metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊5Dic5, (C4×C20)⋊17C4, (C2×C42).6D5, C5⋊3(C42⋊5C4), (C22×C4).398D10, C2.2(C42⋊2D5), C10.4(C42⋊2C2), C22.46(C4○D20), C23.270(C22×D5), C10.62(C42⋊C2), (C22×C20).476C22, (C22×C10).312C23, C22.38(C22×Dic5), C10.10C42.13C2, C2.7(C23.21D10), (C22×Dic5).31C22, (C2×C4×C20).3C2, (C2×C20).451(C2×C4), (C2×C4).63(C2×Dic5), (C2×C10).71(C4○D4), (C2×C10).278(C22×C4), SmallGroup(320,564)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊5Dic5
G = < a,b,c,d | a4=b4=c10=1, d2=c5, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 414 in 138 conjugacy classes, 71 normal (9 characteristic)
C1, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C42, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2.C42, C2×C42, C2×Dic5, C2×C20, C2×C20, C22×C10, C42⋊5C4, C4×C20, C22×Dic5, C22×C20, C10.10C42, C2×C4×C20, C42⋊5Dic5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C4○D4, Dic5, D10, C42⋊C2, C42⋊2C2, C2×Dic5, C22×D5, C42⋊5C4, C4○D20, C22×Dic5, C42⋊2D5, C23.21D10, C42⋊5Dic5
►Regular action on 320 points
Generators in S320
(1 104 12 98)(2 105 13 99)(3 106 14 100)(4 107 15 91)(5 108 16 92)(6 109 17 93)(7 110 18 94)(8 101 19 95)(9 102 20 96)(10 103 11 97)(21 221 316 212)(22 222 317 213)(23 223 318 214)(24 224 319 215)(25 225 320 216)(26 226 311 217)(27 227 312 218)(28 228 313 219)(29 229 314 220)(30 230 315 211)(31 130 50 111)(32 121 41 112)(33 122 42 113)(34 123 43 114)(35 124 44 115)(36 125 45 116)(37 126 46 117)(38 127 47 118)(39 128 48 119)(40 129 49 120)(51 147 67 131)(52 148 68 132)(53 149 69 133)(54 150 70 134)(55 141 61 135)(56 142 62 136)(57 143 63 137)(58 144 64 138)(59 145 65 139)(60 146 66 140)(71 167 90 151)(72 168 81 152)(73 169 82 153)(74 170 83 154)(75 161 84 155)(76 162 85 156)(77 163 86 157)(78 164 87 158)(79 165 88 159)(80 166 89 160)(171 281 190 275)(172 282 181 276)(173 283 182 277)(174 284 183 278)(175 285 184 279)(176 286 185 280)(177 287 186 271)(178 288 187 272)(179 289 188 273)(180 290 189 274)(191 266 207 257)(192 267 208 258)(193 268 209 259)(194 269 210 260)(195 270 201 251)(196 261 202 252)(197 262 203 253)(198 263 204 254)(199 264 205 255)(200 265 206 256)(231 306 247 297)(232 307 248 298)(233 308 249 299)(234 309 250 300)(235 310 241 291)(236 301 242 292)(237 302 243 293)(238 303 244 294)(239 304 245 295)(240 305 246 296)
(1 79 39 58)(2 80 40 59)(3 71 31 60)(4 72 32 51)(5 73 33 52)(6 74 34 53)(7 75 35 54)(8 76 36 55)(9 77 37 56)(10 78 38 57)(11 87 47 63)(12 88 48 64)(13 89 49 65)(14 90 50 66)(15 81 41 67)(16 82 42 68)(17 83 43 69)(18 84 44 70)(19 85 45 61)(20 86 46 62)(21 252 301 276)(22 253 302 277)(23 254 303 278)(24 255 304 279)(25 256 305 280)(26 257 306 271)(27 258 307 272)(28 259 308 273)(29 260 309 274)(30 251 310 275)(91 152 112 131)(92 153 113 132)(93 154 114 133)(94 155 115 134)(95 156 116 135)(96 157 117 136)(97 158 118 137)(98 159 119 138)(99 160 120 139)(100 151 111 140)(101 162 125 141)(102 163 126 142)(103 164 127 143)(104 165 128 144)(105 166 129 145)(106 167 130 146)(107 168 121 147)(108 169 122 148)(109 170 123 149)(110 161 124 150)(171 230 195 241)(172 221 196 242)(173 222 197 243)(174 223 198 244)(175 224 199 245)(176 225 200 246)(177 226 191 247)(178 227 192 248)(179 228 193 249)(180 229 194 250)(181 212 202 236)(182 213 203 237)(183 214 204 238)(184 215 205 239)(185 216 206 240)(186 217 207 231)(187 218 208 232)(188 219 209 233)(189 220 210 234)(190 211 201 235)(261 292 282 316)(262 293 283 317)(263 294 284 318)(264 295 285 319)(265 296 286 320)(266 297 287 311)(267 298 288 312)(268 299 289 313)(269 300 290 314)(270 291 281 315)
(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)
(1 172 6 177)(2 171 7 176)(3 180 8 175)(4 179 9 174)(5 178 10 173)(11 182 16 187)(12 181 17 186)(13 190 18 185)(14 189 19 184)(15 188 20 183)(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)(51 219 56 214)(52 218 57 213)(53 217 58 212)(54 216 59 211)(55 215 60 220)(61 224 66 229)(62 223 67 228)(63 222 68 227)(64 221 69 226)(65 230 70 225)(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)(91 259 96 254)(92 258 97 253)(93 257 98 252)(94 256 99 251)(95 255 100 260)(101 264 106 269)(102 263 107 268)(103 262 108 267)(104 261 109 266)(105 270 110 265)(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)(131 299 136 294)(132 298 137 293)(133 297 138 292)(134 296 139 291)(135 295 140 300)(141 304 146 309)(142 303 147 308)(143 302 148 307)(144 301 149 306)(145 310 150 305)(151 314 156 319)(152 313 157 318)(153 312 158 317)(154 311 159 316)(155 320 160 315)
G:=sub<Sym(320)| (1,104,12,98)(2,105,13,99)(3,106,14,100)(4,107,15,91)(5,108,16,92)(6,109,17,93)(7,110,18,94)(8,101,19,95)(9,102,20,96)(10,103,11,97)(21,221,316,212)(22,222,317,213)(23,223,318,214)(24,224,319,215)(25,225,320,216)(26,226,311,217)(27,227,312,218)(28,228,313,219)(29,229,314,220)(30,230,315,211)(31,130,50,111)(32,121,41,112)(33,122,42,113)(34,123,43,114)(35,124,44,115)(36,125,45,116)(37,126,46,117)(38,127,47,118)(39,128,48,119)(40,129,49,120)(51,147,67,131)(52,148,68,132)(53,149,69,133)(54,150,70,134)(55,141,61,135)(56,142,62,136)(57,143,63,137)(58,144,64,138)(59,145,65,139)(60,146,66,140)(71,167,90,151)(72,168,81,152)(73,169,82,153)(74,170,83,154)(75,161,84,155)(76,162,85,156)(77,163,86,157)(78,164,87,158)(79,165,88,159)(80,166,89,160)(171,281,190,275)(172,282,181,276)(173,283,182,277)(174,284,183,278)(175,285,184,279)(176,286,185,280)(177,287,186,271)(178,288,187,272)(179,289,188,273)(180,290,189,274)(191,266,207,257)(192,267,208,258)(193,268,209,259)(194,269,210,260)(195,270,201,251)(196,261,202,252)(197,262,203,253)(198,263,204,254)(199,264,205,255)(200,265,206,256)(231,306,247,297)(232,307,248,298)(233,308,249,299)(234,309,250,300)(235,310,241,291)(236,301,242,292)(237,302,243,293)(238,303,244,294)(239,304,245,295)(240,305,246,296), (1,79,39,58)(2,80,40,59)(3,71,31,60)(4,72,32,51)(5,73,33,52)(6,74,34,53)(7,75,35,54)(8,76,36,55)(9,77,37,56)(10,78,38,57)(11,87,47,63)(12,88,48,64)(13,89,49,65)(14,90,50,66)(15,81,41,67)(16,82,42,68)(17,83,43,69)(18,84,44,70)(19,85,45,61)(20,86,46,62)(21,252,301,276)(22,253,302,277)(23,254,303,278)(24,255,304,279)(25,256,305,280)(26,257,306,271)(27,258,307,272)(28,259,308,273)(29,260,309,274)(30,251,310,275)(91,152,112,131)(92,153,113,132)(93,154,114,133)(94,155,115,134)(95,156,116,135)(96,157,117,136)(97,158,118,137)(98,159,119,138)(99,160,120,139)(100,151,111,140)(101,162,125,141)(102,163,126,142)(103,164,127,143)(104,165,128,144)(105,166,129,145)(106,167,130,146)(107,168,121,147)(108,169,122,148)(109,170,123,149)(110,161,124,150)(171,230,195,241)(172,221,196,242)(173,222,197,243)(174,223,198,244)(175,224,199,245)(176,225,200,246)(177,226,191,247)(178,227,192,248)(179,228,193,249)(180,229,194,250)(181,212,202,236)(182,213,203,237)(183,214,204,238)(184,215,205,239)(185,216,206,240)(186,217,207,231)(187,218,208,232)(188,219,209,233)(189,220,210,234)(190,211,201,235)(261,292,282,316)(262,293,283,317)(263,294,284,318)(264,295,285,319)(265,296,286,320)(266,297,287,311)(267,298,288,312)(268,299,289,313)(269,300,290,314)(270,291,281,315), (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), (1,172,6,177)(2,171,7,176)(3,180,8,175)(4,179,9,174)(5,178,10,173)(11,182,16,187)(12,181,17,186)(13,190,18,185)(14,189,19,184)(15,188,20,183)(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)(51,219,56,214)(52,218,57,213)(53,217,58,212)(54,216,59,211)(55,215,60,220)(61,224,66,229)(62,223,67,228)(63,222,68,227)(64,221,69,226)(65,230,70,225)(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)(91,259,96,254)(92,258,97,253)(93,257,98,252)(94,256,99,251)(95,255,100,260)(101,264,106,269)(102,263,107,268)(103,262,108,267)(104,261,109,266)(105,270,110,265)(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)(131,299,136,294)(132,298,137,293)(133,297,138,292)(134,296,139,291)(135,295,140,300)(141,304,146,309)(142,303,147,308)(143,302,148,307)(144,301,149,306)(145,310,150,305)(151,314,156,319)(152,313,157,318)(153,312,158,317)(154,311,159,316)(155,320,160,315)>;
G:=Group( (1,104,12,98)(2,105,13,99)(3,106,14,100)(4,107,15,91)(5,108,16,92)(6,109,17,93)(7,110,18,94)(8,101,19,95)(9,102,20,96)(10,103,11,97)(21,221,316,212)(22,222,317,213)(23,223,318,214)(24,224,319,215)(25,225,320,216)(26,226,311,217)(27,227,312,218)(28,228,313,219)(29,229,314,220)(30,230,315,211)(31,130,50,111)(32,121,41,112)(33,122,42,113)(34,123,43,114)(35,124,44,115)(36,125,45,116)(37,126,46,117)(38,127,47,118)(39,128,48,119)(40,129,49,120)(51,147,67,131)(52,148,68,132)(53,149,69,133)(54,150,70,134)(55,141,61,135)(56,142,62,136)(57,143,63,137)(58,144,64,138)(59,145,65,139)(60,146,66,140)(71,167,90,151)(72,168,81,152)(73,169,82,153)(74,170,83,154)(75,161,84,155)(76,162,85,156)(77,163,86,157)(78,164,87,158)(79,165,88,159)(80,166,89,160)(171,281,190,275)(172,282,181,276)(173,283,182,277)(174,284,183,278)(175,285,184,279)(176,286,185,280)(177,287,186,271)(178,288,187,272)(179,289,188,273)(180,290,189,274)(191,266,207,257)(192,267,208,258)(193,268,209,259)(194,269,210,260)(195,270,201,251)(196,261,202,252)(197,262,203,253)(198,263,204,254)(199,264,205,255)(200,265,206,256)(231,306,247,297)(232,307,248,298)(233,308,249,299)(234,309,250,300)(235,310,241,291)(236,301,242,292)(237,302,243,293)(238,303,244,294)(239,304,245,295)(240,305,246,296), (1,79,39,58)(2,80,40,59)(3,71,31,60)(4,72,32,51)(5,73,33,52)(6,74,34,53)(7,75,35,54)(8,76,36,55)(9,77,37,56)(10,78,38,57)(11,87,47,63)(12,88,48,64)(13,89,49,65)(14,90,50,66)(15,81,41,67)(16,82,42,68)(17,83,43,69)(18,84,44,70)(19,85,45,61)(20,86,46,62)(21,252,301,276)(22,253,302,277)(23,254,303,278)(24,255,304,279)(25,256,305,280)(26,257,306,271)(27,258,307,272)(28,259,308,273)(29,260,309,274)(30,251,310,275)(91,152,112,131)(92,153,113,132)(93,154,114,133)(94,155,115,134)(95,156,116,135)(96,157,117,136)(97,158,118,137)(98,159,119,138)(99,160,120,139)(100,151,111,140)(101,162,125,141)(102,163,126,142)(103,164,127,143)(104,165,128,144)(105,166,129,145)(106,167,130,146)(107,168,121,147)(108,169,122,148)(109,170,123,149)(110,161,124,150)(171,230,195,241)(172,221,196,242)(173,222,197,243)(174,223,198,244)(175,224,199,245)(176,225,200,246)(177,226,191,247)(178,227,192,248)(179,228,193,249)(180,229,194,250)(181,212,202,236)(182,213,203,237)(183,214,204,238)(184,215,205,239)(185,216,206,240)(186,217,207,231)(187,218,208,232)(188,219,209,233)(189,220,210,234)(190,211,201,235)(261,292,282,316)(262,293,283,317)(263,294,284,318)(264,295,285,319)(265,296,286,320)(266,297,287,311)(267,298,288,312)(268,299,289,313)(269,300,290,314)(270,291,281,315), (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), (1,172,6,177)(2,171,7,176)(3,180,8,175)(4,179,9,174)(5,178,10,173)(11,182,16,187)(12,181,17,186)(13,190,18,185)(14,189,19,184)(15,188,20,183)(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)(51,219,56,214)(52,218,57,213)(53,217,58,212)(54,216,59,211)(55,215,60,220)(61,224,66,229)(62,223,67,228)(63,222,68,227)(64,221,69,226)(65,230,70,225)(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)(91,259,96,254)(92,258,97,253)(93,257,98,252)(94,256,99,251)(95,255,100,260)(101,264,106,269)(102,263,107,268)(103,262,108,267)(104,261,109,266)(105,270,110,265)(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)(131,299,136,294)(132,298,137,293)(133,297,138,292)(134,296,139,291)(135,295,140,300)(141,304,146,309)(142,303,147,308)(143,302,148,307)(144,301,149,306)(145,310,150,305)(151,314,156,319)(152,313,157,318)(153,312,158,317)(154,311,159,316)(155,320,160,315) );
G=PermutationGroup([[(1,104,12,98),(2,105,13,99),(3,106,14,100),(4,107,15,91),(5,108,16,92),(6,109,17,93),(7,110,18,94),(8,101,19,95),(9,102,20,96),(10,103,11,97),(21,221,316,212),(22,222,317,213),(23,223,318,214),(24,224,319,215),(25,225,320,216),(26,226,311,217),(27,227,312,218),(28,228,313,219),(29,229,314,220),(30,230,315,211),(31,130,50,111),(32,121,41,112),(33,122,42,113),(34,123,43,114),(35,124,44,115),(36,125,45,116),(37,126,46,117),(38,127,47,118),(39,128,48,119),(40,129,49,120),(51,147,67,131),(52,148,68,132),(53,149,69,133),(54,150,70,134),(55,141,61,135),(56,142,62,136),(57,143,63,137),(58,144,64,138),(59,145,65,139),(60,146,66,140),(71,167,90,151),(72,168,81,152),(73,169,82,153),(74,170,83,154),(75,161,84,155),(76,162,85,156),(77,163,86,157),(78,164,87,158),(79,165,88,159),(80,166,89,160),(171,281,190,275),(172,282,181,276),(173,283,182,277),(174,284,183,278),(175,285,184,279),(176,286,185,280),(177,287,186,271),(178,288,187,272),(179,289,188,273),(180,290,189,274),(191,266,207,257),(192,267,208,258),(193,268,209,259),(194,269,210,260),(195,270,201,251),(196,261,202,252),(197,262,203,253),(198,263,204,254),(199,264,205,255),(200,265,206,256),(231,306,247,297),(232,307,248,298),(233,308,249,299),(234,309,250,300),(235,310,241,291),(236,301,242,292),(237,302,243,293),(238,303,244,294),(239,304,245,295),(240,305,246,296)], [(1,79,39,58),(2,80,40,59),(3,71,31,60),(4,72,32,51),(5,73,33,52),(6,74,34,53),(7,75,35,54),(8,76,36,55),(9,77,37,56),(10,78,38,57),(11,87,47,63),(12,88,48,64),(13,89,49,65),(14,90,50,66),(15,81,41,67),(16,82,42,68),(17,83,43,69),(18,84,44,70),(19,85,45,61),(20,86,46,62),(21,252,301,276),(22,253,302,277),(23,254,303,278),(24,255,304,279),(25,256,305,280),(26,257,306,271),(27,258,307,272),(28,259,308,273),(29,260,309,274),(30,251,310,275),(91,152,112,131),(92,153,113,132),(93,154,114,133),(94,155,115,134),(95,156,116,135),(96,157,117,136),(97,158,118,137),(98,159,119,138),(99,160,120,139),(100,151,111,140),(101,162,125,141),(102,163,126,142),(103,164,127,143),(104,165,128,144),(105,166,129,145),(106,167,130,146),(107,168,121,147),(108,169,122,148),(109,170,123,149),(110,161,124,150),(171,230,195,241),(172,221,196,242),(173,222,197,243),(174,223,198,244),(175,224,199,245),(176,225,200,246),(177,226,191,247),(178,227,192,248),(179,228,193,249),(180,229,194,250),(181,212,202,236),(182,213,203,237),(183,214,204,238),(184,215,205,239),(185,216,206,240),(186,217,207,231),(187,218,208,232),(188,219,209,233),(189,220,210,234),(190,211,201,235),(261,292,282,316),(262,293,283,317),(263,294,284,318),(264,295,285,319),(265,296,286,320),(266,297,287,311),(267,298,288,312),(268,299,289,313),(269,300,290,314),(270,291,281,315)], [(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)], [(1,172,6,177),(2,171,7,176),(3,180,8,175),(4,179,9,174),(5,178,10,173),(11,182,16,187),(12,181,17,186),(13,190,18,185),(14,189,19,184),(15,188,20,183),(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),(51,219,56,214),(52,218,57,213),(53,217,58,212),(54,216,59,211),(55,215,60,220),(61,224,66,229),(62,223,67,228),(63,222,68,227),(64,221,69,226),(65,230,70,225),(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),(91,259,96,254),(92,258,97,253),(93,257,98,252),(94,256,99,251),(95,255,100,260),(101,264,106,269),(102,263,107,268),(103,262,108,267),(104,261,109,266),(105,270,110,265),(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),(131,299,136,294),(132,298,137,293),(133,297,138,292),(134,296,139,291),(135,295,140,300),(141,304,146,309),(142,303,147,308),(143,302,148,307),(144,301,149,306),(145,310,150,305),(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 |
| type | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C4 | D5 | C4○D4 | Dic5 | D10 | C4○D20 |
| kernel | C42⋊5Dic5 | C10.10C42 | C2×C4×C20 | C4×C20 | C2×C42 | C2×C10 | C42 | C22×C4 | C22 |
| # reps | 1 | 6 | 1 | 8 | 2 | 12 | 8 | 6 | 48 |
Matrix representation of C42⋊5Dic5 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 39 | 28 | 0 | 0 |
| 0 | 13 | 2 | 0 | 0 |
| 0 | 0 | 0 | 18 | 35 |
| 0 | 0 | 0 | 6 | 23 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 2 | 13 |
| 0 | 0 | 0 | 28 | 39 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 35 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 38 | 36 | 0 | 0 |
| 0 | 18 | 3 | 0 | 0 |
| 0 | 0 | 0 | 31 | 4 |
| 0 | 0 | 0 | 26 | 10 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,39,13,0,0,0,28,2,0,0,0,0,0,18,6,0,0,0,35,23],[40,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,2,28,0,0,0,13,39],[40,0,0,0,0,0,0,40,0,0,0,1,6,0,0,0,0,0,0,1,0,0,0,40,35],[9,0,0,0,0,0,38,18,0,0,0,36,3,0,0,0,0,0,31,26,0,0,0,4,10] >;
C42⋊5Dic5 in GAP, Magma, Sage, TeX
C_4^2\rtimes_5{\rm Dic}_5 % in TeX
G:=Group("C4^2:5Dic5"); // GroupNames label
G:=SmallGroup(320,564);
// by ID
G=gap.SmallGroup(320,564);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,120,1094,184,12550]);
// 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*b^2,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations