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