direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C10.D8, C10⋊2(C2.D8), C10.47(C2×D8), (C2×C10).38D8, C20.45(C4⋊C4), C20.61(C2×Q8), (C2×C20).14Q8, C4⋊C4.223D10, (C2×C20).130D4, (C2×C10).15Q16, C10.31(C2×Q16), C4.26(C2×Dic10), (C2×C4).26Dic10, (C2×C20).315C23, C20.115(C22×C4), C22.19(D4⋊D5), (C22×C4).327D10, (C22×C10).180D4, C23.95(C5⋊D4), C22.7(C5⋊Q16), C4.14(C10.D4), C4⋊Dic5.321C22, (C22×C20).130C22, C22.23(C10.D4), C5⋊3(C2×C2.D8), (C2×C5⋊2C8)⋊7C4, C4.84(C2×C4×D5), (C2×C4⋊C4).4D5, C2.1(C2×D4⋊D5), C5⋊2C8⋊29(C2×C4), (C10×C4⋊C4).3C2, C10.55(C2×C4⋊C4), C2.1(C2×C5⋊Q16), (C2×C4).149(C4×D5), (C2×C10).72(C4⋊C4), (C2×C20).247(C2×C4), (C2×C10).435(C2×D4), (C2×C4⋊Dic5).30C2, (C22×C5⋊2C8).3C2, C2.7(C2×C10.D4), C22.54(C2×C5⋊D4), (C2×C4).122(C5⋊D4), (C5×C4⋊C4).254C22, (C2×C4).415(C22×D5), (C2×C5⋊2C8).247C22, SmallGroup(320,589)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C10.D8
G = < a,b,c,d | a2=b10=c8=1, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=c-1 >
Subgroups: 366 in 130 conjugacy classes, 79 normal (27 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, Dic5, C20, C20, C20, C2×C10, C2×C10, C2.D8, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C2×C20, C22×C10, C2×C2.D8, C2×C5⋊2C8, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, C22×C20, C10.D8, C22×C5⋊2C8, C2×C4⋊Dic5, C10×C4⋊C4, C2×C10.D8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, D10, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, Dic10, C4×D5, C5⋊D4, C22×D5, C2×C2.D8, C10.D4, D4⋊D5, C5⋊Q16, C2×Dic10, C2×C4×D5, C2×C5⋊D4, C10.D8, C2×C10.D4, C2×D4⋊D5, C2×C5⋊Q16, C2×C10.D8
►Regular action on 320 points
Generators in S320
(1 117)(2 118)(3 119)(4 120)(5 111)(6 112)(7 113)(8 114)(9 115)(10 116)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 121)(18 122)(19 123)(20 124)(21 226)(22 227)(23 228)(24 229)(25 230)(26 221)(27 222)(28 223)(29 224)(30 225)(31 95)(32 96)(33 97)(34 98)(35 99)(36 100)(37 91)(38 92)(39 93)(40 94)(41 104)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 101)(49 102)(50 103)(51 157)(52 158)(53 159)(54 160)(55 151)(56 152)(57 153)(58 154)(59 155)(60 156)(61 167)(62 168)(63 169)(64 170)(65 161)(66 162)(67 163)(68 164)(69 165)(70 166)(71 135)(72 136)(73 137)(74 138)(75 139)(76 140)(77 131)(78 132)(79 133)(80 134)(81 144)(82 145)(83 146)(84 147)(85 148)(86 149)(87 150)(88 141)(89 142)(90 143)(171 280)(172 271)(173 272)(174 273)(175 274)(176 275)(177 276)(178 277)(179 278)(180 279)(181 289)(182 290)(183 281)(184 282)(185 283)(186 284)(187 285)(188 286)(189 287)(190 288)(191 252)(192 253)(193 254)(194 255)(195 256)(196 257)(197 258)(198 259)(199 260)(200 251)(201 263)(202 264)(203 265)(204 266)(205 267)(206 268)(207 269)(208 270)(209 261)(210 262)(211 320)(212 311)(213 312)(214 313)(215 314)(216 315)(217 316)(218 317)(219 318)(220 319)(231 292)(232 293)(233 294)(234 295)(235 296)(236 297)(237 298)(238 299)(239 300)(240 291)(241 303)(242 304)(243 305)(244 306)(245 307)(246 308)(247 309)(248 310)(249 301)(250 302)
(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)
(1 210 45 193 37 182 11 174)(2 209 46 192 38 181 12 173)(3 208 47 191 39 190 13 172)(4 207 48 200 40 189 14 171)(5 206 49 199 31 188 15 180)(6 205 50 198 32 187 16 179)(7 204 41 197 33 186 17 178)(8 203 42 196 34 185 18 177)(9 202 43 195 35 184 19 176)(10 201 44 194 36 183 20 175)(21 170 298 137 306 144 317 153)(22 169 299 136 307 143 318 152)(23 168 300 135 308 142 319 151)(24 167 291 134 309 141 320 160)(25 166 292 133 310 150 311 159)(26 165 293 132 301 149 312 158)(27 164 294 131 302 148 313 157)(28 163 295 140 303 147 314 156)(29 162 296 139 304 146 315 155)(30 161 297 138 305 145 316 154)(51 222 68 233 77 250 85 214)(52 221 69 232 78 249 86 213)(53 230 70 231 79 248 87 212)(54 229 61 240 80 247 88 211)(55 228 62 239 71 246 89 220)(56 227 63 238 72 245 90 219)(57 226 64 237 73 244 81 218)(58 225 65 236 74 243 82 217)(59 224 66 235 75 242 83 216)(60 223 67 234 76 241 84 215)(91 290 125 273 117 262 108 254)(92 289 126 272 118 261 109 253)(93 288 127 271 119 270 110 252)(94 287 128 280 120 269 101 251)(95 286 129 279 111 268 102 260)(96 285 130 278 112 267 103 259)(97 284 121 277 113 266 104 258)(98 283 122 276 114 265 105 257)(99 282 123 275 115 264 106 256)(100 281 124 274 116 263 107 255)
(1 299 6 294)(2 298 7 293)(3 297 8 292)(4 296 9 291)(5 295 10 300)(11 307 16 302)(12 306 17 301)(13 305 18 310)(14 304 19 309)(15 303 20 308)(21 41 26 46)(22 50 27 45)(23 49 28 44)(24 48 29 43)(25 47 30 42)(31 314 36 319)(32 313 37 318)(33 312 38 317)(34 311 39 316)(35 320 40 315)(51 254 56 259)(52 253 57 258)(53 252 58 257)(54 251 59 256)(55 260 60 255)(61 269 66 264)(62 268 67 263)(63 267 68 262)(64 266 69 261)(65 265 70 270)(71 279 76 274)(72 278 77 273)(73 277 78 272)(74 276 79 271)(75 275 80 280)(81 284 86 289)(82 283 87 288)(83 282 88 287)(84 281 89 286)(85 290 90 285)(91 219 96 214)(92 218 97 213)(93 217 98 212)(94 216 99 211)(95 215 100 220)(101 224 106 229)(102 223 107 228)(103 222 108 227)(104 221 109 226)(105 230 110 225)(111 234 116 239)(112 233 117 238)(113 232 118 237)(114 231 119 236)(115 240 120 235)(121 249 126 244)(122 248 127 243)(123 247 128 242)(124 246 129 241)(125 245 130 250)(131 174 136 179)(132 173 137 178)(133 172 138 177)(134 171 139 176)(135 180 140 175)(141 189 146 184)(142 188 147 183)(143 187 148 182)(144 186 149 181)(145 185 150 190)(151 199 156 194)(152 198 157 193)(153 197 158 192)(154 196 159 191)(155 195 160 200)(161 203 166 208)(162 202 167 207)(163 201 168 206)(164 210 169 205)(165 209 170 204)
G:=sub<Sym(320)| (1,117)(2,118)(3,119)(4,120)(5,111)(6,112)(7,113)(8,114)(9,115)(10,116)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,121)(18,122)(19,123)(20,124)(21,226)(22,227)(23,228)(24,229)(25,230)(26,221)(27,222)(28,223)(29,224)(30,225)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,91)(38,92)(39,93)(40,94)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,101)(49,102)(50,103)(51,157)(52,158)(53,159)(54,160)(55,151)(56,152)(57,153)(58,154)(59,155)(60,156)(61,167)(62,168)(63,169)(64,170)(65,161)(66,162)(67,163)(68,164)(69,165)(70,166)(71,135)(72,136)(73,137)(74,138)(75,139)(76,140)(77,131)(78,132)(79,133)(80,134)(81,144)(82,145)(83,146)(84,147)(85,148)(86,149)(87,150)(88,141)(89,142)(90,143)(171,280)(172,271)(173,272)(174,273)(175,274)(176,275)(177,276)(178,277)(179,278)(180,279)(181,289)(182,290)(183,281)(184,282)(185,283)(186,284)(187,285)(188,286)(189,287)(190,288)(191,252)(192,253)(193,254)(194,255)(195,256)(196,257)(197,258)(198,259)(199,260)(200,251)(201,263)(202,264)(203,265)(204,266)(205,267)(206,268)(207,269)(208,270)(209,261)(210,262)(211,320)(212,311)(213,312)(214,313)(215,314)(216,315)(217,316)(218,317)(219,318)(220,319)(231,292)(232,293)(233,294)(234,295)(235,296)(236,297)(237,298)(238,299)(239,300)(240,291)(241,303)(242,304)(243,305)(244,306)(245,307)(246,308)(247,309)(248,310)(249,301)(250,302), (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), (1,210,45,193,37,182,11,174)(2,209,46,192,38,181,12,173)(3,208,47,191,39,190,13,172)(4,207,48,200,40,189,14,171)(5,206,49,199,31,188,15,180)(6,205,50,198,32,187,16,179)(7,204,41,197,33,186,17,178)(8,203,42,196,34,185,18,177)(9,202,43,195,35,184,19,176)(10,201,44,194,36,183,20,175)(21,170,298,137,306,144,317,153)(22,169,299,136,307,143,318,152)(23,168,300,135,308,142,319,151)(24,167,291,134,309,141,320,160)(25,166,292,133,310,150,311,159)(26,165,293,132,301,149,312,158)(27,164,294,131,302,148,313,157)(28,163,295,140,303,147,314,156)(29,162,296,139,304,146,315,155)(30,161,297,138,305,145,316,154)(51,222,68,233,77,250,85,214)(52,221,69,232,78,249,86,213)(53,230,70,231,79,248,87,212)(54,229,61,240,80,247,88,211)(55,228,62,239,71,246,89,220)(56,227,63,238,72,245,90,219)(57,226,64,237,73,244,81,218)(58,225,65,236,74,243,82,217)(59,224,66,235,75,242,83,216)(60,223,67,234,76,241,84,215)(91,290,125,273,117,262,108,254)(92,289,126,272,118,261,109,253)(93,288,127,271,119,270,110,252)(94,287,128,280,120,269,101,251)(95,286,129,279,111,268,102,260)(96,285,130,278,112,267,103,259)(97,284,121,277,113,266,104,258)(98,283,122,276,114,265,105,257)(99,282,123,275,115,264,106,256)(100,281,124,274,116,263,107,255), (1,299,6,294)(2,298,7,293)(3,297,8,292)(4,296,9,291)(5,295,10,300)(11,307,16,302)(12,306,17,301)(13,305,18,310)(14,304,19,309)(15,303,20,308)(21,41,26,46)(22,50,27,45)(23,49,28,44)(24,48,29,43)(25,47,30,42)(31,314,36,319)(32,313,37,318)(33,312,38,317)(34,311,39,316)(35,320,40,315)(51,254,56,259)(52,253,57,258)(53,252,58,257)(54,251,59,256)(55,260,60,255)(61,269,66,264)(62,268,67,263)(63,267,68,262)(64,266,69,261)(65,265,70,270)(71,279,76,274)(72,278,77,273)(73,277,78,272)(74,276,79,271)(75,275,80,280)(81,284,86,289)(82,283,87,288)(83,282,88,287)(84,281,89,286)(85,290,90,285)(91,219,96,214)(92,218,97,213)(93,217,98,212)(94,216,99,211)(95,215,100,220)(101,224,106,229)(102,223,107,228)(103,222,108,227)(104,221,109,226)(105,230,110,225)(111,234,116,239)(112,233,117,238)(113,232,118,237)(114,231,119,236)(115,240,120,235)(121,249,126,244)(122,248,127,243)(123,247,128,242)(124,246,129,241)(125,245,130,250)(131,174,136,179)(132,173,137,178)(133,172,138,177)(134,171,139,176)(135,180,140,175)(141,189,146,184)(142,188,147,183)(143,187,148,182)(144,186,149,181)(145,185,150,190)(151,199,156,194)(152,198,157,193)(153,197,158,192)(154,196,159,191)(155,195,160,200)(161,203,166,208)(162,202,167,207)(163,201,168,206)(164,210,169,205)(165,209,170,204)>;
G:=Group( (1,117)(2,118)(3,119)(4,120)(5,111)(6,112)(7,113)(8,114)(9,115)(10,116)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,121)(18,122)(19,123)(20,124)(21,226)(22,227)(23,228)(24,229)(25,230)(26,221)(27,222)(28,223)(29,224)(30,225)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,91)(38,92)(39,93)(40,94)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,101)(49,102)(50,103)(51,157)(52,158)(53,159)(54,160)(55,151)(56,152)(57,153)(58,154)(59,155)(60,156)(61,167)(62,168)(63,169)(64,170)(65,161)(66,162)(67,163)(68,164)(69,165)(70,166)(71,135)(72,136)(73,137)(74,138)(75,139)(76,140)(77,131)(78,132)(79,133)(80,134)(81,144)(82,145)(83,146)(84,147)(85,148)(86,149)(87,150)(88,141)(89,142)(90,143)(171,280)(172,271)(173,272)(174,273)(175,274)(176,275)(177,276)(178,277)(179,278)(180,279)(181,289)(182,290)(183,281)(184,282)(185,283)(186,284)(187,285)(188,286)(189,287)(190,288)(191,252)(192,253)(193,254)(194,255)(195,256)(196,257)(197,258)(198,259)(199,260)(200,251)(201,263)(202,264)(203,265)(204,266)(205,267)(206,268)(207,269)(208,270)(209,261)(210,262)(211,320)(212,311)(213,312)(214,313)(215,314)(216,315)(217,316)(218,317)(219,318)(220,319)(231,292)(232,293)(233,294)(234,295)(235,296)(236,297)(237,298)(238,299)(239,300)(240,291)(241,303)(242,304)(243,305)(244,306)(245,307)(246,308)(247,309)(248,310)(249,301)(250,302), (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), (1,210,45,193,37,182,11,174)(2,209,46,192,38,181,12,173)(3,208,47,191,39,190,13,172)(4,207,48,200,40,189,14,171)(5,206,49,199,31,188,15,180)(6,205,50,198,32,187,16,179)(7,204,41,197,33,186,17,178)(8,203,42,196,34,185,18,177)(9,202,43,195,35,184,19,176)(10,201,44,194,36,183,20,175)(21,170,298,137,306,144,317,153)(22,169,299,136,307,143,318,152)(23,168,300,135,308,142,319,151)(24,167,291,134,309,141,320,160)(25,166,292,133,310,150,311,159)(26,165,293,132,301,149,312,158)(27,164,294,131,302,148,313,157)(28,163,295,140,303,147,314,156)(29,162,296,139,304,146,315,155)(30,161,297,138,305,145,316,154)(51,222,68,233,77,250,85,214)(52,221,69,232,78,249,86,213)(53,230,70,231,79,248,87,212)(54,229,61,240,80,247,88,211)(55,228,62,239,71,246,89,220)(56,227,63,238,72,245,90,219)(57,226,64,237,73,244,81,218)(58,225,65,236,74,243,82,217)(59,224,66,235,75,242,83,216)(60,223,67,234,76,241,84,215)(91,290,125,273,117,262,108,254)(92,289,126,272,118,261,109,253)(93,288,127,271,119,270,110,252)(94,287,128,280,120,269,101,251)(95,286,129,279,111,268,102,260)(96,285,130,278,112,267,103,259)(97,284,121,277,113,266,104,258)(98,283,122,276,114,265,105,257)(99,282,123,275,115,264,106,256)(100,281,124,274,116,263,107,255), (1,299,6,294)(2,298,7,293)(3,297,8,292)(4,296,9,291)(5,295,10,300)(11,307,16,302)(12,306,17,301)(13,305,18,310)(14,304,19,309)(15,303,20,308)(21,41,26,46)(22,50,27,45)(23,49,28,44)(24,48,29,43)(25,47,30,42)(31,314,36,319)(32,313,37,318)(33,312,38,317)(34,311,39,316)(35,320,40,315)(51,254,56,259)(52,253,57,258)(53,252,58,257)(54,251,59,256)(55,260,60,255)(61,269,66,264)(62,268,67,263)(63,267,68,262)(64,266,69,261)(65,265,70,270)(71,279,76,274)(72,278,77,273)(73,277,78,272)(74,276,79,271)(75,275,80,280)(81,284,86,289)(82,283,87,288)(83,282,88,287)(84,281,89,286)(85,290,90,285)(91,219,96,214)(92,218,97,213)(93,217,98,212)(94,216,99,211)(95,215,100,220)(101,224,106,229)(102,223,107,228)(103,222,108,227)(104,221,109,226)(105,230,110,225)(111,234,116,239)(112,233,117,238)(113,232,118,237)(114,231,119,236)(115,240,120,235)(121,249,126,244)(122,248,127,243)(123,247,128,242)(124,246,129,241)(125,245,130,250)(131,174,136,179)(132,173,137,178)(133,172,138,177)(134,171,139,176)(135,180,140,175)(141,189,146,184)(142,188,147,183)(143,187,148,182)(144,186,149,181)(145,185,150,190)(151,199,156,194)(152,198,157,193)(153,197,158,192)(154,196,159,191)(155,195,160,200)(161,203,166,208)(162,202,167,207)(163,201,168,206)(164,210,169,205)(165,209,170,204) );
G=PermutationGroup([[(1,117),(2,118),(3,119),(4,120),(5,111),(6,112),(7,113),(8,114),(9,115),(10,116),(11,125),(12,126),(13,127),(14,128),(15,129),(16,130),(17,121),(18,122),(19,123),(20,124),(21,226),(22,227),(23,228),(24,229),(25,230),(26,221),(27,222),(28,223),(29,224),(30,225),(31,95),(32,96),(33,97),(34,98),(35,99),(36,100),(37,91),(38,92),(39,93),(40,94),(41,104),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,101),(49,102),(50,103),(51,157),(52,158),(53,159),(54,160),(55,151),(56,152),(57,153),(58,154),(59,155),(60,156),(61,167),(62,168),(63,169),(64,170),(65,161),(66,162),(67,163),(68,164),(69,165),(70,166),(71,135),(72,136),(73,137),(74,138),(75,139),(76,140),(77,131),(78,132),(79,133),(80,134),(81,144),(82,145),(83,146),(84,147),(85,148),(86,149),(87,150),(88,141),(89,142),(90,143),(171,280),(172,271),(173,272),(174,273),(175,274),(176,275),(177,276),(178,277),(179,278),(180,279),(181,289),(182,290),(183,281),(184,282),(185,283),(186,284),(187,285),(188,286),(189,287),(190,288),(191,252),(192,253),(193,254),(194,255),(195,256),(196,257),(197,258),(198,259),(199,260),(200,251),(201,263),(202,264),(203,265),(204,266),(205,267),(206,268),(207,269),(208,270),(209,261),(210,262),(211,320),(212,311),(213,312),(214,313),(215,314),(216,315),(217,316),(218,317),(219,318),(220,319),(231,292),(232,293),(233,294),(234,295),(235,296),(236,297),(237,298),(238,299),(239,300),(240,291),(241,303),(242,304),(243,305),(244,306),(245,307),(246,308),(247,309),(248,310),(249,301),(250,302)], [(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)], [(1,210,45,193,37,182,11,174),(2,209,46,192,38,181,12,173),(3,208,47,191,39,190,13,172),(4,207,48,200,40,189,14,171),(5,206,49,199,31,188,15,180),(6,205,50,198,32,187,16,179),(7,204,41,197,33,186,17,178),(8,203,42,196,34,185,18,177),(9,202,43,195,35,184,19,176),(10,201,44,194,36,183,20,175),(21,170,298,137,306,144,317,153),(22,169,299,136,307,143,318,152),(23,168,300,135,308,142,319,151),(24,167,291,134,309,141,320,160),(25,166,292,133,310,150,311,159),(26,165,293,132,301,149,312,158),(27,164,294,131,302,148,313,157),(28,163,295,140,303,147,314,156),(29,162,296,139,304,146,315,155),(30,161,297,138,305,145,316,154),(51,222,68,233,77,250,85,214),(52,221,69,232,78,249,86,213),(53,230,70,231,79,248,87,212),(54,229,61,240,80,247,88,211),(55,228,62,239,71,246,89,220),(56,227,63,238,72,245,90,219),(57,226,64,237,73,244,81,218),(58,225,65,236,74,243,82,217),(59,224,66,235,75,242,83,216),(60,223,67,234,76,241,84,215),(91,290,125,273,117,262,108,254),(92,289,126,272,118,261,109,253),(93,288,127,271,119,270,110,252),(94,287,128,280,120,269,101,251),(95,286,129,279,111,268,102,260),(96,285,130,278,112,267,103,259),(97,284,121,277,113,266,104,258),(98,283,122,276,114,265,105,257),(99,282,123,275,115,264,106,256),(100,281,124,274,116,263,107,255)], [(1,299,6,294),(2,298,7,293),(3,297,8,292),(4,296,9,291),(5,295,10,300),(11,307,16,302),(12,306,17,301),(13,305,18,310),(14,304,19,309),(15,303,20,308),(21,41,26,46),(22,50,27,45),(23,49,28,44),(24,48,29,43),(25,47,30,42),(31,314,36,319),(32,313,37,318),(33,312,38,317),(34,311,39,316),(35,320,40,315),(51,254,56,259),(52,253,57,258),(53,252,58,257),(54,251,59,256),(55,260,60,255),(61,269,66,264),(62,268,67,263),(63,267,68,262),(64,266,69,261),(65,265,70,270),(71,279,76,274),(72,278,77,273),(73,277,78,272),(74,276,79,271),(75,275,80,280),(81,284,86,289),(82,283,87,288),(83,282,88,287),(84,281,89,286),(85,290,90,285),(91,219,96,214),(92,218,97,213),(93,217,98,212),(94,216,99,211),(95,215,100,220),(101,224,106,229),(102,223,107,228),(103,222,108,227),(104,221,109,226),(105,230,110,225),(111,234,116,239),(112,233,117,238),(113,232,118,237),(114,231,119,236),(115,240,120,235),(121,249,126,244),(122,248,127,243),(123,247,128,242),(124,246,129,241),(125,245,130,250),(131,174,136,179),(132,173,137,178),(133,172,138,177),(134,171,139,176),(135,180,140,175),(141,189,146,184),(142,188,147,183),(143,187,148,182),(144,186,149,181),(145,185,150,190),(151,199,156,194),(152,198,157,193),(153,197,158,192),(154,196,159,191),(155,195,160,200),(161,203,166,208),(162,202,167,207),(163,201,168,206),(164,210,169,205),(165,209,170,204)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D5 | D8 | Q16 | D10 | D10 | Dic10 | C4×D5 | C5⋊D4 | C5⋊D4 | D4⋊D5 | C5⋊Q16 |
| kernel | C2×C10.D8 | C10.D8 | C22×C5⋊2C8 | C2×C4⋊Dic5 | C10×C4⋊C4 | C2×C5⋊2C8 | C2×C20 | C2×C20 | C22×C10 | C2×C4⋊C4 | C2×C10 | C2×C10 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 8 | 8 | 4 | 4 | 4 | 4 |
Matrix representation of C2×C10.D8 ►in GL6(𝔽41)
| 40 | 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 7 | 0 | 0 | 0 | 0 |
| 34 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 34 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 39 | 0 | 0 |
| 0 | 0 | 36 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 11 |
| 0 | 0 | 0 | 0 | 26 | 0 |
| 19 | 32 | 0 | 0 | 0 | 0 |
| 22 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 2 | 0 | 0 |
| 0 | 0 | 5 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 35 |
| 0 | 0 | 0 | 0 | 19 | 21 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,34,0,0,0,0,7,34,0,0,0,0,0,0,0,34,0,0,0,0,6,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[7,34,0,0,0,0,1,34,0,0,0,0,0,0,14,36,0,0,0,0,39,27,0,0,0,0,0,0,24,26,0,0,0,0,11,0],[19,22,0,0,0,0,32,22,0,0,0,0,0,0,27,5,0,0,0,0,2,14,0,0,0,0,0,0,20,19,0,0,0,0,35,21] >;
C2×C10.D8 in GAP, Magma, Sage, TeX
C_2\times C_{10}.D_8 % in TeX
G:=Group("C2xC10.D8"); // GroupNames label
G:=SmallGroup(320,589);
// by ID
G=gap.SmallGroup(320,589);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,422,58,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^8=1,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations