metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C20)⋊6Q8, (C2×Dic5)⋊5Q8, (C2×C4)⋊5Dic10, C2.1(C20⋊Q8), C10.9(C4⋊Q8), C10.10(C4×Q8), C22.9(Q8×D5), C2.4(C4×Dic10), C22.50(D4×D5), (C22×C4).5D10, (C2×Dic10)⋊15C4, C10.1(C22⋊Q8), C2.1(D10⋊Q8), (C2×Dic5).181D4, C2.C42.7D5, C2.4(Dic5⋊3Q8), C10.14(C4.4D4), C22.25(C4○D20), (C22×Dic10).1C2, C22.12(C2×Dic10), C23.243(C22×D5), C22.25(D4⋊2D5), Dic5.18(C22⋊C4), (C22×C20).326C22, (C22×C10).271C23, C5⋊2(C23.67C23), C2.1(Dic5.5D4), (C22×Dic5).1C22, C10.10C42.30C2, C2.1(Dic5.14D4), (C2×C4).22(C4×D5), C2.5(D5×C22⋊C4), C22.80(C2×C4×D5), (C2×C10).56(C2×Q8), (C2×C4×Dic5).20C2, (C2×C20).203(C2×C4), (C2×C10).187(C2×D4), C10.42(C2×C22⋊C4), (C2×Dic5).84(C2×C4), (C2×C10.D4).3C2, (C2×C10).120(C4○D4), (C2×C10).138(C22×C4), (C5×C2.C42).14C2, SmallGroup(320,273)
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, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b9, dcd-1=c-1 >
Subgroups: 622 in 186 conjugacy classes, 77 normal (51 characteristic)
C1, C2, C4, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic5, Dic5, C20, C2×C10, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C22×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.67C23, C4×Dic5, C10.D4, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, C10.10C42, C5×C2.C42, C2×C4×Dic5, C2×C10.D4, C22×Dic10, (C2×C20)⋊Q8
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, C22×D5, C23.67C23, C2×Dic10, C2×C4×D5, C4○D20, D4×D5, D4⋊2D5, Q8×D5, C4×Dic10, Dic5.14D4, D5×C22⋊C4, Dic5.5D4, Dic5⋊3Q8, C20⋊Q8, D10⋊Q8, (C2×C20)⋊Q8
►Regular action on 320 points
Generators in S320
(1 171)(2 172)(3 173)(4 174)(5 175)(6 176)(7 177)(8 178)(9 179)(10 180)(11 161)(12 162)(13 163)(14 164)(15 165)(16 166)(17 167)(18 168)(19 169)(20 170)(21 270)(22 271)(23 272)(24 273)(25 274)(26 275)(27 276)(28 277)(29 278)(30 279)(31 280)(32 261)(33 262)(34 263)(35 264)(36 265)(37 266)(38 267)(39 268)(40 269)(41 208)(42 209)(43 210)(44 211)(45 212)(46 213)(47 214)(48 215)(49 216)(50 217)(51 218)(52 219)(53 220)(54 201)(55 202)(56 203)(57 204)(58 205)(59 206)(60 207)(61 303)(62 304)(63 305)(64 306)(65 307)(66 308)(67 309)(68 310)(69 311)(70 312)(71 313)(72 314)(73 315)(74 316)(75 317)(76 318)(77 319)(78 320)(79 301)(80 302)(81 254)(82 255)(83 256)(84 257)(85 258)(86 259)(87 260)(88 241)(89 242)(90 243)(91 244)(92 245)(93 246)(94 247)(95 248)(96 249)(97 250)(98 251)(99 252)(100 253)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 129)(110 130)(111 131)(112 132)(113 133)(114 134)(115 135)(116 136)(117 137)(118 138)(119 139)(120 140)(141 282)(142 283)(143 284)(144 285)(145 286)(146 287)(147 288)(148 289)(149 290)(150 291)(151 292)(152 293)(153 294)(154 295)(155 296)(156 297)(157 298)(158 299)(159 300)(160 281)(181 236)(182 237)(183 238)(184 239)(185 240)(186 221)(187 222)(188 223)(189 224)(190 225)(191 226)(192 227)(193 228)(194 229)(195 230)(196 231)(197 232)(198 233)(199 234)(200 235)
(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 27 204 304)(2 277 205 63)(3 29 206 306)(4 279 207 65)(5 31 208 308)(6 261 209 67)(7 33 210 310)(8 263 211 69)(9 35 212 312)(10 265 213 71)(11 37 214 314)(12 267 215 73)(13 39 216 316)(14 269 217 75)(15 21 218 318)(16 271 219 77)(17 23 220 320)(18 273 201 79)(19 25 202 302)(20 275 203 61)(22 52 319 166)(24 54 301 168)(26 56 303 170)(28 58 305 172)(30 60 307 174)(32 42 309 176)(34 44 311 178)(36 46 313 180)(38 48 315 162)(40 50 317 164)(41 66 175 280)(43 68 177 262)(45 70 179 264)(47 72 161 266)(49 74 163 268)(51 76 165 270)(53 78 167 272)(55 80 169 274)(57 62 171 276)(59 64 173 278)(81 285 235 114)(82 145 236 135)(83 287 237 116)(84 147 238 137)(85 289 239 118)(86 149 240 139)(87 291 221 120)(88 151 222 121)(89 293 223 102)(90 153 224 123)(91 295 225 104)(92 155 226 125)(93 297 227 106)(94 157 228 127)(95 299 229 108)(96 159 230 129)(97 281 231 110)(98 141 232 131)(99 283 233 112)(100 143 234 133)(101 241 292 187)(103 243 294 189)(105 245 296 191)(107 247 298 193)(109 249 300 195)(111 251 282 197)(113 253 284 199)(115 255 286 181)(117 257 288 183)(119 259 290 185)(122 242 152 188)(124 244 154 190)(126 246 156 192)(128 248 158 194)(130 250 160 196)(132 252 142 198)(134 254 144 200)(136 256 146 182)(138 258 148 184)(140 260 150 186)
(1 120 204 291)(2 109 205 300)(3 118 206 289)(4 107 207 298)(5 116 208 287)(6 105 209 296)(7 114 210 285)(8 103 211 294)(9 112 212 283)(10 101 213 292)(11 110 214 281)(12 119 215 290)(13 108 216 299)(14 117 217 288)(15 106 218 297)(16 115 219 286)(17 104 220 295)(18 113 201 284)(19 102 202 293)(20 111 203 282)(21 227 318 93)(22 236 319 82)(23 225 320 91)(24 234 301 100)(25 223 302 89)(26 232 303 98)(27 221 304 87)(28 230 305 96)(29 239 306 85)(30 228 307 94)(31 237 308 83)(32 226 309 92)(33 235 310 81)(34 224 311 90)(35 233 312 99)(36 222 313 88)(37 231 314 97)(38 240 315 86)(39 229 316 95)(40 238 317 84)(41 146 175 136)(42 155 176 125)(43 144 177 134)(44 153 178 123)(45 142 179 132)(46 151 180 121)(47 160 161 130)(48 149 162 139)(49 158 163 128)(50 147 164 137)(51 156 165 126)(52 145 166 135)(53 154 167 124)(54 143 168 133)(55 152 169 122)(56 141 170 131)(57 150 171 140)(58 159 172 129)(59 148 173 138)(60 157 174 127)(61 251 275 197)(62 260 276 186)(63 249 277 195)(64 258 278 184)(65 247 279 193)(66 256 280 182)(67 245 261 191)(68 254 262 200)(69 243 263 189)(70 252 264 198)(71 241 265 187)(72 250 266 196)(73 259 267 185)(74 248 268 194)(75 257 269 183)(76 246 270 192)(77 255 271 181)(78 244 272 190)(79 253 273 199)(80 242 274 188)
G:=sub<Sym(320)| (1,171)(2,172)(3,173)(4,174)(5,175)(6,176)(7,177)(8,178)(9,179)(10,180)(11,161)(12,162)(13,163)(14,164)(15,165)(16,166)(17,167)(18,168)(19,169)(20,170)(21,270)(22,271)(23,272)(24,273)(25,274)(26,275)(27,276)(28,277)(29,278)(30,279)(31,280)(32,261)(33,262)(34,263)(35,264)(36,265)(37,266)(38,267)(39,268)(40,269)(41,208)(42,209)(43,210)(44,211)(45,212)(46,213)(47,214)(48,215)(49,216)(50,217)(51,218)(52,219)(53,220)(54,201)(55,202)(56,203)(57,204)(58,205)(59,206)(60,207)(61,303)(62,304)(63,305)(64,306)(65,307)(66,308)(67,309)(68,310)(69,311)(70,312)(71,313)(72,314)(73,315)(74,316)(75,317)(76,318)(77,319)(78,320)(79,301)(80,302)(81,254)(82,255)(83,256)(84,257)(85,258)(86,259)(87,260)(88,241)(89,242)(90,243)(91,244)(92,245)(93,246)(94,247)(95,248)(96,249)(97,250)(98,251)(99,252)(100,253)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140)(141,282)(142,283)(143,284)(144,285)(145,286)(146,287)(147,288)(148,289)(149,290)(150,291)(151,292)(152,293)(153,294)(154,295)(155,296)(156,297)(157,298)(158,299)(159,300)(160,281)(181,236)(182,237)(183,238)(184,239)(185,240)(186,221)(187,222)(188,223)(189,224)(190,225)(191,226)(192,227)(193,228)(194,229)(195,230)(196,231)(197,232)(198,233)(199,234)(200,235), (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), (1,27,204,304)(2,277,205,63)(3,29,206,306)(4,279,207,65)(5,31,208,308)(6,261,209,67)(7,33,210,310)(8,263,211,69)(9,35,212,312)(10,265,213,71)(11,37,214,314)(12,267,215,73)(13,39,216,316)(14,269,217,75)(15,21,218,318)(16,271,219,77)(17,23,220,320)(18,273,201,79)(19,25,202,302)(20,275,203,61)(22,52,319,166)(24,54,301,168)(26,56,303,170)(28,58,305,172)(30,60,307,174)(32,42,309,176)(34,44,311,178)(36,46,313,180)(38,48,315,162)(40,50,317,164)(41,66,175,280)(43,68,177,262)(45,70,179,264)(47,72,161,266)(49,74,163,268)(51,76,165,270)(53,78,167,272)(55,80,169,274)(57,62,171,276)(59,64,173,278)(81,285,235,114)(82,145,236,135)(83,287,237,116)(84,147,238,137)(85,289,239,118)(86,149,240,139)(87,291,221,120)(88,151,222,121)(89,293,223,102)(90,153,224,123)(91,295,225,104)(92,155,226,125)(93,297,227,106)(94,157,228,127)(95,299,229,108)(96,159,230,129)(97,281,231,110)(98,141,232,131)(99,283,233,112)(100,143,234,133)(101,241,292,187)(103,243,294,189)(105,245,296,191)(107,247,298,193)(109,249,300,195)(111,251,282,197)(113,253,284,199)(115,255,286,181)(117,257,288,183)(119,259,290,185)(122,242,152,188)(124,244,154,190)(126,246,156,192)(128,248,158,194)(130,250,160,196)(132,252,142,198)(134,254,144,200)(136,256,146,182)(138,258,148,184)(140,260,150,186), (1,120,204,291)(2,109,205,300)(3,118,206,289)(4,107,207,298)(5,116,208,287)(6,105,209,296)(7,114,210,285)(8,103,211,294)(9,112,212,283)(10,101,213,292)(11,110,214,281)(12,119,215,290)(13,108,216,299)(14,117,217,288)(15,106,218,297)(16,115,219,286)(17,104,220,295)(18,113,201,284)(19,102,202,293)(20,111,203,282)(21,227,318,93)(22,236,319,82)(23,225,320,91)(24,234,301,100)(25,223,302,89)(26,232,303,98)(27,221,304,87)(28,230,305,96)(29,239,306,85)(30,228,307,94)(31,237,308,83)(32,226,309,92)(33,235,310,81)(34,224,311,90)(35,233,312,99)(36,222,313,88)(37,231,314,97)(38,240,315,86)(39,229,316,95)(40,238,317,84)(41,146,175,136)(42,155,176,125)(43,144,177,134)(44,153,178,123)(45,142,179,132)(46,151,180,121)(47,160,161,130)(48,149,162,139)(49,158,163,128)(50,147,164,137)(51,156,165,126)(52,145,166,135)(53,154,167,124)(54,143,168,133)(55,152,169,122)(56,141,170,131)(57,150,171,140)(58,159,172,129)(59,148,173,138)(60,157,174,127)(61,251,275,197)(62,260,276,186)(63,249,277,195)(64,258,278,184)(65,247,279,193)(66,256,280,182)(67,245,261,191)(68,254,262,200)(69,243,263,189)(70,252,264,198)(71,241,265,187)(72,250,266,196)(73,259,267,185)(74,248,268,194)(75,257,269,183)(76,246,270,192)(77,255,271,181)(78,244,272,190)(79,253,273,199)(80,242,274,188)>;
G:=Group( (1,171)(2,172)(3,173)(4,174)(5,175)(6,176)(7,177)(8,178)(9,179)(10,180)(11,161)(12,162)(13,163)(14,164)(15,165)(16,166)(17,167)(18,168)(19,169)(20,170)(21,270)(22,271)(23,272)(24,273)(25,274)(26,275)(27,276)(28,277)(29,278)(30,279)(31,280)(32,261)(33,262)(34,263)(35,264)(36,265)(37,266)(38,267)(39,268)(40,269)(41,208)(42,209)(43,210)(44,211)(45,212)(46,213)(47,214)(48,215)(49,216)(50,217)(51,218)(52,219)(53,220)(54,201)(55,202)(56,203)(57,204)(58,205)(59,206)(60,207)(61,303)(62,304)(63,305)(64,306)(65,307)(66,308)(67,309)(68,310)(69,311)(70,312)(71,313)(72,314)(73,315)(74,316)(75,317)(76,318)(77,319)(78,320)(79,301)(80,302)(81,254)(82,255)(83,256)(84,257)(85,258)(86,259)(87,260)(88,241)(89,242)(90,243)(91,244)(92,245)(93,246)(94,247)(95,248)(96,249)(97,250)(98,251)(99,252)(100,253)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140)(141,282)(142,283)(143,284)(144,285)(145,286)(146,287)(147,288)(148,289)(149,290)(150,291)(151,292)(152,293)(153,294)(154,295)(155,296)(156,297)(157,298)(158,299)(159,300)(160,281)(181,236)(182,237)(183,238)(184,239)(185,240)(186,221)(187,222)(188,223)(189,224)(190,225)(191,226)(192,227)(193,228)(194,229)(195,230)(196,231)(197,232)(198,233)(199,234)(200,235), (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), (1,27,204,304)(2,277,205,63)(3,29,206,306)(4,279,207,65)(5,31,208,308)(6,261,209,67)(7,33,210,310)(8,263,211,69)(9,35,212,312)(10,265,213,71)(11,37,214,314)(12,267,215,73)(13,39,216,316)(14,269,217,75)(15,21,218,318)(16,271,219,77)(17,23,220,320)(18,273,201,79)(19,25,202,302)(20,275,203,61)(22,52,319,166)(24,54,301,168)(26,56,303,170)(28,58,305,172)(30,60,307,174)(32,42,309,176)(34,44,311,178)(36,46,313,180)(38,48,315,162)(40,50,317,164)(41,66,175,280)(43,68,177,262)(45,70,179,264)(47,72,161,266)(49,74,163,268)(51,76,165,270)(53,78,167,272)(55,80,169,274)(57,62,171,276)(59,64,173,278)(81,285,235,114)(82,145,236,135)(83,287,237,116)(84,147,238,137)(85,289,239,118)(86,149,240,139)(87,291,221,120)(88,151,222,121)(89,293,223,102)(90,153,224,123)(91,295,225,104)(92,155,226,125)(93,297,227,106)(94,157,228,127)(95,299,229,108)(96,159,230,129)(97,281,231,110)(98,141,232,131)(99,283,233,112)(100,143,234,133)(101,241,292,187)(103,243,294,189)(105,245,296,191)(107,247,298,193)(109,249,300,195)(111,251,282,197)(113,253,284,199)(115,255,286,181)(117,257,288,183)(119,259,290,185)(122,242,152,188)(124,244,154,190)(126,246,156,192)(128,248,158,194)(130,250,160,196)(132,252,142,198)(134,254,144,200)(136,256,146,182)(138,258,148,184)(140,260,150,186), (1,120,204,291)(2,109,205,300)(3,118,206,289)(4,107,207,298)(5,116,208,287)(6,105,209,296)(7,114,210,285)(8,103,211,294)(9,112,212,283)(10,101,213,292)(11,110,214,281)(12,119,215,290)(13,108,216,299)(14,117,217,288)(15,106,218,297)(16,115,219,286)(17,104,220,295)(18,113,201,284)(19,102,202,293)(20,111,203,282)(21,227,318,93)(22,236,319,82)(23,225,320,91)(24,234,301,100)(25,223,302,89)(26,232,303,98)(27,221,304,87)(28,230,305,96)(29,239,306,85)(30,228,307,94)(31,237,308,83)(32,226,309,92)(33,235,310,81)(34,224,311,90)(35,233,312,99)(36,222,313,88)(37,231,314,97)(38,240,315,86)(39,229,316,95)(40,238,317,84)(41,146,175,136)(42,155,176,125)(43,144,177,134)(44,153,178,123)(45,142,179,132)(46,151,180,121)(47,160,161,130)(48,149,162,139)(49,158,163,128)(50,147,164,137)(51,156,165,126)(52,145,166,135)(53,154,167,124)(54,143,168,133)(55,152,169,122)(56,141,170,131)(57,150,171,140)(58,159,172,129)(59,148,173,138)(60,157,174,127)(61,251,275,197)(62,260,276,186)(63,249,277,195)(64,258,278,184)(65,247,279,193)(66,256,280,182)(67,245,261,191)(68,254,262,200)(69,243,263,189)(70,252,264,198)(71,241,265,187)(72,250,266,196)(73,259,267,185)(74,248,268,194)(75,257,269,183)(76,246,270,192)(77,255,271,181)(78,244,272,190)(79,253,273,199)(80,242,274,188) );
G=PermutationGroup([[(1,171),(2,172),(3,173),(4,174),(5,175),(6,176),(7,177),(8,178),(9,179),(10,180),(11,161),(12,162),(13,163),(14,164),(15,165),(16,166),(17,167),(18,168),(19,169),(20,170),(21,270),(22,271),(23,272),(24,273),(25,274),(26,275),(27,276),(28,277),(29,278),(30,279),(31,280),(32,261),(33,262),(34,263),(35,264),(36,265),(37,266),(38,267),(39,268),(40,269),(41,208),(42,209),(43,210),(44,211),(45,212),(46,213),(47,214),(48,215),(49,216),(50,217),(51,218),(52,219),(53,220),(54,201),(55,202),(56,203),(57,204),(58,205),(59,206),(60,207),(61,303),(62,304),(63,305),(64,306),(65,307),(66,308),(67,309),(68,310),(69,311),(70,312),(71,313),(72,314),(73,315),(74,316),(75,317),(76,318),(77,319),(78,320),(79,301),(80,302),(81,254),(82,255),(83,256),(84,257),(85,258),(86,259),(87,260),(88,241),(89,242),(90,243),(91,244),(92,245),(93,246),(94,247),(95,248),(96,249),(97,250),(98,251),(99,252),(100,253),(101,121),(102,122),(103,123),(104,124),(105,125),(106,126),(107,127),(108,128),(109,129),(110,130),(111,131),(112,132),(113,133),(114,134),(115,135),(116,136),(117,137),(118,138),(119,139),(120,140),(141,282),(142,283),(143,284),(144,285),(145,286),(146,287),(147,288),(148,289),(149,290),(150,291),(151,292),(152,293),(153,294),(154,295),(155,296),(156,297),(157,298),(158,299),(159,300),(160,281),(181,236),(182,237),(183,238),(184,239),(185,240),(186,221),(187,222),(188,223),(189,224),(190,225),(191,226),(192,227),(193,228),(194,229),(195,230),(196,231),(197,232),(198,233),(199,234),(200,235)], [(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)], [(1,27,204,304),(2,277,205,63),(3,29,206,306),(4,279,207,65),(5,31,208,308),(6,261,209,67),(7,33,210,310),(8,263,211,69),(9,35,212,312),(10,265,213,71),(11,37,214,314),(12,267,215,73),(13,39,216,316),(14,269,217,75),(15,21,218,318),(16,271,219,77),(17,23,220,320),(18,273,201,79),(19,25,202,302),(20,275,203,61),(22,52,319,166),(24,54,301,168),(26,56,303,170),(28,58,305,172),(30,60,307,174),(32,42,309,176),(34,44,311,178),(36,46,313,180),(38,48,315,162),(40,50,317,164),(41,66,175,280),(43,68,177,262),(45,70,179,264),(47,72,161,266),(49,74,163,268),(51,76,165,270),(53,78,167,272),(55,80,169,274),(57,62,171,276),(59,64,173,278),(81,285,235,114),(82,145,236,135),(83,287,237,116),(84,147,238,137),(85,289,239,118),(86,149,240,139),(87,291,221,120),(88,151,222,121),(89,293,223,102),(90,153,224,123),(91,295,225,104),(92,155,226,125),(93,297,227,106),(94,157,228,127),(95,299,229,108),(96,159,230,129),(97,281,231,110),(98,141,232,131),(99,283,233,112),(100,143,234,133),(101,241,292,187),(103,243,294,189),(105,245,296,191),(107,247,298,193),(109,249,300,195),(111,251,282,197),(113,253,284,199),(115,255,286,181),(117,257,288,183),(119,259,290,185),(122,242,152,188),(124,244,154,190),(126,246,156,192),(128,248,158,194),(130,250,160,196),(132,252,142,198),(134,254,144,200),(136,256,146,182),(138,258,148,184),(140,260,150,186)], [(1,120,204,291),(2,109,205,300),(3,118,206,289),(4,107,207,298),(5,116,208,287),(6,105,209,296),(7,114,210,285),(8,103,211,294),(9,112,212,283),(10,101,213,292),(11,110,214,281),(12,119,215,290),(13,108,216,299),(14,117,217,288),(15,106,218,297),(16,115,219,286),(17,104,220,295),(18,113,201,284),(19,102,202,293),(20,111,203,282),(21,227,318,93),(22,236,319,82),(23,225,320,91),(24,234,301,100),(25,223,302,89),(26,232,303,98),(27,221,304,87),(28,230,305,96),(29,239,306,85),(30,228,307,94),(31,237,308,83),(32,226,309,92),(33,235,310,81),(34,224,311,90),(35,233,312,99),(36,222,313,88),(37,231,314,97),(38,240,315,86),(39,229,316,95),(40,238,317,84),(41,146,175,136),(42,155,176,125),(43,144,177,134),(44,153,178,123),(45,142,179,132),(46,151,180,121),(47,160,161,130),(48,149,162,139),(49,158,163,128),(50,147,164,137),(51,156,165,126),(52,145,166,135),(53,154,167,124),(54,143,168,133),(55,152,169,122),(56,141,170,131),(57,150,171,140),(58,159,172,129),(59,148,173,138),(60,157,174,127),(61,251,275,197),(62,260,276,186),(63,249,277,195),(64,258,278,184),(65,247,279,193),(66,256,280,182),(67,245,261,191),(68,254,262,200),(69,243,263,189),(70,252,264,198),(71,241,265,187),(72,250,266,196),(73,259,267,185),(74,248,268,194),(75,257,269,183),(76,246,270,192),(77,255,271,181),(78,244,272,190),(79,253,273,199),(80,242,274,188)]])
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 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | - | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | Q8 | D5 | C4○D4 | D10 | Dic10 | C4×D5 | C4○D20 | D4×D5 | D4⋊2D5 | Q8×D5 |
| kernel | (C2×C20)⋊Q8 | C10.10C42 | C5×C2.C42 | C2×C4×Dic5 | C2×C10.D4 | C22×Dic10 | C2×Dic10 | C2×Dic5 | C2×Dic5 | C2×C20 | C2.C42 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 3 | 1 | 1 | 1 | 1 | 8 | 4 | 2 | 2 | 2 | 4 | 6 | 8 | 8 | 8 | 4 | 2 | 2 |
Matrix representation of (C2×C20)⋊Q8 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 32 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 18 |
| 0 | 0 | 0 | 0 | 9 | 1 |
| 26 | 8 | 0 | 0 | 0 | 0 |
| 23 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 12 |
| 0 | 0 | 0 | 0 | 39 | 5 |
| 2 | 35 | 0 | 0 | 0 | 0 |
| 35 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 25 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,9,0,0,0,0,32,0,0,0,0,0,0,0,40,9,0,0,0,0,18,1],[26,23,0,0,0,0,8,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,39,0,0,0,0,12,5],[2,35,0,0,0,0,35,39,0,0,0,0,0,0,2,25,0,0,0,0,13,39,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;
(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(320,273);
// by ID
G=gap.SmallGroup(320,273);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,387,58,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=c^2,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b^9,d*c*d^-1=c^-1>;
// generators/relations