)(140,182)(201,271)(202,272)(203,273)(204,274)(205,275)(206,276)(207,277)(208,278)(209,279)(210,280)(211,261)(212,262)(213,263)(214,264)(215,265)(216,266)(217,267)(218,268)(219,269)(220,270), (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), (1,280,246,72)(2,261,247,73)(3,262,248,74)(4,263,249,75)(5,264,250,76)(6,265,251,77)(7,266,252,78)(8,267,253,79)(9,268,254,80)(10,269,255,61)(11,270,256,62)(12,271,257,63)(13,272,258,64)(14,273,259,65)(15,274,260,66)(16,275,241,67)(17,276,242,68)(18,277,243,69)(19,278,244,70)(20,279,245,71)(21,138,116,87)(22,139,117,88)(23,140,118,89)(24,121,119,90)(25,122,120,91)(26,123,101,92)(27,124,102,93)(28,125,103,94)(29,126,104,95)(30,127,105,96)(31,128,106,97)(32,129,107,98)(33,130,108,99)(34,131,109,100)(35,132,110,81)(36,133,111,82)(37,134,112,83)(38,135,113,84)(39,136,114,85)(40,137,115,86)(41,166,306,212)(42,167,307,213)(43,168,308,214)(44,169,309,215)(45,170,310,216)(46,171,311,217)(47,172,312,218)(48,173,313,219)(49,174,314,220)(50,175,315,201)(51,176,316,202)(52,177,317,203)(53,178,318,204)(54,179,319,205)(55,180,320,206)(56,161,301,207)(57,162,302,208)(58,163,303,209)(59,164,304,210)(60,165,305,211)(141,293,183,227)(142,294,184,228)(143,295,185,229)(144,296,186,230)(145,297,187,231)(146,298,188,232)(147,299,189,233)(148,300,190,234)(149,281,191,235)(150,282,192,236)(151,283,193,237)(152,284,194,238)(153,285,195,239)(154,286,196,240)(155,287,197,221)(156,288,198,222)(157,289,199,223)(158,290,200,224)(159,291,181,225)(160,292,182,226), (1,40,246,115)(2,21,247,116)(3,22,248,117)(4,23,249,118)(5,24,250,119)(6,25,251,120)(7,26,252,101)(8,27,253,102)(9,28,254,103)(10,29,255,104)(11,30,256,105)(12,31,257,106)(13,32,258,107)(14,33,259,108)(15,34,260,109)(16,35,241,110)(17,36,242,111)(18,37,243,112)(19,38,244,113)(20,39,245,114)(41,225,306,291)(42,226,307,292)(43,227,308,293)(44,228,309,294)(45,229,310,295)(46,230,311,296)(47,231,312,297)(48,232,313,298)(49,233,314,299)(50,234,315,300)(51,235,316,281)(52,236,317,282)(53,237,318,283)(54,238,319,284)(55,239,320,285)(56,240,301,286)(57,221,302,287)(58,222,303,288)(59,223,304,289)(60,224,305,290)(61,126,269,95)(62,127,270,96)(63,128,271,97)(64,129,272,98)(65,130,273,99)(66,131,274,100)(67,132,275,81)(68,133,276,82)(69,134,277,83)(70,135,278,84)(71,136,279,85)(72,137,280,86)(73,138,261,87)(74,139,262,88)(75,140,263,89)(76,121,264,90)(77,122,265,91)(78,123,266,92)(79,124,267,93)(80,125,268,94)(141,168,183,214)(142,169,184,215)(143,170,185,216)(144,171,186,217)(145,172,187,218)(146,173,188,219)(147,174,189,220)(148,175,190,201)(149,176,191,202)(150,177,192,203)(151,178,193,204)(152,179,194,205)(153,180,195,206)(154,161,196,207)(155,162,197,208)(156,163,198,209)(157,164,199,210)(158,165,200,211)(159,166,181,212)(160,167,182,213) );
G=PermutationGroup([[(1,304),(2,305),(3,306),(4,307),(5,308),(6,309),(7,310),(8,311),(9,312),(10,313),(11,314),(12,315),(13,316),(14,317),(15,318),(16,319),(17,320),(18,301),(19,302),(20,303),(21,290),(22,291),(23,292),(24,293),(25,294),(26,295),(27,296),(28,297),(29,298),(30,299),(31,300),(32,281),(33,282),(34,283),(35,284),(36,285),(37,286),(38,287),(39,288),(40,289),(41,248),(42,249),(43,250),(44,251),(45,252),(46,253),(47,254),(48,255),(49,256),(50,257),(51,258),(52,259),(53,260),(54,241),(55,242),(56,243),(57,244),(58,245),(59,246),(60,247),(61,173),(62,174),(63,175),(64,176),(65,177),(66,178),(67,179),(68,180),(69,161),(70,162),(71,163),(72,164),(73,165),(74,166),(75,167),(76,168),(77,169),(78,170),(79,171),(80,172),(81,152),(82,153),(83,154),(84,155),(85,156),(86,157),(87,158),(88,159),(89,160),(90,141),(91,142),(92,143),(93,144),(94,145),(95,146),(96,147),(97,148),(98,149),(99,150),(100,151),(101,229),(102,230),(103,231),(104,232),(105,233),(106,234),(107,235),(108,236),(109,237),(110,238),(111,239),(112,240),(113,221),(114,222),(115,223),(116,224),(117,225),(118,226),(119,227),(120,228),(121,183),(122,184),(123,185),(124,186),(125,187),(126,188),(127,189),(128,190),(129,191),(130,192),(131,193),(132,194),(133,195),(134,196),(135,197),(136,198),(137,199),(138,200),(139,181),(140,182),(201,271),(202,272),(203,273),(204,274),(205,275),(206,276),(207,277),(208,278),(209,279),(210,280),(211,261),(212,262),(213,263),(214,264),(215,265),(216,266),(217,267),(218,268),(219,269),(220,270)], [(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)], [(1,280,246,72),(2,261,247,73),(3,262,248,74),(4,263,249,75),(5,264,250,76),(6,265,251,77),(7,266,252,78),(8,267,253,79),(9,268,254,80),(10,269,255,61),(11,270,256,62),(12,271,257,63),(13,272,258,64),(14,273,259,65),(15,274,260,66),(16,275,241,67),(17,276,242,68),(18,277,243,69),(19,278,244,70),(20,279,245,71),(21,138,116,87),(22,139,117,88),(23,140,118,89),(24,121,119,90),(25,122,120,91),(26,123,101,92),(27,124,102,93),(28,125,103,94),(29,126,104,95),(30,127,105,96),(31,128,106,97),(32,129,107,98),(33,130,108,99),(34,131,109,100),(35,132,110,81),(36,133,111,82),(37,134,112,83),(38,135,113,84),(39,136,114,85),(40,137,115,86),(41,166,306,212),(42,167,307,213),(43,168,308,214),(44,169,309,215),(45,170,310,216),(46,171,311,217),(47,172,312,218),(48,173,313,219),(49,174,314,220),(50,175,315,201),(51,176,316,202),(52,177,317,203),(53,178,318,204),(54,179,319,205),(55,180,320,206),(56,161,301,207),(57,162,302,208),(58,163,303,209),(59,164,304,210),(60,165,305,211),(141,293,183,227),(142,294,184,228),(143,295,185,229),(144,296,186,230),(145,297,187,231),(146,298,188,232),(147,299,189,233),(148,300,190,234),(149,281,191,235),(150,282,192,236),(151,283,193,237),(152,284,194,238),(153,285,195,239),(154,286,196,240),(155,287,197,221),(156,288,198,222),(157,289,199,223),(158,290,200,224),(159,291,181,225),(160,292,182,226)], [(1,40,246,115),(2,21,247,116),(3,22,248,117),(4,23,249,118),(5,24,250,119),(6,25,251,120),(7,26,252,101),(8,27,253,102),(9,28,254,103),(10,29,255,104),(11,30,256,105),(12,31,257,106),(13,32,258,107),(14,33,259,108),(15,34,260,109),(16,35,241,110),(17,36,242,111),(18,37,243,112),(19,38,244,113),(20,39,245,114),(41,225,306,291),(42,226,307,292),(43,227,308,293),(44,228,309,294),(45,229,310,295),(46,230,311,296),(47,231,312,297),(48,232,313,298),(49,233,314,299),(50,234,315,300),(51,235,316,281),(52,236,317,282),(53,237,318,283),(54,238,319,284),(55,239,320,285),(56,240,301,286),(57,221,302,287),(58,222,303,288),(59,223,304,289),(60,224,305,290),(61,126,269,95),(62,127,270,96),(63,128,271,97),(64,129,272,98),(65,130,273,99),(66,131,274,100),(67,132,275,81),(68,133,276,82),(69,134,277,83),(70,135,278,84),(71,136,279,85),(72,137,280,86),(73,138,261,87),(74,139,262,88),(75,140,263,89),(76,121,264,90),(77,122,265,91),(78,123,266,92),(79,124,267,93),(80,125,268,94),(141,168,183,214),(142,169,184,215),(143,170,185,216),(144,171,186,217),(145,172,187,218),(146,173,188,219),(147,174,189,220),(148,175,190,201),(149,176,191,202),(150,177,192,203),(151,178,193,204),(152,179,194,205),(153,180,195,206),(154,161,196,207),(155,162,197,208),(156,163,198,209),(157,164,199,210),(158,165,200,211),(159,166,181,212),(160,167,182,213)]])
200 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4AF | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 20A | ··· | 20AF | 20AG | ··· | 20DX |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | Q8 | C4○D4 | C5×Q8 | C5×C4○D4 |
| kernel | Q8×C2×C20 | C2×C4×C20 | C10×C4⋊C4 | Q8×C20 | Q8×C2×C10 | Q8×C10 | C2×C4×Q8 | C2×C42 | C2×C4⋊C4 | C4×Q8 | C22×Q8 | C2×Q8 | C2×C20 | C2×C10 | C2×C4 | C22 |
| # reps | 1 | 3 | 3 | 8 | 1 | 16 | 4 | 12 | 12 | 32 | 4 | 64 | 4 | 4 | 16 | 16 |
Matrix representation of Q8×C2×C20 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[9,0,0,0,0,40,0,0,0,0,10,0,0,0,0,10],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[1,0,0,0,0,40,0,0,0,0,9,0,0,0,0,32] >;
Q8×C2×C20 in GAP, Magma, Sage, TeX
Q_8\times C_2\times C_{20} % in TeX
G:=Group("Q8xC2xC20"); // GroupNames label
G:=SmallGroup(320,1518);
// by ID
G=gap.SmallGroup(320,1518);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,568,1276]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations