direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic5⋊C8, C10⋊2(C4⋊C8), (C2×Dic5)⋊8C8, Dic5⋊9(C2×C8), C23.61(C2×F5), (C22×C4).13F5, (C22×C20).19C4, C10.13(C22×C8), (C2×Dic5).37Q8, (C4×Dic5).30C4, Dic5.18(C2×Q8), Dic5.36(C2×D4), C22.27(C4⋊F5), Dic5.35(C4⋊C4), C22.16(D5⋊C8), (C2×Dic5).179D4, C10.25(C2×M4(2)), (C2×C10).23M4(2), C22.46(C22×F5), (C22×Dic5).31C4, C22.9(C22.F5), (C2×Dic5).343C23, (C4×Dic5).326C22, (C22×Dic5).271C22, C5⋊3(C2×C4⋊C8), C2.4(C2×C4⋊F5), C10.20(C2×C4⋊C4), C2.14(C2×D5⋊C8), (C22×C5⋊C8).6C2, (C2×C10).14(C2×C8), (C2×C5⋊C8).37C22, (C2×C4).106(C2×F5), (C2×C4×Dic5).41C2, (C2×C10).26(C4⋊C4), (C2×C20).106(C2×C4), C2.3(C2×C22.F5), (C2×C10).59(C22×C4), (C22×C10).59(C2×C4), (C2×Dic5).181(C2×C4), SmallGroup(320,1090)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic5⋊C8
G = < a,b,c,d | a2=b10=d8=1, c2=b5, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b3, dcd-1=b5c >
Subgroups: 378 in 138 conjugacy classes, 76 normal (22 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C22×C4, C22×C4, Dic5, Dic5, C20, C2×C10, C2×C10, C4⋊C8, C2×C42, C22×C8, C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C2×C4⋊C8, C4×Dic5, C2×C5⋊C8, C2×C5⋊C8, C22×Dic5, C22×C20, Dic5⋊C8, C2×C4×Dic5, C22×C5⋊C8, C2×Dic5⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, F5, C4⋊C8, C2×C4⋊C4, C22×C8, C2×M4(2), C2×F5, C2×C4⋊C8, D5⋊C8, C4⋊F5, C22.F5, C22×F5, Dic5⋊C8, C2×D5⋊C8, C2×C4⋊F5, C2×C22.F5, C2×Dic5⋊C8
►Regular action on 320 points
Generators in S320
(1 110)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 240)(12 231)(13 232)(14 233)(15 234)(16 235)(17 236)(18 237)(19 238)(20 239)(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(27 91)(28 92)(29 93)(30 94)(31 130)(32 121)(33 122)(34 123)(35 124)(36 125)(37 126)(38 127)(39 128)(40 129)(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 120)(50 111)(51 147)(52 148)(53 149)(54 150)(55 141)(56 142)(57 143)(58 144)(59 145)(60 146)(61 135)(62 136)(63 137)(64 138)(65 139)(66 140)(67 131)(68 132)(69 133)(70 134)(71 161)(72 162)(73 163)(74 164)(75 165)(76 166)(77 167)(78 168)(79 169)(80 170)(81 152)(82 153)(83 154)(84 155)(85 156)(86 157)(87 158)(88 159)(89 160)(90 151)(171 270)(172 261)(173 262)(174 263)(175 264)(176 265)(177 266)(178 267)(179 268)(180 269)(181 252)(182 253)(183 254)(184 255)(185 256)(186 257)(187 258)(188 259)(189 260)(190 251)(191 287)(192 288)(193 289)(194 290)(195 281)(196 282)(197 283)(198 284)(199 285)(200 286)(201 275)(202 276)(203 277)(204 278)(205 279)(206 280)(207 271)(208 272)(209 273)(210 274)(211 310)(212 301)(213 302)(214 303)(215 304)(216 305)(217 306)(218 307)(219 308)(220 309)(221 292)(222 293)(223 294)(224 295)(225 296)(226 297)(227 298)(228 299)(229 300)(230 291)(241 315)(242 316)(243 317)(244 318)(245 319)(246 320)(247 311)(248 312)(249 313)(250 314)
(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)
(1 151 6 156)(2 160 7 155)(3 159 8 154)(4 158 9 153)(5 157 10 152)(11 184 16 189)(12 183 17 188)(13 182 18 187)(14 181 19 186)(15 190 20 185)(21 170 26 165)(22 169 27 164)(23 168 28 163)(24 167 29 162)(25 166 30 161)(31 150 36 145)(32 149 37 144)(33 148 38 143)(34 147 39 142)(35 146 40 141)(41 133 46 138)(42 132 47 137)(43 131 48 136)(44 140 49 135)(45 139 50 134)(51 128 56 123)(52 127 57 122)(53 126 58 121)(54 125 59 130)(55 124 60 129)(61 115 66 120)(62 114 67 119)(63 113 68 118)(64 112 69 117)(65 111 70 116)(71 99 76 94)(72 98 77 93)(73 97 78 92)(74 96 79 91)(75 95 80 100)(81 104 86 109)(82 103 87 108)(83 102 88 107)(84 101 89 106)(85 110 90 105)(171 319 176 314)(172 318 177 313)(173 317 178 312)(174 316 179 311)(175 315 180 320)(191 308 196 303)(192 307 197 302)(193 306 198 301)(194 305 199 310)(195 304 200 309)(201 295 206 300)(202 294 207 299)(203 293 208 298)(204 292 209 297)(205 291 210 296)(211 290 216 285)(212 289 217 284)(213 288 218 283)(214 287 219 282)(215 286 220 281)(221 273 226 278)(222 272 227 277)(223 271 228 276)(224 280 229 275)(225 279 230 274)(231 254 236 259)(232 253 237 258)(233 252 238 257)(234 251 239 256)(235 260 240 255)(241 269 246 264)(242 268 247 263)(243 267 248 262)(244 266 249 261)(245 265 250 270)
(1 238 36 221 30 244 45 212)(2 235 35 224 21 241 44 215)(3 232 34 227 22 248 43 218)(4 239 33 230 23 245 42 211)(5 236 32 223 24 242 41 214)(6 233 31 226 25 249 50 217)(7 240 40 229 26 246 49 220)(8 237 39 222 27 243 48 213)(9 234 38 225 28 250 47 216)(10 231 37 228 29 247 46 219)(11 129 300 100 320 120 309 106)(12 126 299 93 311 117 308 109)(13 123 298 96 312 114 307 102)(14 130 297 99 313 111 306 105)(15 127 296 92 314 118 305 108)(16 124 295 95 315 115 304 101)(17 121 294 98 316 112 303 104)(18 128 293 91 317 119 302 107)(19 125 292 94 318 116 301 110)(20 122 291 97 319 113 310 103)(51 208 79 178 67 192 88 187)(52 205 78 171 68 199 87 190)(53 202 77 174 69 196 86 183)(54 209 76 177 70 193 85 186)(55 206 75 180 61 200 84 189)(56 203 74 173 62 197 83 182)(57 210 73 176 63 194 82 185)(58 207 72 179 64 191 81 188)(59 204 71 172 65 198 90 181)(60 201 80 175 66 195 89 184)(131 288 159 258 147 272 169 267)(132 285 158 251 148 279 168 270)(133 282 157 254 149 276 167 263)(134 289 156 257 150 273 166 266)(135 286 155 260 141 280 165 269)(136 283 154 253 142 277 164 262)(137 290 153 256 143 274 163 265)(138 287 152 259 144 271 162 268)(139 284 151 252 145 278 161 261)(140 281 160 255 146 275 170 264)
G:=sub<Sym(320)| (1,110)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,240)(12,231)(13,232)(14,233)(15,234)(16,235)(17,236)(18,237)(19,238)(20,239)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,91)(28,92)(29,93)(30,94)(31,130)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,120)(50,111)(51,147)(52,148)(53,149)(54,150)(55,141)(56,142)(57,143)(58,144)(59,145)(60,146)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,131)(68,132)(69,133)(70,134)(71,161)(72,162)(73,163)(74,164)(75,165)(76,166)(77,167)(78,168)(79,169)(80,170)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,151)(171,270)(172,261)(173,262)(174,263)(175,264)(176,265)(177,266)(178,267)(179,268)(180,269)(181,252)(182,253)(183,254)(184,255)(185,256)(186,257)(187,258)(188,259)(189,260)(190,251)(191,287)(192,288)(193,289)(194,290)(195,281)(196,282)(197,283)(198,284)(199,285)(200,286)(201,275)(202,276)(203,277)(204,278)(205,279)(206,280)(207,271)(208,272)(209,273)(210,274)(211,310)(212,301)(213,302)(214,303)(215,304)(216,305)(217,306)(218,307)(219,308)(220,309)(221,292)(222,293)(223,294)(224,295)(225,296)(226,297)(227,298)(228,299)(229,300)(230,291)(241,315)(242,316)(243,317)(244,318)(245,319)(246,320)(247,311)(248,312)(249,313)(250,314), (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), (1,151,6,156)(2,160,7,155)(3,159,8,154)(4,158,9,153)(5,157,10,152)(11,184,16,189)(12,183,17,188)(13,182,18,187)(14,181,19,186)(15,190,20,185)(21,170,26,165)(22,169,27,164)(23,168,28,163)(24,167,29,162)(25,166,30,161)(31,150,36,145)(32,149,37,144)(33,148,38,143)(34,147,39,142)(35,146,40,141)(41,133,46,138)(42,132,47,137)(43,131,48,136)(44,140,49,135)(45,139,50,134)(51,128,56,123)(52,127,57,122)(53,126,58,121)(54,125,59,130)(55,124,60,129)(61,115,66,120)(62,114,67,119)(63,113,68,118)(64,112,69,117)(65,111,70,116)(71,99,76,94)(72,98,77,93)(73,97,78,92)(74,96,79,91)(75,95,80,100)(81,104,86,109)(82,103,87,108)(83,102,88,107)(84,101,89,106)(85,110,90,105)(171,319,176,314)(172,318,177,313)(173,317,178,312)(174,316,179,311)(175,315,180,320)(191,308,196,303)(192,307,197,302)(193,306,198,301)(194,305,199,310)(195,304,200,309)(201,295,206,300)(202,294,207,299)(203,293,208,298)(204,292,209,297)(205,291,210,296)(211,290,216,285)(212,289,217,284)(213,288,218,283)(214,287,219,282)(215,286,220,281)(221,273,226,278)(222,272,227,277)(223,271,228,276)(224,280,229,275)(225,279,230,274)(231,254,236,259)(232,253,237,258)(233,252,238,257)(234,251,239,256)(235,260,240,255)(241,269,246,264)(242,268,247,263)(243,267,248,262)(244,266,249,261)(245,265,250,270), (1,238,36,221,30,244,45,212)(2,235,35,224,21,241,44,215)(3,232,34,227,22,248,43,218)(4,239,33,230,23,245,42,211)(5,236,32,223,24,242,41,214)(6,233,31,226,25,249,50,217)(7,240,40,229,26,246,49,220)(8,237,39,222,27,243,48,213)(9,234,38,225,28,250,47,216)(10,231,37,228,29,247,46,219)(11,129,300,100,320,120,309,106)(12,126,299,93,311,117,308,109)(13,123,298,96,312,114,307,102)(14,130,297,99,313,111,306,105)(15,127,296,92,314,118,305,108)(16,124,295,95,315,115,304,101)(17,121,294,98,316,112,303,104)(18,128,293,91,317,119,302,107)(19,125,292,94,318,116,301,110)(20,122,291,97,319,113,310,103)(51,208,79,178,67,192,88,187)(52,205,78,171,68,199,87,190)(53,202,77,174,69,196,86,183)(54,209,76,177,70,193,85,186)(55,206,75,180,61,200,84,189)(56,203,74,173,62,197,83,182)(57,210,73,176,63,194,82,185)(58,207,72,179,64,191,81,188)(59,204,71,172,65,198,90,181)(60,201,80,175,66,195,89,184)(131,288,159,258,147,272,169,267)(132,285,158,251,148,279,168,270)(133,282,157,254,149,276,167,263)(134,289,156,257,150,273,166,266)(135,286,155,260,141,280,165,269)(136,283,154,253,142,277,164,262)(137,290,153,256,143,274,163,265)(138,287,152,259,144,271,162,268)(139,284,151,252,145,278,161,261)(140,281,160,255,146,275,170,264)>;
G:=Group( (1,110)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,240)(12,231)(13,232)(14,233)(15,234)(16,235)(17,236)(18,237)(19,238)(20,239)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,91)(28,92)(29,93)(30,94)(31,130)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,120)(50,111)(51,147)(52,148)(53,149)(54,150)(55,141)(56,142)(57,143)(58,144)(59,145)(60,146)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,131)(68,132)(69,133)(70,134)(71,161)(72,162)(73,163)(74,164)(75,165)(76,166)(77,167)(78,168)(79,169)(80,170)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,151)(171,270)(172,261)(173,262)(174,263)(175,264)(176,265)(177,266)(178,267)(179,268)(180,269)(181,252)(182,253)(183,254)(184,255)(185,256)(186,257)(187,258)(188,259)(189,260)(190,251)(191,287)(192,288)(193,289)(194,290)(195,281)(196,282)(197,283)(198,284)(199,285)(200,286)(201,275)(202,276)(203,277)(204,278)(205,279)(206,280)(207,271)(208,272)(209,273)(210,274)(211,310)(212,301)(213,302)(214,303)(215,304)(216,305)(217,306)(218,307)(219,308)(220,309)(221,292)(222,293)(223,294)(224,295)(225,296)(226,297)(227,298)(228,299)(229,300)(230,291)(241,315)(242,316)(243,317)(244,318)(245,319)(246,320)(247,311)(248,312)(249,313)(250,314), (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), (1,151,6,156)(2,160,7,155)(3,159,8,154)(4,158,9,153)(5,157,10,152)(11,184,16,189)(12,183,17,188)(13,182,18,187)(14,181,19,186)(15,190,20,185)(21,170,26,165)(22,169,27,164)(23,168,28,163)(24,167,29,162)(25,166,30,161)(31,150,36,145)(32,149,37,144)(33,148,38,143)(34,147,39,142)(35,146,40,141)(41,133,46,138)(42,132,47,137)(43,131,48,136)(44,140,49,135)(45,139,50,134)(51,128,56,123)(52,127,57,122)(53,126,58,121)(54,125,59,130)(55,124,60,129)(61,115,66,120)(62,114,67,119)(63,113,68,118)(64,112,69,117)(65,111,70,116)(71,99,76,94)(72,98,77,93)(73,97,78,92)(74,96,79,91)(75,95,80,100)(81,104,86,109)(82,103,87,108)(83,102,88,107)(84,101,89,106)(85,110,90,105)(171,319,176,314)(172,318,177,313)(173,317,178,312)(174,316,179,311)(175,315,180,320)(191,308,196,303)(192,307,197,302)(193,306,198,301)(194,305,199,310)(195,304,200,309)(201,295,206,300)(202,294,207,299)(203,293,208,298)(204,292,209,297)(205,291,210,296)(211,290,216,285)(212,289,217,284)(213,288,218,283)(214,287,219,282)(215,286,220,281)(221,273,226,278)(222,272,227,277)(223,271,228,276)(224,280,229,275)(225,279,230,274)(231,254,236,259)(232,253,237,258)(233,252,238,257)(234,251,239,256)(235,260,240,255)(241,269,246,264)(242,268,247,263)(243,267,248,262)(244,266,249,261)(245,265,250,270), (1,238,36,221,30,244,45,212)(2,235,35,224,21,241,44,215)(3,232,34,227,22,248,43,218)(4,239,33,230,23,245,42,211)(5,236,32,223,24,242,41,214)(6,233,31,226,25,249,50,217)(7,240,40,229,26,246,49,220)(8,237,39,222,27,243,48,213)(9,234,38,225,28,250,47,216)(10,231,37,228,29,247,46,219)(11,129,300,100,320,120,309,106)(12,126,299,93,311,117,308,109)(13,123,298,96,312,114,307,102)(14,130,297,99,313,111,306,105)(15,127,296,92,314,118,305,108)(16,124,295,95,315,115,304,101)(17,121,294,98,316,112,303,104)(18,128,293,91,317,119,302,107)(19,125,292,94,318,116,301,110)(20,122,291,97,319,113,310,103)(51,208,79,178,67,192,88,187)(52,205,78,171,68,199,87,190)(53,202,77,174,69,196,86,183)(54,209,76,177,70,193,85,186)(55,206,75,180,61,200,84,189)(56,203,74,173,62,197,83,182)(57,210,73,176,63,194,82,185)(58,207,72,179,64,191,81,188)(59,204,71,172,65,198,90,181)(60,201,80,175,66,195,89,184)(131,288,159,258,147,272,169,267)(132,285,158,251,148,279,168,270)(133,282,157,254,149,276,167,263)(134,289,156,257,150,273,166,266)(135,286,155,260,141,280,165,269)(136,283,154,253,142,277,164,262)(137,290,153,256,143,274,163,265)(138,287,152,259,144,271,162,268)(139,284,151,252,145,278,161,261)(140,281,160,255,146,275,170,264) );
