metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊8Dic5, C20⋊8(C4⋊C4), (C4×C20)⋊20C4, C4⋊2(C4⋊Dic5), (C2×C4).90D20, C10.8(C4⋊Q8), (C2×C20).53Q8, C5⋊2(C42⋊9C4), (C2×C20).383D4, (C2×C42).16D5, C10.9(C4⋊1D4), C2.1(C20⋊4D4), C2.3(C20⋊2Q8), (C2×C4).45Dic10, C22.36(C2×D20), (C22×C4).419D10, C22.21(C2×Dic10), C23.268(C22×D5), (C22×C10).310C23, (C22×C20).513C22, C22.36(C22×Dic5), (C22×Dic5).29C22, (C2×C4×C20).11C2, C10.50(C2×C4⋊C4), C2.5(C2×C4⋊Dic5), (C2×C10).28(C2×Q8), (C2×C20).469(C2×C4), (C2×C10).146(C2×D4), (C2×C4⋊Dic5).16C2, (C2×C4).79(C2×Dic5), (C2×C10).276(C22×C4), SmallGroup(320,562)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊8Dic5
G = < a,b,c,d | a4=b4=c10=1, d2=c5, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 558 in 186 conjugacy classes, 119 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C42, C4⋊C4, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×C42, C2×C4⋊C4, C2×Dic5, C2×C20, C22×C10, C42⋊9C4, C4⋊Dic5, C4×C20, C22×Dic5, C22×C20, C2×C4⋊Dic5, C2×C4×C20, C42⋊8Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, D10, C2×C4⋊C4, C4⋊1D4, C4⋊Q8, Dic10, D20, C2×Dic5, C22×D5, C42⋊9C4, C4⋊Dic5, C2×Dic10, C2×D20, C22×Dic5, C20⋊2Q8, C20⋊4D4, C2×C4⋊Dic5, C42⋊8Dic5
►Regular action on 320 points
Generators in S320
(1 119 39 98)(2 120 40 99)(3 111 31 100)(4 112 32 91)(5 113 33 92)(6 114 34 93)(7 115 35 94)(8 116 36 95)(9 117 37 96)(10 118 38 97)(11 124 44 106)(12 125 45 107)(13 126 46 108)(14 127 47 109)(15 128 48 110)(16 129 49 101)(17 130 50 102)(18 121 41 103)(19 122 42 104)(20 123 43 105)(21 215 295 242)(22 216 296 243)(23 217 297 244)(24 218 298 245)(25 219 299 246)(26 220 300 247)(27 211 291 248)(28 212 292 249)(29 213 293 250)(30 214 294 241)(51 152 72 131)(52 153 73 132)(53 154 74 133)(54 155 75 134)(55 156 76 135)(56 157 77 136)(57 158 78 137)(58 159 79 138)(59 160 80 139)(60 151 71 140)(61 166 89 141)(62 167 90 142)(63 168 81 143)(64 169 82 144)(65 170 83 145)(66 161 84 146)(67 162 85 147)(68 163 86 148)(69 164 87 149)(70 165 88 150)(171 251 208 288)(172 252 209 289)(173 253 210 290)(174 254 201 281)(175 255 202 282)(176 256 203 283)(177 257 204 284)(178 258 205 285)(179 259 206 286)(180 260 207 287)(181 261 191 271)(182 262 192 272)(183 263 193 273)(184 264 194 274)(185 265 195 275)(186 266 196 276)(187 267 197 277)(188 268 198 278)(189 269 199 279)(190 270 200 280)(221 301 231 311)(222 302 232 312)(223 303 233 313)(224 304 234 314)(225 305 235 315)(226 306 236 316)(227 307 237 317)(228 308 238 318)(229 309 239 319)(230 310 240 320)
(1 65 15 53)(2 66 16 54)(3 67 17 55)(4 68 18 56)(5 69 19 57)(6 70 20 58)(7 61 11 59)(8 62 12 60)(9 63 13 51)(10 64 14 52)(21 271 316 287)(22 272 317 288)(23 273 318 289)(24 274 319 290)(25 275 320 281)(26 276 311 282)(27 277 312 283)(28 278 313 284)(29 279 314 285)(30 280 315 286)(31 85 50 76)(32 86 41 77)(33 87 42 78)(34 88 43 79)(35 89 44 80)(36 90 45 71)(37 81 46 72)(38 82 47 73)(39 83 48 74)(40 84 49 75)(91 148 103 136)(92 149 104 137)(93 150 105 138)(94 141 106 139)(95 142 107 140)(96 143 108 131)(97 144 109 132)(98 