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