direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20⋊Q8, (C2×C20)⋊2Q8, C20⋊4(C2×Q8), C10⋊2(C4⋊Q8), Dic5⋊1(C2×Q8), (C2×Dic5)⋊8Q8, (C2×C4)⋊6Dic10, C4⋊2(C2×Dic10), C4⋊C4.258D10, C22.30(Q8×D5), C10.7(C22×Q8), (C2×C10).41C24, Dic5.14(C2×D4), C22.129(D4×D5), C10.39(C22×D4), (C2×C20).132C23, (C2×Dic5).165D4, (C22×C4).173D10, C2.9(C22×Dic10), C22.79(C23×D5), C4⋊Dic5.356C22, (C22×Dic10).9C2, (C2×Dic5).13C23, C22.36(C2×Dic10), C23.321(C22×D5), (C22×C10).390C23, (C22×C20).214C22, (C4×Dic5).278C22, (C2×Dic10).234C22, C10.D4.102C22, (C22×Dic5).229C22, C5⋊2(C2×C4⋊Q8), C2.5(C2×Q8×D5), C2.13(C2×D4×D5), (C2×C4⋊C4).24D5, (C10×C4⋊C4).17C2, (C2×C10).50(C2×Q8), (C2×C4×Dic5).12C2, (C2×C10).385(C2×D4), (C2×C4⋊Dic5).43C2, (C5×C4⋊C4).290C22, (C2×C4).137(C22×D5), (C2×C10.D4).23C2, SmallGroup(320,1169)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 894 in 290 conjugacy classes, 143 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×16], C22, C22 [×6], C5, C2×C4 [×10], C2×C4 [×24], Q8 [×16], C23, C10 [×3], C10 [×4], C42 [×4], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×16], Dic5 [×8], Dic5 [×4], C20 [×4], C20 [×4], C2×C10, C2×C10 [×6], C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×3], C4⋊Q8 [×8], C22×Q8 [×2], Dic10 [×16], C2×Dic5 [×16], C2×Dic5 [×4], C2×C20 [×10], C2×C20 [×4], C22×C10, C2×C4⋊Q8, C4×Dic5 [×4], C10.D4 [×8], C4⋊Dic5 [×4], C5×C4⋊C4 [×4], C2×Dic10 [×8], C2×Dic10 [×8], C22×Dic5 [×2], C22×Dic5 [×2], C22×C20, C22×C20 [×2], C20⋊Q8 [×8], C2×C4×Dic5, C2×C10.D4 [×2], C2×C4⋊Dic5, C10×C4⋊C4, C22×Dic10 [×2], C2×C20⋊Q8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×12], C24, D10 [×7], C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], Dic10 [×4], C22×D5 [×7], C2×C4⋊Q8, C2×Dic10 [×6], D4×D5 [×2], Q8×D5 [×2], C23×D5, C20⋊Q8 [×4], C22×Dic10, C2×D4×D5, C2×Q8×D5, C2×C20⋊Q8
Generators and relations
G = < a,b,c,d | a2=b20=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b11, dbd-1=b9, dcd-1=c-1 >
►Regular action on 320 points
Generators in S320
(1 188)(2 189)(3 190)(4 191)(5 192)(6 193)(7 194)(8 195)(9 196)(10 197)(11 198)(12 199)(13 200)(14 181)(15 182)(16 183)(17 184)(18 185)(19 186)(20 187)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 61)(38 62)(39 63)(40 64)(41 143)(42 144)(43 145)(44 146)(45 147)(46 148)(47 149)(48 150)(49 151)(50 152)(51 153)(52 154)(53 155)(54 156)(55 157)(56 158)(57 159)(58 160)(59 141)(60 142)(81 290)(82 291)(83 292)(84 293)(85 294)(86 295)(87 296)(88 297)(89 298)(90 299)(91 300)(92 281)(93 282)(94 283)(95 284)(96 285)(97 286)(98 287)(99 288)(100 289)(101 260)(102 241)(103 242)(104 243)(105 244)(106 245)(107 246)(108 247)(109 248)(110 249)(111 250)(112 251)(113 252)(114 253)(115 254)(116 255)(117 256)(118 257)(119 258)(120 259)(121 235)(122 236)(123 237)(124 238)(125 239)(126 240)(127 221)(128 222)(129 223)(130 224)(131 225)(132 226)(133 227)(134 228)(135 229)(136 230)(137 231)(138 232)(139 233)(140 234)(161 208)(162 209)(163 210)(164 211)(165 212)(166 213)(167 214)(168 215)(169 216)(170 217)(171 218)(172 219)(173 220)(174 201)(175 202)(176 203)(177 204)(178 205)(179 206)(180 207)(261 303)(262 304)(263 305)(264 306)(265 307)(266 308)(267 309)(268 310)(269 311)(270 312)(271 313)(272 314)(273 315)(274 316)(275 317)(276 318)(277 319)(278 320)(279 301)(280 302)
(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 79 106 165)(2 70 107 176)(3 61 108 167)(4 72 109 178)(5 63 110 169)(6 74 111 180)(7 65 112 171)(8 76 113 162)(9 67 114 173)(10 78 115 164)(11 69 116 175)(12 80 117 166)(13 71 118 177)(14 62 119 168)(15 73 120 179)(16 64 101 170)(17 75 102 161)(18 66 103 172)(19 77 104 163)(20 68 105 174)(21 251 218 194)(22 242 219 185)(23 253 220 196)(24 244 201 187)(25 255 202 198)(26 246 203 189)(27 257 204 200)(28 248 205 191)(29 259 206 182)(30 250 207 193)(31 241 208 184)(32 252 209 195)(33 243 210 186)(34 254 211 197)(35 245 212 188)(36 256 213 199)(37 247 214 190)(38 258 215 181)(39 249 216 192)(40 260 217 183)(41 136 289 264)(42 127 290 275)(43 138 291 266)(44 129 292 277)(45 140 293 268)(46 131 294 279)(47 122 295 270)(48 133 296 261)(49 124 297 272)(50 135 298 263)(51 126 299 274)(52 137 300 265)(53 128 281 276)(54 139 282 267)(55 130 283 278)(56 121 284 269)(57 132 285 280)(58 123 286 271)(59 134 287 262)(60 125 288 273)(81 317 144 221)(82 308 145 232)(83 319 146 223)(84 310 147 234)(85 301 148 225)(86 312 149 236)(87 303 150 227)(88 314 151 238)(89 305 152 229)(90 316 153 240)(91 307 154 231)(92 318 155 222)(93 309 156 233)(94 320 157 224)(95 311 158 235)(96 302 159 226)(97 313 160 237)(98 304 141 228)(99 315 142 239)(100 306 143 230)
(1 141 106 98)(2 150 107 87)(3 159 108 96)(4 148 109 85)(5 157 110 94)(6 146 111 83)(7 155 112 92)(8 144 113 81)(9 153 114 90)(10 142 115 99)(11 151 116 88)(12 160 117 97)(13 149 118 86)(14 158 119 95)(15 147 120 84)(16 156 101 93)(17 145 102 82)(18 154 103 91)(19 143 104 100)(20 152 105 89)(21 276 218 128)(22 265 219 137)(23 274 220 126)(24 263 201 135)(25 272 202 124)(26 261 203 133)(27 270 204 122)(28 279 205 131)(29 268 206 140)(30 277 207 129)(31 266 208 138)(32 275 209 127)(33 264 210 136)(34 273 211 125)(35 262 212 134)(36 271 213 123)(37 280 214 132)(38 269 215 121)(39 278 216 130)(40 267 217 139)(41 243 289 186)(42 252 290 195)(43 241 291 184)(44 250 292 193)(45 259 293 182)(46 248 294 191)(47 257 295 200)(48 246 296 189)(49 255 297 198)(50 244 298 187)(51 253 299 196)(52 242 300 185)(53 251 281 194)(54 260 282 183)(55 249 283 192)(56 258 284 181)(57 247 285 190)(58 256 286 199)(59 245 287 188)(60 254 288 197)(61 302 167 226)(62 311 168 235)(63 320 169 224)(64 309 170 233)(65 318 171 222)(66 307 172 231)(67 316 173 240)(68 305 174 229)(69 314 175 238)(70 303 176 227)(71 312 177 236)(72 301 178 225)(73 310 179 234)(74 319 180 223)(75 308 161 232)(76 317 162 221)(77 306 163 230)(78 315 164 239)(79 304 165 228)(80 313 166 237)
G:=sub<Sym(320)| (1,188)(2,189)(3,190)(4,191)(5,192)(6,193)(7,194)(8,195)(9,196)(10,197)(11,198)(12,199)(13,200)(14,181)(15,182)(16,183)(17,184)(18,185)(19,186)(20,187)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,61)(38,62)(39,63)(40,64)(41,143)(42,144)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,159)(58,160)(59,141)(60,142)(81,290)(82,291)(83,292)(84,293)(85,294)(86,295)(87,296)(88,297)(89,298)(90,299)(91,300)(92,281)(93,282)(94,283)(95,284)(96,285)(97,286)(98,287)(99,288)(100,289)(101,260)(102,241)(103,242)(104,243)(105,244)(106,245)(107,246)(108,247)(109,248)(110,249)(111,250)(112,251)(113,252)(114,253)(115,254)(116,255)(117,256)(118,257)(119,258)(120,259)(121,235)(122,236)(123,237)(124,238)(125,239)(126,240)(127,221)(128,222)(129,223)(130,224)(131,225)(132,226)(133,227)(134,228)(135,229)(136,230)(137,231)(138,232)(139,233)(140,234)(161,208)(162,209)(163,210)(164,211)(165,212)(166,213)(167,214)(168,215)(169,216)(170,217)(171,218)(172,219)(173,220)(174,201)(175,202)(176,203)(177,204)(178,205)(179,206)(180,207)(261,303)(262,304)(263,305)(264,306)(265,307)(266,308)(267,309)(268,310)(269,311)(270,312)(271,313)(272,314)(273,315)(274,316)(275,317)(276,318)(277,319)(278,320)(279,301)(280,302), (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), (1,79,106,165)(2,70,107,176)(3,61,108,167)(4,72,109,178)(5,63,110,169)(6,74,111,180)(7,65,112,171)(8,76,113,162)(9,67,114,173)(10,78,115,164)(11,69,116,175)(12,80,117,166)(13,71,118,177)(14,62,119,168)(15,73,120,179)(16,64,101,170)(17,75,102,161)(18,66,103,172)(19,77,104,163)(20,68,105,174)(21,251,218,194)(22,242,219,185)(23,253,220,196)(24,244,201,187)(25,255,202,198)(26,246,203,189)(27,257,204,200)(28,248,205,191)(29,259,206,182)(30,250,207,193)(31,241,208,184)(32,252,209,195)(33,243,210,186)(34,254,211,197)(35,245,212,188)(36,256,213,199)(37,247,214,190)(38,258,215,181)(39,249,216,192)(40,260,217,183)(41,136,289,264)(42,127,290,275)(43,138,291,266)(44,129,292,277)(45,140,293,268)(46,131,294,279)(47,122,295,270)(48,133,296,261)(49,124,297,272)(50,135,298,263)(51,126,299,274)(52,137,300,265)(53,128,281,276)(54,139,282,267)(55,130,283,278)(56,121,284,269)(57,132,285,280)(58,123,286,271)(59,134,287,262)(60,125,288,273)(81,317,144,221)(82,308,145,232)(83,319,146,223)(84,310,147,234)(85,301,148,225)(86,312,149,236)(87,303,150,227)(88,314,151,238)(89,305,152,229)(90,316,153,240)(91,307,154,231)(92,318,155,222)(93,309,156,233)(94,320,157,224)(95,311,158,235)(96,302,159,226)(97,313,160,237)(98,304,141,228)(99,315,142,239)(100,306,143,230), (1,141,106,98)(2,150,107,87)(3,159,108,96)(4,148,109,85)(5,157,110,94)(6,146,111,83)(7,155,112,92)(8,144,113,81)(9,153,114,90)(10,142,115,99)(11,151,116,88)(12,160,117,97)(13,149,118,86)(14,158,119,95)(15,147,120,84)(16,156,101,93)(17,145,102,82)(18,154,103,91)(19,143,104,100)(20,152,105,89)(21,276,218,128)(22,265,219,137)(23,274,220,126)(24,263,201,135)(25,272,202,124)(26,261,203,133)(27,270,204,122)(28,279,205,131)(29,268,206,140)(30,277,207,129)(31,266,208,138)(32,275,209,127)(33,264,210,136)(34,273,211,125)(35,262,212,134)(36,271,213,123)(37,280,214,132)(38,269,215,121)(39,278,216,130)(40,267,217,139)(41,243,289,186)(42,252,290,195)(43,241,291,184)(44,250,292,193)(45,259,293,182)(46,248,294,191)(47,257,295,200)(48,246,296,189)(49,255,297,198)(50,244,298,187)(51,253,299,196)(52,242,300,185)(53,251,281,194)(54,260,282,183)(55,249,283,192)(56,258,284,181)(57,247,285,190)(58,256,286,199)(59,245,287,188)(60,254,288,197)(61,302,167,226)(62,311,168,235)(63,320,169,224)(64,309,170,233)(65,318,171,222)(66,307,172,231)(67,316,173,240)(68,305,174,229)(69,314,175,238)(70,303,176,227)(71,312,177,236)(72,301,178,225)(73,310,179,234)(74,319,180,223)(75,308,161,232)(76,317,162,221)(77,306,163,230)(78,315,164,239)(79,304,165,228)(80,313,166,237)>;
G:=Group( (1,188)(2,189)(3,190)(4,191)(5,192)(6,193)(7,194)(8,195)(9,196)(10,197)(11,198)(12,199)(13,200)(14,181)(15,182)(16,183)(17,184)(18,185)(19,186)(20,187)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,61)(38,62)(39,63)(40,64)(41,143)(42,144)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,159)(58,160)(59,141)(60,142)(81,290)(82,291)(83,292)(84,293)(85,294)(86,295)(87,296)(88,297)(89,298)(90,299)(91,300)(92,281)(93,282)(94,283)(95,284)(96,285)(97,286)(98,287)(99,288)(100,289)(101,260)(102,241)(103,242)(104,243)(105,244)(106,245)(107,246)(108,247)(109,248)(110,249)(111,250)(112,251)(113,252)(114,253)(115,254)(116,255)(117,256)(118,257)(119,258)(120,259)(121,235)(122,236)(123,237)(124,238)(125,239)(126,240)(127,221)(128,222)(129,223)(130,224)(131,225)(132,226)(133,227)(134,228)(135,229)(136,230)(137,231)(138,232)(139,233)(140,234)(161,208)(162,209)(163,210)(164,211)(165,212)(166,213)(167,214)(168,215)(169,216)(170,217)(171,218)(172,219)(173,220)(174,201)(175,202)(176,203)(177,204)(178,205)(179,206)(180,207)(261,303)(262,304)(263,305)(264,306)(265,307)(266,308)(267,309)(268,310)(269,311)(270,312)(271,313)(272,314)(273,315)(274,316)(275,317)(276,318)(277,319)(278,320)(279,301)(280,302), (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), (1,79,106,165)(2,70,107,176)(3,61,108,167)(4,72,109,178)(5,63,110,169)(6,74,111,180)(7,65,112,171)(8,76,113,162)(9,67,114,173)(10,78,115,164)(11,69,116,175)(12,80,117,166)(13,71,118,177)(14,62,119,168)(15,73,120,179)(16,64,101,170)(17,75,102,161)(18,66,103,172)(19,77,104,163)(20,68,105,174)(21,251,218,194)(22,242,219,185)(23,253,220,196)(24,244,201,187)(25,255,202,198)(26,246,203,189)(27,257,204,200)(28,248,205,191)(29,259,206,182)(30,250,207,193)(31,241,208,184)(32,252,209,195)(33,243,210,186)(34,254,211,197)(35,245,212,188)(36,256,213,199)(37,247,214,190)(38,258,215,181)(39,249,216,192)(40,260,217,183)(41,136,289,264)(42,127,290,275)(43,138,291,266)(44,129,292,277)(45,140,293,268)(46,131,294,279)(47,122,295,270)(48,133,296,261)(49,124,297,272)(50,135,298,263)(51,126,299,274)(52,137,300,265)(53,128,281,276)(54,139,282,267)(55,130,283,278)(56,121,284,269)(57,132,285,280)(58,123,286,271)(59,134,287,262)(60,125,288,273)(81,317,144,221)(82,308,145,232)(83,319,146,223)(84,310,147,234)(85,301,148,225)(86,312,149,236)(87,303,150,227)(88,314,151,238)(89,305,152,229)(90,316,153,240)(91,307,154,231)(92,318,155,222)(93,309,156,233)(94,320,157,224)(95,311,158,235)(96,302,159,226)(97,313,160,237)(98,304,141,228)(99,315,142,239)(100,306,143,230), (1,141,106,98)(2,150,107,87)(3,159,108,96)(4,148,109,85)(5,157,110,94)(6,146,111,83)(7,155,112,92)(8,144,113,81)(9,153,114,90)(10,142,115,99)(11,151,116,88)(12,160,117,97)(13,149,118,86)(14,158,119,95)(15,147,120,84)(16,156,101,93)(17,145,102,82)(18,154,103,91)(19,143,104,100)(20,152,105,89)(21,276,218,128)(22,265,219,137)(23,274,220,126)(24,263,201,135)(25,272,202,124)(26,261,203,133)(27,270,204,122)(28,279,205,131)(29,268,206,140)(30,277,207,129)(31,266,208,138)(32,275,209,127)(33,264,210,136)(34,273,211,125)(35,262,212,134)(36,271,213,123)(37,280,214,132)(38,269,215,121)(39,278,216,130)(40,267,217,139)(41,243,289,186)(42,252,290,195)(43,241,291,184)(44,250,292,193)(45,259,293,182)(46,248,294,191)(47,257,295,200)(48,246,296,189)(49,255,297,198)(50,244,298,187)(51,253,299,196)(52,242,300,185)(53,251,281,194)(54,260,282,183)(55,249,283,192)(56,258,284,181)(57,247,285,190)(58,256,286,199)(59,245,287,188)(60,254,288,197)(61,302,167,226)(62,311,168,235)(63,320,169,224)(64,309,170,233)(65,318,171,222)(66,307,172,231)(67,316,173,240)(68,305,174,229)(69,314,175,238)(70,303,176,227)(71,312,177,236)(72,301,178,225)(73,310,179,234)(74,319,180,223)(75,308,161,232)(76,317,162,221)(77,306,163,230)(78,315,164,239)(79,304,165,228)(80,313,166,237) );
G=PermutationGroup([(1,188),(2,189),(3,190),(4,191),(5,192),(6,193),(7,194),(8,195),(9,196),(10,197),(11,198),(12,199),(13,200),(14,181),(15,182),(16,183),(17,184),(18,185),(19,186),(20,187),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,72),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(37,61),(38,62),(39,63),(40,64),(41,143),(42,144),(43,145),(44,146),(45,147),(46,148),(47,149),(48,150),(49,151),(50,152),(51,153),(52,154),(53,155),(54,156),(55,157),(56,158),(57,159),(58,160),(59,141),(60,142),(81,290),(82,291),(83,292),(84,293),(85,294),(86,295),(87,296),(88,297),(89,298),(90,299),(91,300),(92,281),(93,282),(94,283),(95,284),(96,285),(97,286),(98,287),(99,288),(100,289),(101,260),(102,241),(103,242),(104,243),(105,244),(106,245),(107,246),(108,247),(109,248),(110,249),(111,250),(112,251),(113,252),(114,253),(115,254),(116,255),(117,256),(118,257),(119,258),(120,259),(121,235),(122,236),(123,237),(124,238),(125,239),(126,240),(127,221),(128,222),(129,223),(130,224),(131,225),(132,226),(133,227),(134,228),(135,229),(136,230),(137,231),(138,232),(139,233),(140,234),(161,208),(162,209),(163,210),(164,211),(165,212),(166,213),(167,214),(168,215),(169,216),(170,217),(171,218),(172,219),(173,220),(174,201),(175,202),(176,203),(177,204),(178,205),(179,206),(180,207),(261,303),(262,304),(263,305),(264,306),(265,307),(266,308),(267,309),(268,310),(269,311),(270,312),(271,313),(272,314),(273,315),(274,316),(275,317),(276,318),(277,319),(278,320),(279,301),(280,302)], [(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)], [(1,79,106,165),(2,70,107,176),(3,61,108,167),(4,72,109,178),(5,63,110,169),(6,74,111,180),(7,65,112,171),(8,76,113,162),(9,67,114,173),(10,78,115,164),(11,69,116,175),(12,80,117,166),(13,71,118,177),(14,62,119,168),(15,73,120,179),(16,64,101,170),(17,75,102,161),(18,66,103,172),(19,77,104,163),(20,68,105,174),(21,251,218,194),(22,242,219,185),(23,253,220,196),(24,244,201,187),(25,255,202,198),(26,246,203,189),(27,257,204,200),(28,248,205,191),(29,259,206,182),(30,250,207,193),(31,241,208,184),(32,252,209,195),(33,243,210,186),(34,254,211,197),(35,245,212,188),(36,256,213,199),(37,247,214,190),(38,258,215,181),(39,249,216,192),(40,260,217,183),(41,136,289,264),(42,127,290,275),(43,138,291,266),(44,129,292,277),(45,140,293,268),(46,131,294,279),(47,122,295,270),(48,133,296,261),(49,124,297,272),(50,135,298,263),(51,126,299,274),(52,137,300,265),(53,128,281,276),(54,139,282,267),(55,130,283,278),(56,121,284,269),(57,132,285,280),(58,123,286,271),(59,134,287,262),(60,125,288,273),(81,317,144,221),(82,308,145,232),(83,319,146,223),(84,310,147,234),(85,301,148,225),(86,312,149,236),(87,303,150,227),(88,314,151,238),(89,305,152,229),(90,316,153,240),(91,307,154,231),(92,318,155,222),(93,309,156,233),(94,320,157,224),(95,311,158,235),(96,302,159,226),(97,313,160,237),(98,304,141,228),(99,315,142,239),(100,306,143,230)], [(1,141,106,98),(2,150,107,87),(3,159,108,96),(4,148,109,85),(5,157,110,94),(6,146,111,83),(7,155,112,92),(8,144,113,81),(9,153,114,90),(10,142,115,99),(11,151,116,88),(12,160,117,97),(13,149,118,86),(14,158,119,95),(15,147,120,84),(16,156,101,93),(17,145,102,82),(18,154,103,91),(19,143,104,100),(20,152,105,89),(21,276,218,128),(22,265,219,137),(23,274,220,126),(24,263,201,135),(25,272,202,124),(26,261,203,133),(27,270,204,122),(28,279,205,131),(29,268,206,140),(30,277,207,129),(31,266,208,138),(32,275,209,127),(33,264,210,136),(34,273,211,125),(35,262,212,134),(36,271,213,123),(37,280,214,132),(38,269,215,121),(39,278,216,130),(40,267,217,139),(41,243,289,186),(42,252,290,195),(43,241,291,184),(44,250,292,193),(45,259,293,182),(46,248,294,191),(47,257,295,200),(48,246,296,189),(49,255,297,198),(50,244,298,187),(51,253,299,196),(52,242,300,185),(53,251,281,194),(54,260,282,183),(55,249,283,192),(56,258,284,181),(57,247,285,190),(58,256,286,199),(59,245,287,188),(60,254,288,197),(61,302,167,226),(62,311,168,235),(63,320,169,224),(64,309,170,233),(65,318,171,222),(66,307,172,231),(67,316,173,240),(68,305,174,229),(69,314,175,238),(70,303,176,227),(71,312,177,236),(72,301,178,225),(73,310,179,234),(74,319,180,223),(75,308,161,232),(76,317,162,221),(77,306,163,230),(78,315,164,239),(79,304,165,228),(80,313,166,237)])
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 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 33 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 35 |
| 0 | 0 | 0 | 0 | 35 | 2 |
| 11 | 28 | 0 | 0 | 0 | 0 |
| 22 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 2 |
| 0 | 0 | 0 | 0 | 2 | 6 |
| 38 | 21 | 0 | 0 | 0 | 0 |
| 21 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 39 | 0 | 0 |
| 0 | 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 6 |
| 0 | 0 | 0 | 0 | 6 | 39 |
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],[40,33,0,0,0,0,1,7,0,0,0,0,0,0,0,1,0,0,0,0,40,35,0,0,0,0,0,0,39,35,0,0,0,0,35,2],[11,22,0,0,0,0,28,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,2,0,0,0,0,2,6],[38,21,0,0,0,0,21,3,0,0,0,0,0,0,28,2,0,0,0,0,39,13,0,0,0,0,0,0,2,6,0,0,0,0,6,39] >;
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | Q8 | D5 | D10 | D10 | Dic10 | D4×D5 | Q8×D5 |
| kernel | C2×C20⋊Q8 | C20⋊Q8 | C2×C4×Dic5 | C2×C10.D4 | C2×C4⋊Dic5 | C10×C4⋊C4 | C22×Dic10 | C2×Dic5 | C2×Dic5 | C2×C20 | C2×C4⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 8 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 8 | 6 | 16 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_2\times C_{20}\rtimes Q_8 % in TeX
G:=Group("C2xC20:Q8"); // GroupNames label
G:=SmallGroup(320,1169);
// by ID
G=gap.SmallGroup(320,1169);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,675,297,80,12550]);
// 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,c*b*c^-1=b^11,d*b*d^-1=b^9,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
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≀