145 110 133)(99 146 101 134)(100 147 102 135)(111 162 130 156)(112 163 121 157)(113 164 122 158)(114 165 123 159)(115 166 124 160)(116 167 125 151)(117 168 126 152)(118 169 127 153)(119 170 128 154)(120 161 129 155)(171 216 182 227)(172 217 183 228)(173 218 184 229)(174 219 185 230)(175 220 186 221)(176 211 187 222)(177 212 188 223)(178 213 189 224)(179 214 190 225)(180 215 181 226)(191 236 207 242)(192 237 208 243)(193 238 209 244)(194 239 210 245)(195 240 201 246)(196 231 202 247)(197 232 203 248)(198 233 204 249)(199 234 205 250)(200 235 206 241)(251 296 262 307)(252 297 263 308)(253 298 264 309)(254 299 265 310)(255 300 266 301)(256 291 267 302)(257 292 268 303)(258 293 269 304)(259 294 270 305)(260 295 261 306)
(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)
(1 186 6 181)(2 185 7 190)(3 184 8 189)(4 183 9 188)(5 182 10 187)(11 179 16 174)(12 178 17 173)(13 177 18 172)(14 176 19 171)(15 175 20 180)(21 170 26 165)(22 169 27 164)(23 168 28 163)(24 167 29 162)(25 166 30 161)(31 194 36 199)(32 193 37 198)(33 192 38 197)(34 191 39 196)(35 200 40 195)(41 209 46 204)(42 208 47 203)(43 207 48 202)(44 206 49 201)(45 205 50 210)(51 223 56 228)(52 222 57 227)(53 221 58 226)(54 230 59 225)(55 229 60 224)(61 214 66 219)(62 213 67 218)(63 212 68 217)(64 211 69 216)(65 220 70 215)(71 234 76 239)(72 233 77 238)(73 232 78 237)(74 231 79 236)(75 240 80 235)(81 249 86 244)(82 248 87 243)(83 247 88 242)(84 246 89 241)(85 245 90 250)(91 263 96 268)(92 262 97 267)(93 261 98 266)(94 270 99 265)(95 269 100 264)(101 254 106 259)(102 253 107 258)(103 252 108 257)(104 251 109 256)(105 260 110 255)(111 274 116 279)(112 273 117 278)(113 272 118 277)(114 271 119 276)(115 280 120 275)(121 289 126 284)(122 288 127 283)(123 287 128 282)(124 286 129 281)(125 285 130 290)(131 303 136 308)(132 302 137 307)(133 301 138 306)(134 310 139 305)(135 309 140 304)(141 294 146 299)(142 293 147 298)(143 292 148 297)(144 291 149 296)(145 300 150 295)(151 314 156 319)(152 313 157 318)(153 312 158 317)(154 311 159 316)(155 320 160 315)
G:=sub<Sym(320)| (1,119,39,98)(2,120,40,99)(3,111,31,100)(4,112,32,91)(5,113,33,92)(6,114,34,93)(7,115,35,94)(8,116,36,95)(9,117,37,96)(10,118,38,97)(11,124,44,106)(12,125,45,107)(13,126,46,108)(14,127,47,109)(15,128,48,110)(16,129,49,101)(17,130,50,102)(18,121,41,103)(19,122,42,104)(20,123,43,105)(21,215,295,242)(22,216,296,243)(23,217,297,244)(24,218,298,245)(25,219,299,246)(26,220,300,247)(27,211,291,248)(28,212,292,249)(29,213,293,250)(30,214,294,241)(51,152,72,131)(52,153,73,132)(53,154,74,133)(54,155,75,134)(55,156,76,135)(56,157,77,136)(57,158,78,137)(58,159,79,138)(59,160,80,139)(60,151,71,140)(61,166,89,141)(62,167,90,142)(63,168,81,143)(64,169,82,144)(65,170,83,145)(66,161,84,146)(67,162,85,147)(68,163,86,148)(69,164,87,149)(70,165,88,150)(171,251,208,288)(172,252,209,289)(173,253,210,290)(174,254,201,281)(175,255,202,282)(176,256,203,283)(177,257,204,284)(178,258,205,285)(179,259,206,286)(180,260,207,287)(181,261,191,271)(182,262,192,272)(183,263,193,273)(184,264,194,274)(185,265,195,275)(186,266,196,276)(187,267,197,277)(188,268,198,278)(189,269,199,279)(190,270,200,280)(221,301,231,311)(222,302,232,312)(223,303,233,313)(224,304,234,314)(225,305,235,315)(226,306,236,316)(227,307,237,317)(228,308,238,318)(229,309,239,319)(230,310,240,320), (1,65,15,53)(2,66,16,54)(3,67,17,55)(4,68,18,56)(5,69,19,57)(6,70,20,58)(7,61,11,59)(8,62,12,60)(9,63,13,51)(10,64,14,52)(21,271,316,287)(22,272,317,288)(23,273,318,289)(24,274,319,290)(25,275,320,281)(26,276,311,282)(27,277,312,283)(28,278,313,284)(29,279,314,285)(30,280,315,286)(31,85,50,76)(32,86,41,77)(33,87,42,78)(34,88,43,79)(35,89,44,80)(36,90,45,71)(37,81,46,72)(38,82,47,73)(39,83,48,74)(40,84,49,75)(91,148,103,136)(92,149,104,137)(93,150,105,138)(94,141,106,139)(95,142,107,140)(96,143,108,131)(97,144,109,132)(98,145,110,133)(99,146,101,134)(100,147,102,135)(111,162,130,156)(112,163,121,157)(113,164,122,158)(114,165,123,159)(115,166,124,160)(116,167,125,151)(117,168,126,152)(118,169,127,153)(119,170,128,154)(120,161,129,155)(171,216,182,227)(172,217,183,228)(173,218,184,229)(174,219,185,230)(175,220,186,221)(176,211,187,222)(177,212,188,223)(178,213,189,224)(179,214,190,225)(180,215,181,226)(191,236,207,242)(192,237,208,243)(193,238,209,244)(194,239,210,245)(195,240,201,246)(196,231,202,247)(197,232,203,248)(198,233,204,249)(199,234,205,250)(200,235,206,241)(251,296,262,307)(252,297,263,308)(253,298,264,309)(254,299,265,310)(255,300,266,301)(256,291,267,302)(257,292,268,303)(258,293,269,304)(259,294,270,305)(260,295,261,306), (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), (1,186,6,181)(2,185,7,190)(3,184,8,189)(4,183,9,188)(5,182,10,187)(11,179,16,174)(12,178,17,173)(13,177,18,172)(14,176,19,171)(15,175,20,180)(21,170,26,165)(22,169,27,164)(23,168,28,163)(24,167,29,162)(25,166,30,161)(31,194,36,199)(32,193,37,198)(33,192,38,197)(34,191,39,196)(35,200,40,195)(41,209,46,204)(42,208,47,203)(43,207,48,202)(44,206,49,201)(45,205,50,210)(51,223,56,228)(52,222,57,227)(53,221,58,226)(54,230,59,225)(55,229,60,224)(61,214,66,219)(62,213,67,218)(63,212,68,217)(64,211,69,216)(65,220,70,215)(71,234,76,239)(72,233,77,238)(73,232,78,237)(74,231,79,236)(75,240,80,235)(81,249,86,244)(82,248,87,243)(83,247,88,242)(84,246,89,241)(85,245,90,250)(91,263,96,268)(92,262,97,267)(93,261,98,266)(94,270,99,265)(95,269,100,264)(101,254,106,259)(102,253,107,258)(103,252,108,257)(104,251,109,256)(105,260,110,255)(111,274,116,279)(112,273,117,278)(113,272,118,277)(114,271,119,276)(115,280,120,275)(121,289,126,284)(122,288,127,283)(123,287,128,282)(124,286,129,281)(125,285,130,290)(131,303,136,308)(132,302,137,307)(133,301,138,306)(134,310,139,305)(135,309,140,304)(141,294,146,299)(142,293,147,298)(143,292,148,297)(144,291,149,296)(145,300,150,295)(151,314,156,319)(152,313,157,318)(153,312,158,317)(154,311,159,316)(155,320,160,315)>;
G:=Group( (1,119,39,98)(2,120,40,99)(3,111,31,100)(4,112,32,91)(5,113,33,92)(6,114,34,93)(7,115,35,94)(8,116,36,95)(9,117,37,96)(10,118,38,97)(11,124,44,106)(12,125,45,107)(13,126,46,108)(14,127,47,109)(15,128,48,110)(16,129,49,101)(17,130,50,102)(18,121,41,103)(19,122,42,104)(20,123,43,105)(21,215,295,242)(22,216,296,243)(23,217,297,244)(24,218,298,245)(25,219,299,246)(26,220,300,247)(27,211,291,248)(28,212,292,249)(29,213,293,250)(30,214,294,241)(51,152,72,131)(52,153,73,132)(53,154,74,133)(54,155,75,134)(55,156,76,135)(56,157,77,136)(57,158,78,137)(58,159,79,138)(59,160,80,139)(60,151,71,140)(61,166,89,141)(62,167,90,142)(63,168,81,143)(64,169,82,144)(65,170,83,145)(66,161,84,146)(67,162,85,147)(68,163,86,148)(69,164,87,149)(70,165,88,150)(171,251,208,288)(172,252,209,289)(173,253,210,290)(174,254,201,281)(175,255,202,282)(176,256,203,283)(177,257,204,284)(178,258,205,285)(179,259,206,286)(180,260,207,287)(181,261,191,271)(182,262,192,272)(183,263,193,273)(184,264,194,274)(185,265,195,275)(186,266,196,276)(187,267,197,277)(188,268,198,278)(189,269,199,279)(190,270,200,280)(221,301,231,311)(222,302,232,312)(223,303,233,313)(224,304,234,314)(225,305,235,315)(226,306,236,316)(227,307,237,317)(228,308,238,318)(229,309,239,319)(230,310,240,320), (1,65,15,53)(2,66,16,54)(3,67,17,55)(4,68,18,56)(5,69,19,57)(6,70,20,58)(7,61,11,59)(8,62,12,60)(9,63,13,51)(10,64,14,52)(21,271,316,287)(22,272,317,288)(23,273,318,289)(24,274,319,290)(25,275,320,281)(26,276,311,282)(27,277,312,283)(28,278,313,284)(29,279,314,285)(30,280,315,286)(31,85,50,76)(32,86,41,77)(33,87,42,78)(34,88,43,79)(35,89,44,80)(36,90,45,71)(37,81,46,72)(38,82,47,73)(39,83,48,74)(40,84,49,75)(91,148,103,136)(92,149,104,137)(93,150,105,138)(94,141,106,139)(95,142,107,140)(96,143,108,131)(97,144,109,132)(98,145,110,133)(99,146,101,134)(100,147,102,135)(111,162,130,156)(112,163,121,157)(113,164,122,158)(114,165,123,159)(115,166,124,160)(116,167,125,151)(117,168,126,152)(118,169,127,153)(119,170,128,154)(120,161,129,155)(171,216,182,227)(172,217,183,228)(173,218,184,229)(174,219,185,230)(175,220,186,221)(176,211,187,222)(177,212,188,223)(178,213,189,224)(179,214,190,225)(180,215,181,226)(191,236,207,242)(192,237,208,243)(193,238,209,244)(194,239,210,245)(195,240,201,246)(196,231,202,247)(197,232,203,248)(198,233,204,249)(199,234,205,250)(200,235,206,241)(251,296,262,307)(252,297,263,308)(253,298,264,309)(254,299,265,310)(255,300,266,301)(256,291,267,302)(257,292,268,303)(258,293,269,304)(259,294,270,305)(260,295,261,306), (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), (1,186,6,181)(2,185,7,190)(3,184,8,189)(4,183,9,188)(5,182,10,187)(11,179,16,174)(12,178,17,173)(13,177,18,172)(14,176,19,171)(15,175,20,180)(21,170,26,165)(22,169,27,164)(23,168,28,163)(24,167,29,162)(25,166,30,161)(31,194,36,199)(32,193,37,198)(33,192,38,197)(34,191,39,196)(35,200,40,195)(41,209,46,204)(42,208,47,203)(43,207,48,202)(44,206,49,201)(45,205,50,210)(51,223,56,228)(52,222,57,227)(53,221,58,226)(54,230,59,225)(55,229,60,224)(61,214,66,219)(62,213,67,218)(63,212,68,217)(64,211,69,216)(65,220,70,215)(71,234,76,239)(72,233,77,238)(73,232,78,237)(74,231,79,236)(75,240,80,235)(81,249,86,244)(82,248,87,243)(83,247,88,242)(84,246,89,241)(85,245,90,250)(91,263,96,268)(92,262,97,267)(93,261,98,266)(94,270,99,265)(95,269,100,264)(101,254,106,259)(102,253,107,258)(103,252,108,257)(104,251,109,256)(105,260,110,255)(111,274,116,279)(112,273,117,278)(113,272,118,277)(114,271,119,276)(115,280,120,275)(121,289,126,284)(122,288,127,283)(123,287,128,282)(124,286,129,281)(125,285,130,290)(131,303,136,308)(132,302,137,307)(133,301,138,306)(134,310,139,305)(135,309,140,304)(141,294,146,299)(142,293,147,298)(143,292,148,297)(144,291,149,296)(145,300,150,295)(151,314,156,319)(152,313,157,318)(153,312,158,317)(154,311,159,316)(155,320,160,315) );
G=PermutationGroup([[(1,119,39,98),(2,120,40,99),(3,111,31,100),(4,112,32,91),(5,113,33,92),(6,114,34,93),(7,115,35,94),(8,116,36,95),(9,117,37,96),(10,118,38,97),(11,124,44,106),(12,125,45,107),(13,126,46,108),(14,127,47,109),(15,128,48,110),(16,129,49,101),(17,130,50,102),(18,121,41,103),(19,122,42,104),(20,123,43,105),(21,215,295,242),(22,216,296,243),(23,217,297,244),(24,218,298,245),(25,219,299,246),(26,220,300,247),(27,211,291,248),(28,212,292,249),(29,213,293,250),(30,214,294,241),(51,152,72,131),(52,153,73,132),(53,154,74,133),(54,155,75,134),(55,156,76,135),(56,157,77,136),(57,158,78,137),(58,159,79,138),(59,160,80,139),(60,151,71,140),(61,166,89,141),(62,167,90,142),(63,168,81,143),(64,169,82,144),(65,170,83,145),(66,161,84,146),(67,162,85,147),(68,163,86,148),(69,164,87,149),(70,165,88,150),(171,251,208,288),(172,252,209,289),(173,253,210,290),(174,254,201,281),(175,255,202,282),(176,256,203,283),(177,257,204,284),(178,258,205,285),(179,259,206,286),(180,260,207,287),(181,261,191,271),(182,262,192,272),(183,263,193,273),(184,264,194,274),(185,265,195,275),(186,266,196,276),(187,267,197,277),(188,268,198,278),(189,269,199,279),(190,270,200,280),(221,301,231,311),(222,302,232,312),(223,303,233,313),(224,304,234,314),(225,305,235,315),(226,306,236,316),(227,307,237,317),(228,308,238,318),(229,309,239,319),(230,310,240,320)], [(1,65,15,53),(2,66,16,54),(3,67,17,55),(4,68,18,56),(5,69,19,57),(6,70,20,58),(7,61,11,59),(8,62,12,60),(9,63,13,51),(10,64,14,52),(21,271,316,287),(22,272,317,288),(23,273,318,289),(24,274,319,290),(25,275,320,281),(26,276,311,282),(27,277,312,283),(28,278,313,284),(29,279,314,285),(30,280,315,286),(31,85,50,76),(32,86,41,77),(33,87,42,78),(34,88,43,79),(35,89,44,80),(36,90,45,71),(37,81,46,72),(38,82,47,73),(39,83,48,74),(40,84,49,75),(91,148,103,136),(92,149,104,137),(93,150,105,138),(94,141,106,139),(95,142,107,140),(96,143,108,131),(97,144,109,132),(98,145,110,133),(99,146,101,134),(100,147,102,135),(111,162,130,156),(112,163,121,157),(113,164,122,158),(114,165,123,159),(115,166,124,160),(116,167,125,151),(117,168,126,152),(118,169,127,153),(119,170,128,154),(120,161,129,155),(171,216,182,227),(172,217,183,228),(173,218,184,229),(174,219,185,230),(175,220,186,221),(176,211,187,222),(177,212,188,223),(178,213,189,224),(179,214,190,225),(180,215,181,226),(191,236,207,242),(192,237,208,243),(193,238,209,244),(194,239,210,245),(195,240,201,246),(196,231,202,247),(197,232,203,248),(198,233,204,249),(199,234,205,250),(200,235,206,241),(251,296,262,307),(252,297,263,308),(253,298,264,309),(254,299,265,310),(255,300,266,301),(256,291,267,302),(257,292,268,303),(258,293,269,304),(259,294,270,305),(260,295,261,306)], [(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)], [(1,186,6,181),(2,185,7,190),(3,184,8,189),(4,183,9,188),(5,182,10,187),(11,179,16,174),(12,178,17,173),(13,177,18,172),(14,176,19,171),(15,175,20,180),(21,170,26,165),(22,169,27,164),(23,168,28,163),(24,167,29,162),(25,166,30,161),(31,194,36,199),(32,193,37,198),(33,192,38,197),(34,191,39,196),(35,200,40,195),(41,209,46,204),(42,208,47,203),(43,207,48,202),(44,206,49,201),(45,205,50,210),(51,223,56,228),(52,222,57,227),(53,221,58,226),(54,230,59,225),(55,229,60,224),(61,214,66,219),(62,213,67,218),(63,212,68,217),(64,211,69,216),(65,220,70,215),(71,234,76,239),(72,233,77,238),(73,232,78,237),(74,231,79,236),(75,240,80,235),(81,249,86,244),(82,248,87,243),(83,247,88,242),(84,246,89,241),(85,245,90,250),(91,263,96,268),(92,262,97,267),(93,261,98,266),(94,270,99,265),(95,269,100,264),(101,254,106,259),(102,253,107,258),(103,252,108,257),(104,251,109,256),(105,260,110,255),(111,274,116,279),(112,273,117,278),(113,272,118,277),(114,271,119,276),(115,280,120,275),(121,289,126,284),(122,288,127,283),(123,287,128,282),(124,286,129,281),(125,285,130,290),(131,303,136,308),(132,302,137,307),(133,301,138,306),(134,310,139,305),(135,309,140,304),(141,294,146,299),(142,293,147,298),(143,292,148,297),(144,291,149,296),(145,300,150,295),(151,314,156,319),(152,313,157,318),(153,312,158,317),(154,311,159,316),(155,320,160,315)]])
92 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | |
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D5 | Dic5 | D10 | Dic10 | D20 |
| kernel | C42⋊8Dic5 | C2×C4⋊Dic5 | C2×C4×C20 | C4×C20 | C2×C20 | C2×C20 | C2×C42 | C42 | C22×C4 | C2×C4 | C2×C4 |
| # reps | 1 | 6 | 1 | 8 | 6 | 6 | 2 | 8 | 6 | 24 | 24 |
Matrix representation of C42⋊8Dic5 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 2 | 9 | 0 | 0 |
| 0 | 4 | 39 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 40 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 40 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 40 | 0 | 0 |
| 0 | 36 | 6 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 37 | 16 | 0 | 0 |
| 0 | 22 | 4 | 0 | 0 |
| 0 | 0 | 0 | 37 | 29 |
| 0 | 0 | 0 | 39 | 4 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,2,4,0,0,0,9,39,0,0,0,0,0,1,40,0,0,0,2,40],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,40,0,0,0,2,40],[40,0,0,0,0,0,1,36,0,0,0,40,6,0,0,0,0,0,40,0,0,0,0,0,40],[9,0,0,0,0,0,37,22,0,0,0,16,4,0,0,0,0,0,37,39,0,0,0,29,4] >;
C42⋊8Dic5 in GAP, Magma, Sage, TeX
C_4^2\rtimes_8{\rm Dic}_5 % in TeX
G:=Group("C4^2:8Dic5"); // GroupNames label
G:=SmallGroup(320,562);
// by ID
G=gap.SmallGroup(320,562);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,232,422,100,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=c^5,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations