metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C20)⋊10Q8, (C2×C4)⋊8Dic10, (C2×C4).89D20, C10.7(C4⋊Q8), C10.18(C4×Q8), (C2×C20).470D4, (C2×C42).14D5, (C2×Dic10)⋊17C4, C2.2(C20⋊2Q8), C2.11(C4×Dic10), C22.34(C2×D20), C20.93(C22⋊C4), C2.1(C4.D20), (C22×C4).395D10, C10.53(C22⋊Q8), C4.22(D10⋊C4), C2.2(C20.48D4), C10.10(C4.4D4), C22.40(C4○D20), (C22×Dic10).4C2, C22.17(C2×Dic10), C23.262(C22×D5), (C22×C20).470C22, (C22×C10).304C23, C5⋊3(C23.67C23), C10.10C42.10C2, (C22×Dic5).27C22, (C2×C4×C20).10C2, (C2×C4).109(C4×D5), (C2×C10).24(C2×Q8), C22.117(C2×C4×D5), (C2×C20).397(C2×C4), C2.5(C2×D10⋊C4), (C2×C10).424(C2×D4), C10.72(C2×C22⋊C4), (C2×C4⋊Dic5).15C2, C22.39(C2×C5⋊D4), (C2×C10).65(C4○D4), (C2×C4).238(C5⋊D4), (C2×Dic5).27(C2×C4), (C2×C10).195(C22×C4), SmallGroup(320,556)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C20)⋊10Q8
G = < a,b,c,d | a2=b20=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=ab9, dcd-1=c-1 >
Subgroups: 606 in 186 conjugacy classes, 87 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C10, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2.C42, C2×C42, C2×C4⋊C4, C22×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.67C23, C4⋊Dic5, C4×C20, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, C22×C20, C10.10C42, C2×C4⋊Dic5, C2×C4×C20, C22×Dic10, (C2×C20)⋊10Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C22⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, D10, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, Dic10, C4×D5, D20, C5⋊D4, C22×D5, C23.67C23, D10⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C4×Dic10, C20⋊2Q8, C4.D20, C20.48D4, C2×D10⋊C4, (C2×C20)⋊10Q8
►Regular action on 320 points
Generators in S320
(1 96)(2 97)(3 98)(4 99)(5 100)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 92)(18 93)(19 94)(20 95)(21 205)(22 206)(23 207)(24 208)(25 209)(26 210)(27 211)(28 212)(29 213)(30 214)(31 215)(32 216)(33 217)(34 218)(35 219)(36 220)(37 201)(38 202)(39 203)(40 204)(41 158)(42 159)(43 160)(44 141)(45 142)(46 143)(47 144)(48 145)(49 146)(50 147)(51 148)(52 149)(53 150)(54 151)(55 152)(56 153)(57 154)(58 155)(59 156)(60 157)(61 276)(62 277)(63 278)(64 279)(65 280)(66 261)(67 262)(68 263)(69 264)(70 265)(71 266)(72 267)(73 268)(74 269)(75 270)(76 271)(77 272)(78 273)(79 274)(80 275)(101 177)(102 178)(103 179)(104 180)(105 161)(106 162)(107 163)(108 164)(109 165)(110 166)(111 167)(112 168)(113 169)(114 170)(115 171)(116 172)(117 173)(118 174)(119 175)(120 176)(121 222)(122 223)(123 224)(124 225)(125 226)(126 227)(127 228)(128 229)(129 230)(130 231)(131 232)(132 233)(133 234)(134 235)(135 236)(136 237)(137 238)(138 239)(139 240)(140 221)(181 313)(182 314)(183 315)(184 316)(185 317)(186 318)(187 319)(188 320)(189 301)(190 302)(191 303)(192 304)(193 305)(194 306)(195 307)(196 308)(197 309)(198 310)(199 311)(200 312)(241 299)(242 300)(243 281)(244 282)(245 283)(246 284)(247 285)(248 286)(249 287)(250 288)(251 289)(252 290)(253 291)(254 292)(255 293)(256 294)(257 295)(258 296)(259 297)(260 298)
(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)
(1 310 132 176)(2 311 133 177)(3 312 134 178)(4 313 135 179)(5 314 136 180)(6 315 137 161)(7 316 138 162)(8 317 139 163)(9 318 140 164)(10 319 121 165)(11 320 122 166)(12 301 123 167)(13 302 124 168)(14 303 125 169)(15 304 126 170)(16 305 127 171)(17 306 128 172)(18 307 129 173)(19 308 130 174)(20 309 131 175)(21 255 158 61)(22 256 159 62)(23 257 160 63)(24 258 141 64)(25 259 142 65)(26 260 143 66)(27 241 144 67)(28 242 145 68)(29 243 146 69)(30 244 147 70)(31 245 148 71)(32 246 149 72)(33 247 150 73)(34 248 151 74)(35 249 152 75)(36 250 153 76)(37 251 154 77)(38 252 155 78)(39 253 156 79)(40 254 157 80)(41 276 205 293)(42 277 206 294)(43 278 207 295)(44 279 208 296)(45 280 209 297)(46 261 210 298)(47 262 211 299)(48 263 212 300)(49 264 213 281)(50 265 214 282)(51 266 215 283)(52 267 216 284)(53 268 217 285)(54 269 218 286)(55 270 219 287)(56 271 220 288)(57 272 201 289)(58 273 202 290)(59 274 203 291)(60 275 204 292)(81 183 238 105)(82 184 239 106)(83 185 240 107)(84 186 221 108)(85 187 222 109)(86 188 223 110)(87 189 224 111)(88 190 225 112)(89 191 226 113)(90 192 227 114)(91 193 228 115)(92 194 229 116)(93 195 230 117)(94 196 231 118)(95 197 232 119)(96 198 233 120)(97 199 234 101)(98 200 235 102)(99 181 236 103)(100 182 237 104)
(1 147 132 30)(2 59 133 203)(3 145 134 28)(4 57 135 201)(5 143 136 26)(6 55 137 219)(7 141 138 24)(8 53 139 217)(9 159 140 22)(10 51 121 215)(11 157 122 40)(12 49 123 213)(13 155 124 38)(14 47 125 211)(15 153 126 36)(16 45 127 209)(17 151 128 34)(18 43 129 207)(19 149 130 32)(20 41 131 205)(21 95 158 232)(23 93 160 230)(25 91 142 228)(27 89 144 226)(29 87 146 224)(31 85 148 222)(33 83 150 240)(35 81 152 238)(37 99 154 236)(39 97 156 234)(42 221 206 84)(44 239 208 82)(46 237 210 100)(48 235 212 98)(50 233 214 96)(52 231 216 94)(54 229 218 92)(56 227 220 90)(58 225 202 88)(60 223 204 86)(61 197 255 119)(62 318 256 164)(63 195 257 117)(64 316 258 162)(65 193 259 115)(66 314 260 180)(67 191 241 113)(68 312 242 178)(69 189 243 111)(70 310 244 176)(71 187 245 109)(72 308 246 174)(73 185 247 107)(74 306 248 172)(75 183 249 105)(76 304 250 170)(77 181 251 103)(78 302 252 168)(79 199 253 101)(80 320 254 166)(102 263 200 300)(104 261 182 298)(106 279 184 296)(108 277 186 294)(110 275 188 292)(112 273 190 290)(114 271 192 288)(116 269 194 286)(118 267 196 284)(120 265 198 282)(161 270 315 287)(163 268 317 285)(165 266 319 283)(167 264 301 281)(169 262 303 299)(171 280 305 297)(173 278 307 295)(175 276 309 293)(177 274 311 291)(179 272 313 289)
G:=sub<Sym(320)| (1,96)(2,97)(3,98)(4,99)(5,100)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,92)(18,93)(19,94)(20,95)(21,205)(22,206)(23,207)(24,208)(25,209)(26,210)(27,211)(28,212)(29,213)(30,214)(31,215)(32,216)(33,217)(34,218)(35,219)(36,220)(37,201)(38,202)(39,203)(40,204)(41,158)(42,159)(43,160)(44,141)(45,142)(46,143)(47,144)(48,145)(49,146)(50,147)(51,148)(52,149)(53,150)(54,151)(55,152)(56,153)(57,154)(58,155)(59,156)(60,157)(61,276)(62,277)(63,278)(64,279)(65,280)(66,261)(67,262)(68,263)(69,264)(70,265)(71,266)(72,267)(73,268)(74,269)(75,270)(76,271)(77,272)(78,273)(79,274)(80,275)(101,177)(102,178)(103,179)(104,180)(105,161)(106,162)(107,163)(108,164)(109,165)(110,166)(111,167)(112,168)(113,169)(114,170)(115,171)(116,172)(117,173)(118,174)(119,175)(120,176)(121,222)(122,223)(123,224)(124,225)(125,226)(126,227)(127,228)(128,229)(129,230)(130,231)(131,232)(132,233)(133,234)(134,235)(135,236)(136,237)(137,238)(138,239)(139,240)(140,221)(181,313)(182,314)(183,315)(184,316)(185,317)(186,318)(187,319)(188,320)(189,301)(190,302)(191,303)(192,304)(193,305)(194,306)(195,307)(196,308)(197,309)(198,310)(199,311)(200,312)(241,299)(242,300)(243,281)(244,282)(245,283)(246,284)(247,285)(248,286)(249,287)(250,288)(251,289)(252,290)(253,291)(254,292)(255,293)(256,294)(257,295)(258,296)(259,297)(260,298), (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), (1,310,132,176)(2,311,133,177)(3,312,134,178)(4,313,135,179)(5,314,136,180)(6,315,137,161)(7,316,138,162)(8,317,139,163)(9,318,140,164)(10,319,121,165)(11,320,122,166)(12,301,123,167)(13,302,124,168)(14,303,125,169)(15,304,126,170)(16,305,127,171)(17,306,128,172)(18,307,129,173)(19,308,130,174)(20,309,131,175)(21,255,158,61)(22,256,159,62)(23,257,160,63)(24,258,141,64)(25,259,142,65)(26,260,143,66)(27,241,144,67)(28,242,145,68)(29,243,146,69)(30,244,147,70)(31,245,148,71)(32,246,149,72)(33,247,150,73)(34,248,151,74)(35,249,152,75)(36,250,153,76)(37,251,154,77)(38,252,155,78)(39,253,156,79)(40,254,157,80)(41,276,205,293)(42,277,206,294)(43,278,207,295)(44,279,208,296)(45,280,209,297)(46,261,210,298)(47,262,211,299)(48,263,212,300)(49,264,213,281)(50,265,214,282)(51,266,215,283)(52,267,216,284)(53,268,217,285)(54,269,218,286)(55,270,219,287)(56,271,220,288)(57,272,201,289)(58,273,202,290)(59,274,203,291)(60,275,204,292)(81,183,238,105)(82,184,239,106)(83,185,240,107)(84,186,221,108)(85,187,222,109)(86,188,223,110)(87,189,224,111)(88,190,225,112)(89,191,226,113)(90,192,227,114)(91,193,228,115)(92,194,229,116)(93,195,230,117)(94,196,231,118)(95,197,232,119)(96,198,233,120)(97,199,234,101)(98,200,235,102)(99,181,236,103)(100,182,237,104), (1,147,132,30)(2,59,133,203)(3,145,134,28)(4,57,135,201)(5,143,136,26)(6,55,137,219)(7,141,138,24)(8,53,139,217)(9,159,140,22)(10,51,121,215)(11,157,122,40)(12,49,123,213)(13,155,124,38)(14,47,125,211)(15,153,126,36)(16,45,127,209)(17,151,128,34)(18,43,129,207)(19,149,130,32)(20,41,131,205)(21,95,158,232)(23,93,160,230)(25,91,142,228)(27,89,144,226)(29,87,146,224)(31,85,148,222)(33,83,150,240)(35,81,152,238)(37,99,154,236)(39,97,156,234)(42,221,206,84)(44,239,208,82)(46,237,210,100)(48,235,212,98)(50,233,214,96)(52,231,216,94)(54,229,218,92)(56,227,220,90)(58,225,202,88)(60,223,204,86)(61,197,255,119)(62,318,256,164)(63,195,257,117)(64,316,258,162)(65,193,259,115)(66,314,260,180)(67,191,241,113)(68,312,242,178)(69,189,243,111)(70,310,244,176)(71,187,245,109)(72,308,246,174)(73,185,247,107)(74,306,248,172)(75,183,249,105)(76,304,250,170)(77,181,251,103)(78,302,252,168)(79,199,253,101)(80,320,254,166)(102,263,200,300)(104,261,182,298)(106,279,184,296)(108,277,186,294)(110,275,188,292)(112,273,190,290)(114,271,192,288)(116,269,194,286)(118,267,196,284)(120,265,198,282)(161,270,315,287)(163,268,317,285)(165,266,319,283)(167,264,301,281)(169,262,303,299)(171,280,305,297)(173,278,307,295)(175,276,309,293)(177,274,311,291)(179,272,313,289)>;
G:=Group( (1,96)(2,97)(3,98)(4,99)(5,100)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,92)(18,93)(19,94)(20,95)(21,205)(22,206)(23,207)(24,208)(25,209)(26,210)(27,211)(28,212)(29,213)(30,214)(31,215)(32,216)(33,217)(34,218)(35,219)(36,220)(37,201)(38,202)(39,203)(40,204)(41,158)(42,159)(43,160)(44,141)(45,142)(46,143)(47,144)(48,145)(49,146)(50,147)(51,148)(52,149)(53,150)(54,151)(55,152)(56,153)(57,154)(58,155)(59,156)(60,157)(61,276)(62,277)(63,278)(64,279)(65,280)(66,261)(67,262)(68,263)(69,264)(70,265)(71,266)(72,267)(73,268)(74,269)(75,270)(76,271)(77,272)(78,273)(79,274)(80,275)(101,177)(102,178)(103,179)(104,180)(105,161)(106,162)(107,163)(108,164)(109,165)(110,166)(111,167)(112,168)(113,169)(114,170)(115,171)(116,172)(117,173)(118,174)(119,175)(120,176)(121,222)(122,223)(123,224)(124,225)(125,226)(126,227)(127,228)(128,229)(129,230)(130,231)(131,232)(132,233)(133,234)(134,235)(135,236)(136,237)(137,238)(138,239)(139,240)(140,221)(181,313)(182,314)(183,315)(184,316)(185,317)(186,318)(187,319)(188,320)(189,301)(190,302)(191,303)(192,304)(193,305)(194,306)(195,307)(196,308)(197,309)(198,310)(199,311)(200,312)(241,299)(242,300)(243,281)(244,282)(245,283)(246,284)(247,285)(248,286)(249,287)(250,288)(251,289)(252,290)(253,291)(254,292)(255,293)(256,294)(257,295)(258,296)(259,297)(260,298), (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), (1,310,132,176)(2,311,133,177)(3,312,134,178)(4,313,135,179)(5,314,136,180)(6,315,137,161)(7,316,138,162)(8,317,139,163)(9,318,140,164)(10,319,121,165)(11,320,122,166)(12,301,123,167)(13,302,124,168)(14,303,125,169)(15,304,126,170)(16,305,127,171)(17,306,128,172)(18,307,129,173)(19,308,130,174)(20,309,131,175)(21,255,158,61)(22,256,159,62)(23,257,160,63)(24,258,141,64)(25,259,142,65)(26,260,143,66)(27,241,144,67)(28,242,145,68)(29,243,146,69)(30,244,147,70)(31,245,148,71)(32,246,149,72)(33,247,150,73)(34,248,151,74)(35,249,152,75)(36,250,153,76)(37,251,154,77)(38,252,155,78)(39,253,156,79)(40,254,157,80)(41,276,205,293)(42,277,206,294)(43,278,207,295)(44,279,208,296)(45,280,209,297)(46,261,210,298)(47,262,211,299)(48,263,212,300)(49,264,213,281)(50,265,214,282)(51,266,215,283)(52,267,216,284)(53,268,217,285)(54,269,218,286)(55,270,219,287)(56,271,220,288)(57,272,201,289)(58,273,202,290)(59,274,203,291)(60,275,204,292)(81,183,238,105)(82,184,239,106)(83,185,240,107)(84,186,221,108)(85,187,222,109)(86,188,223,110)(87,189,224,111)(88,190,225,112)(89,191,226,113)(90,192,227,114)(91,193,228,115)(92,194,229,116)(93,195,230,117)(94,196,231,118)(95,197,232,119)(96,198,233,120)(97,199,234,101)(98,200,235,102)(99,181,236,103)(100,182,237,104), (1,147,132,30)(2,59,133,203)(3,145,134,28)(4,57,135,201)(5,143,136,26)(6,55,137,219)(7,141,138,24)(8,53,139,217)(9,159,140,22)(10,51,121,215)(11,157,122,40)(12,49,123,213)(13,155,124,38)(14,47,125,211)(15,153,126,36)(16,45,127,209)(17,151,128,34)(18,43,129,207)(19,149,130,32)(20,41,131,205)(21,95,158,232)(23,93,160,230)(25,91,142,228)(27,89,144,226)(29,87,146,224)(31,85,148,222)(33,83,150,240)(35,81,152,238)(37,99,154,236)(39,97,156,234)(42,221,206,84)(44,239,208,82)(46,237,210,100)(48,235,212,98)(50,233,214,96)(52,231,216,94)(54,229,218,92)(56,227,220,90)(58,225,202,88)(60,223,204,86)(61,197,255,119)(62,318,256,164)(63,195,257,117)(64,316,258,162)(65,193,259,115)(66,314,260,180)(67,191,241,113)(68,312,242,178)(69,189,243,111)(70,310,244,176)(71,187,245,109)(72,308,246,174)(73,185,247,107)(74,306,248,172)(75,183,249,105)(76,304,250,170)(77,181,251,103)(78,302,252,168)(79,199,253,101)(80,320,254,166)(102,263,200,300)(104,261,182,298)(106,279,184,296)(108,277,186,294)(110,275,188,292)(112,273,190,290)(114,271,192,288)(116,269,194,286)(118,267,196,284)(120,265,198,282)(161,270,315,287)(163,268,317,285)(165,266,319,283)(167,264,301,281)(169,262,303,299)(171,280,305,297)(173,278,307,295)(175,276,309,293)(177,274,311,291)(179,272,313,289) );
G=PermutationGroup([[(1,96),(2,97),(3,98),(4,99),(5,100),(6,81),(7,82),(8,83),(9,84),(10,85),(11,86),(12,87),(13,88),(14,89),(15,90),(16,91),(17,92),(18,93),(19,94),(20,95),(21,205),(22,206),(23,207),(24,208),(25,209),(26,210),(27,211),(28,212),(29,213),(30,214),(31,215),(32,216),(33,217),(34,218),(35,219),(36,220),(37,201),(38,202),(39,203),(40,204),(41,158),(42,159),(43,160),(44,141),(45,142),(46,143),(47,144),(48,145),(49,146),(50,147),(51,148),(52,149),(53,150),(54,151),(55,152),(56,153),(57,154),(58,155),(59,156),(60,157),(61,276),(62,277),(63,278),(64,279),(65,280),(66,261),(67,262),(68,263),(69,264),(70,265),(71,266),(72,267),(73,268),(74,269),(75,270),(76,271),(77,272),(78,273),(79,274),(80,275),(101,177),(102,178),(103,179),(104,180),(105,161),(106,162),(107,163),(108,164),(109,165),(110,166),(111,167),(112,168),(113,169),(114,170),(115,171),(116,172),(117,173),(118,174),(119,175),(120,176),(121,222),(122,223),(123,224),(124,225),(125,226),(126,227),(127,228),(128,229),(129,230),(130,231),(131,232),(132,233),(133,234),(134,235),(135,236),(136,237),(137,238),(138,239),(139,240),(140,221),(181,313),(182,314),(183,315),(184,316),(185,317),(186,318),(187,319),(188,320),(189,301),(190,302),(191,303),(192,304),(193,305),(194,306),(195,307),(196,308),(197,309),(198,310),(199,311),(200,312),(241,299),(242,300),(243,281),(244,282),(245,283),(246,284),(247,285),(248,286),(249,287),(250,288),(251,289),(252,290),(253,291),(254,292),(255,293),(256,294),(257,295),(258,296),(259,297),(260,298)], [(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)], [(1,310,132,176),(2,311,133,177),(3,312,134,178),(4,313,135,179),(5,314,136,180),(6,315,137,161),(7,316,138,162),(8,317,139,163),(9,318,140,164),(10,319,121,165),(11,320,122,166),(12,301,123,167),(13,302,124,168),(14,303,125,169),(15,304,126,170),(16,305,127,171),(17,306,128,172),(18,307,129,173),(19,308,130,174),(20,309,131,175),(21,255,158,61),(22,256,159,62),(23,257,160,63),(24,258,141,64),(25,259,142,65),(26,260,143,66),(27,241,144,67),(28,242,145,68),(29,243,146,69),(30,244,147,70),(31,245,148,71),(32,246,149,72),(33,247,150,73),(34,248,151,74),(35,249,152,75),(36,250,153,76),(37,251,154,77),(38,252,155,78),(39,253,156,79),(40,254,157,80),(41,276,205,293),(42,277,206,294),(43,278,207,295),(44,279,208,296),(45,280,209,297),(46,261,210,298),(47,262,211,299),(48,263,212,300),(49,264,213,281),(50,265,214,282),(51,266,215,283),(52,267,216,284),(53,268,217,285),(54,269,218,286),(55,270,219,287),(56,271,220,288),(57,272,201,289),(58,273,202,290),(59,274,203,291),(60,275,204,292),(81,183,238,105),(82,184,239,106),(83,185,240,107),(84,186,221,108),(85,187,222,109),(86,188,223,110),(87,189,224,111),(88,190,225,112),(89,191,226,113),(90,192,227,114),(91,193,228,115),(92,194,229,116),(93,195,230,117),(94,196,231,118),(95,197,232,119),(96,198,233,120),(97,199,234,101),(98,200,235,102),(99,181,236,103),(100,182,237,104)], [(1,147,132,30),(2,59,133,203),(3,145,134,28),(4,57,135,201),(5,143,136,26),(6,55,137,219),(7,141,138,24),(8,53,139,217),(9,159,140,22),(10,51,121,215),(11,157,122,40),(12,49,123,213),(13,155,124,38),(14,47,125,211),(15,153,126,36),(16,45,127,209),(17,151,128,34),(18,43,129,207),(19,149,130,32),(20,41,131,205),(21,95,158,232),(23,93,160,230),(25,91,142,228),(27,89,144,226),(29,87,146,224),(31,85,148,222),(33,83,150,240),(35,81,152,238),(37,99,154,236),(39,97,156,234),(42,221,206,84),(44,239,208,82),(46,237,210,100),(48,235,212,98),(50,233,214,96),(52,231,216,94),(54,229,218,92),(56,227,220,90),(58,225,202,88),(60,223,204,86),(61,197,255,119),(62,318,256,164),(63,195,257,117),(64,316,258,162),(65,193,259,115),(66,314,260,180),(67,191,241,113),(68,312,242,178),(69,189,243,111),(70,310,244,176),(71,187,245,109),(72,308,246,174),(73,185,247,107),(74,306,248,172),(75,183,249,105),(76,304,250,170),(77,181,251,103),(78,302,252,168),(79,199,253,101),(80,320,254,166),(102,263,200,300),(104,261,182,298),(106,279,184,296),(108,277,186,294),(110,275,188,292),(112,273,190,290),(114,271,192,288),(116,269,194,286),(118,267,196,284),(120,265,198,282),(161,270,315,287),(163,268,317,285),(165,266,319,283),(167,264,301,281),(169,262,303,299),(171,280,305,297),(173,278,307,295),(175,276,309,293),(177,274,311,291),(179,272,313,289)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D5 | C4○D4 | D10 | Dic10 | C4×D5 | D20 | C5⋊D4 | C4○D20 |
| kernel | (C2×C20)⋊10Q8 | C10.10C42 | C2×C4⋊Dic5 | C2×C4×C20 | C22×Dic10 | C2×Dic10 | C2×C20 | C2×C20 | C2×C42 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 4 | 2 | 4 | 6 | 16 | 8 | 8 | 8 | 16 |
Matrix representation of (C2×C20)⋊10Q8 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 22 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 2 | 0 | 0 |
| 0 | 0 | 25 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 11 |
| 0 | 0 | 0 | 0 | 30 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 28 | 0 | 0 |
| 0 | 0 | 22 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 20 | 2 | 0 | 0 | 0 | 0 |
| 26 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 31 | 0 | 0 |
| 0 | 0 | 38 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 21 |
| 0 | 0 | 0 | 0 | 3 | 15 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[22,32,0,0,0,0,9,0,0,0,0,0,0,0,30,25,0,0,0,0,2,14,0,0,0,0,0,0,27,30,0,0,0,0,11,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,22,0,0,0,0,28,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[20,26,0,0,0,0,2,21,0,0,0,0,0,0,16,38,0,0,0,0,31,25,0,0,0,0,0,0,26,3,0,0,0,0,21,15] >;
(C2×C20)⋊10Q8 in GAP, Magma, Sage, TeX
(C_2\times C_{20})\rtimes_{10}Q_8 % in TeX
G:=Group("(C2xC20):10Q8"); // GroupNames label
G:=SmallGroup(320,556);
// by ID
G=gap.SmallGroup(320,556);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,758,58,12550]);
// 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,b*c=c*b,d*b*d^-1=a*b^9,d*c*d^-1=c^-1>;
// generators/relations