G=PermutationGroup([[(1,110),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,109),(11,240),(12,231),(13,232),(14,233),(15,234),(16,235),(17,236),(18,237),(19,238),(20,239),(21,95),(22,96),(23,97),(24,98),(25,99),(26,100),(27,91),(28,92),(29,93),(30,94),(31,130),(32,121),(33,122),(34,123),(35,124),(36,125),(37,126),(38,127),(39,128),(40,129),(41,112),(42,113),(43,114),(44,115),(45,116),(46,117),(47,118),(48,119),(49,120),(50,111),(51,147),(52,148),(53,149),(54,150),(55,141),(56,142),(57,143),(58,144),(59,145),(60,146),(61,135),(62,136),(63,137),(64,138),(65,139),(66,140),(67,131),(68,132),(69,133),(70,134),(71,161),(72,162),(73,163),(74,164),(75,165),(76,166),(77,167),(78,168),(79,169),(80,170),(81,152),(82,153),(83,154),(84,155),(85,156),(86,157),(87,158),(88,159),(89,160),(90,151),(171,270),(172,261),(173,262),(174,263),(175,264),(176,265),(177,266),(178,267),(179,268),(180,269),(181,252),(182,253),(183,254),(184,255),(185,256),(186,257),(187,258),(188,259),(189,260),(190,251),(191,287),(192,288),(193,289),(194,290),(195,281),(196,282),(197,283),(198,284),(199,285),(200,286),(201,275),(202,276),(203,277),(204,278),(205,279),(206,280),(207,271),(208,272),(209,273),(210,274),(211,310),(212,301),(213,302),(214,303),(215,304),(216,305),(217,306),(218,307),(219,308),(220,309),(221,292),(222,293),(223,294),(224,295),(225,296),(226,297),(227,298),(228,299),(229,300),(230,291),(241,315),(242,316),(243,317),(244,318),(245,319),(246,320),(247,311),(248,312),(249,313),(250,314)], [(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)], [(1,151,6,156),(2,160,7,155),(3,159,8,154),(4,158,9,153),(5,157,10,152),(11,184,16,189),(12,183,17,188),(13,182,18,187),(14,181,19,186),(15,190,20,185),(21,170,26,165),(22,169,27,164),(23,168,28,163),(24,167,29,162),(25,166,30,161),(31,150,36,145),(32,149,37,144),(33,148,38,143),(34,147,39,142),(35,146,40,141),(41,133,46,138),(42,132,47,137),(43,131,48,136),(44,140,49,135),(45,139,50,134),(51,128,56,123),(52,127,57,122),(53,126,58,121),(54,125,59,130),(55,124,60,129),(61,115,66,120),(62,114,67,119),(63,113,68,118),(64,112,69,117),(65,111,70,116),(71,99,76,94),(72,98,77,93),(73,97,78,92),(74,96,79,91),(75,95,80,100),(81,104,86,109),(82,103,87,108),(83,102,88,107),(84,101,89,106),(85,110,90,105),(171,319,176,314),(172,318,177,313),(173,317,178,312),(174,316,179,311),(175,315,180,320),(191,308,196,303),(192,307,197,302),(193,306,198,301),(194,305,199,310),(195,304,200,309),(201,295,206,300),(202,294,207,299),(203,293,208,298),(204,292,209,297),(205,291,210,296),(211,290,216,285),(212,289,217,284),(213,288,218,283),(214,287,219,282),(215,286,220,281),(221,273,226,278),(222,272,227,277),(223,271,228,276),(224,280,229,275),(225,279,230,274),(231,254,236,259),(232,253,237,258),(233,252,238,257),(234,251,239,256),(235,260,240,255),(241,269,246,264),(242,268,247,263),(243,267,248,262),(244,266,249,261),(245,265,250,270)], [(1,238,36,221,30,244,45,212),(2,235,35,224,21,241,44,215),(3,232,34,227,22,248,43,218),(4,239,33,230,23,245,42,211),(5,236,32,223,24,242,41,214),(6,233,31,226,25,249,50,217),(7,240,40,229,26,246,49,220),(8,237,39,222,27,243,48,213),(9,234,38,225,28,250,47,216),(10,231,37,228,29,247,46,219),(11,129,300,100,320,120,309,106),(12,126,299,93,311,117,308,109),(13,123,298,96,312,114,307,102),(14,130,297,99,313,111,306,105),(15,127,296,92,314,118,305,108),(16,124,295,95,315,115,304,101),(17,121,294,98,316,112,303,104),(18,128,293,91,317,119,302,107),(19,125,292,94,318,116,301,110),(20,122,291,97,319,113,310,103),(51,208,79,178,67,192,88,187),(52,205,78,171,68,199,87,190),(53,202,77,174,69,196,86,183),(54,209,76,177,70,193,85,186),(55,206,75,180,61,200,84,189),(56,203,74,173,62,197,83,182),(57,210,73,176,63,194,82,185),(58,207,72,179,64,191,81,188),(59,204,71,172,65,198,90,181),(60,201,80,175,66,195,89,184),(131,288,159,258,147,272,169,267),(132,285,158,251,148,279,168,270),(133,282,157,254,149,276,167,263),(134,289,156,257,150,273,166,266),(135,286,155,260,141,280,165,269),(136,283,154,253,142,277,164,262),(137,290,153,256,143,274,163,265),(138,287,152,259,144,271,162,268),(139,284,151,252,145,278,161,261),(140,281,160,255,146,275,170,264)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 5 | 8A | ··· | 8P | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | Q8 | M4(2) | F5 | C2×F5 | C2×F5 | D5⋊C8 | C4⋊F5 | C22.F5 |
| kernel | C2×Dic5⋊C8 | Dic5⋊C8 | C2×C4×Dic5 | C22×C5⋊C8 | C4×Dic5 | C22×Dic5 | C22×C20 | C2×Dic5 | C2×Dic5 | C2×Dic5 | C2×C10 | C22×C4 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 2 | 16 | 2 | 2 | 4 | 1 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of C2×Dic5⋊C8 ►in GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 40 |
| 1 | 23 | 0 | 0 | 0 | 0 | 0 | 0 |
| 32 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 25 | 9 | 35 |
| 0 | 0 | 0 | 0 | 4 | 34 | 3 | 28 |
| 0 | 0 | 0 | 0 | 13 | 28 | 37 | 7 |
| 0 | 0 | 0 | 0 | 7 | 21 | 16 | 32 |
| 27 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 14 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 0 | 11 | 8 |
| 0 | 0 | 0 | 0 | 4 | 8 | 40 | 1 |
| 0 | 0 | 0 | 0 | 33 | 1 | 40 | 12 |
| 0 | 0 | 0 | 0 | 33 | 12 | 7 | 0 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[1,32,0,0,0,0,0,0,23,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,20,4,13,7,0,0,0,0,25,34,28,21,0,0,0,0,9,3,37,16,0,0,0,0,35,28,7,32],[27,3,0,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,29,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,34,4,33,33,0,0,0,0,0,8,1,12,0,0,0,0,11,40,40,7,0,0,0,0,8,1,12,0] >;
C2×Dic5⋊C8 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_5\rtimes C_8 % in TeX
G:=Group("C2xDic5:C8"); // GroupNames label
G:=SmallGroup(320,1090);
// by ID
G=gap.SmallGroup(320,1090);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,268,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=d^8=1,c^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^5*c>;
// generators/relations