direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C4⋊Dic5, C20⋊5C42, C42⋊7Dic5, (C4×C20)⋊23C4, C20⋊10(C4⋊C4), C4⋊2(C4×Dic5), C2.2(C4×D20), C10.46(C4×D4), C10.21(C4×Q8), (C2×C20).67Q8, (C2×C4).167D20, (C2×C20).400D4, C2.3(C4×Dic10), (C2×C42).15D5, C10.38(C2×C42), (C2×C4).57Dic10, C22.35(C2×D20), (C22×C4).418D10, C22.44(C4○D20), C22.20(C2×Dic10), C23.267(C22×D5), C10.60(C42⋊C2), (C22×C10).309C23, (C22×C20).474C22, C22.18(C22×Dic5), C10.10C42.43C2, C2.3(C23.21D10), (C22×Dic5).203C22, C5⋊5(C4×C4⋊C4), (C2×C4×C20).21C2, C10.49(C2×C4⋊C4), C2.7(C2×C4×Dic5), C2.2(C2×C4⋊Dic5), C22.52(C2×C4×D5), (C2×C4).110(C4×D5), (C2×C10).27(C2×Q8), (C2×C4×Dic5).34C2, (C2×C20).490(C2×C4), (C2×C10).145(C2×D4), (C2×C4⋊Dic5).47C2, (C2×C4).61(C2×Dic5), (C2×C10).69(C4○D4), (C2×C10).199(C22×C4), (C2×Dic5).102(C2×C4), SmallGroup(320,561)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C4⋊Dic5
G = < a,b,c,d | a4=b4=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 510 in 194 conjugacy classes, 119 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C42, C42, C4⋊C4, C22×C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C4×C4⋊C4, C4×Dic5, C4⋊Dic5, C4×C20, C22×Dic5, C22×C20, C10.10C42, C2×C4×Dic5, C2×C4⋊Dic5, C2×C4×C20, C4×C4⋊Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C42, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, D10, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, Dic10, C4×D5, D20, C2×Dic5, C22×D5, C4×C4⋊C4, C4×Dic5, C4⋊Dic5, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C22×Dic5, C4×Dic10, C4×D20, C2×C4×Dic5, C2×C4⋊Dic5, C23.21D10, C4×C4⋊Dic5
►Regular action on 320 points
Generators in S320
(1 66 13 57)(2 67 14 58)(3 68 15 59)(4 69 16 60)(5 70 17 51)(6 61 18 52)(7 62 19 53)(8 63 20 54)(9 64 11 55)(10 65 12 56)(21 288 319 279)(22 289 320 280)(23 290 311 271)(24 281 312 272)(25 282 313 273)(26 283 314 274)(27 284 315 275)(28 285 316 276)(29 286 317 277)(30 287 318 278)(31 87 47 71)(32 88 48 72)(33 89 49 73)(34 90 50 74)(35 81 41 75)(36 82 42 76)(37 83 43 77)(38 84 44 78)(39 85 45 79)(40 86 46 80)(91 150 110 131)(92 141 101 132)(93 142 102 133)(94 143 103 134)(95 144 104 135)(96 145 105 136)(97 146 106 137)(98 147 107 138)(99 148 108 139)(100 149 109 140)(111 170 127 151)(112 161 128 152)(113 162 129 153)(114 163 130 154)(115 164 121 155)(116 165 122 156)(117 166 123 157)(118 167 124 158)(119 168 125 159)(120 169 126 160)(171 227 187 211)(172 228 188 212)(173 229 189 213)(174 230 190 214)(175 221 181 215)(176 222 182 216)(177 223 183 217)(178 224 184 218)(179 225 185 219)(180 226 186 220)(191 250 210 231)(192 241 201 232)(193 242 202 233)(194 243 203 234)(195 244 204 235)(196 245 205 236)(197 246 206 237)(198 247 207 238)(199 248 208 239)(200 249 209 240)(251 307 267 291)(252 308 268 292)(253 309 269 293)(254 310 270 294)(255 301 261 295)(256 302 262 296)(257 303 263 297)(258 304 264 298)(259 305 265 299)(260 306 266 300)
(1 111 31 97)(2 112 32 98)(3 113 33 99)(4 114 34 100)(5 115 35 91)(6 116 36 92)(7 117 37 93)(8 118 38 94)(9 119 39 95)(10 120 40 96)(11 125 45 104)(12 126 46 105)(13 127 47 106)(14 128 48 107)(15 129 49 108)(16 130 50 109)(17 121 41 110)(18 122 42 101)(19 123 43 102)(20 124 44 103)(21 224 304 248)(22 225 305 249)(23 226 306 250)(24 227 307 241)(25 228 308 242)(26 229 309 243)(27 230 310 244)(28 221 301 245)(29 222 302 246)(30 223 303 247)(51 155 75 131)(52 156 76 132)(53 157 77 133)(54 158 78 134)(55 159 79 135)(56 160 80 136)(57 151 71 137)(58 152 72 138)(59 153 73 139)(60 154 74 140)(61 165 82 141)(62 166 83 142)(63 167 84 143)(64 168 85 144)(65 169 86 145)(66 170 87 146)(67 161 88 147)(68 162 89 148)(69 163 90 149)(70 164 81 150)(171 251 192 272)(172 252 193 273)(173 253 194 274)(174 254 195 275)(175 255 196 276)(176 256 197 277)(177 257 198 278)(178 258 199 279)(179 259 200 280)(180 260 191 271)(181 261 205 285)(182 262 206 286)(183 263 207 287)(184 264 208 288)(185 265 209 289)(186 266 210 290)(187 267 201 281)(188 268 202 282)(189 269 203 283)(190 270 204 284)(211 291 232 312)(212 292 233 313)(213 293 234 314)(214 294 235 315)(215 295 236 316)(216 296 237 317)(217 297 238 318)(218 298 239 319)(219 299 240 320)(220 300 231 311)
(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)
(1 173 6 178)(2 172 7 177)(3 171 8 176)(4 180 9 175)(5 179 10 174)(11 181 16 186)(12 190 17 185)(13 189 18 184)(14 188 19 183)(15 187 20 182)(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)| (1,66,13,57)(2,67,14,58)(3,68,15,59)(4,69,16,60)(5,70,17,51)(6,61,18,52)(7,62,19,53)(8,63,20,54)(9,64,11,55)(10,65,12,56)(21,288,319,279)(22,289,320,280)(23,290,311,271)(24,281,312,272)(25,282,313,273)(26,283,314,274)(27,284,315,275)(28,285,316,276)(29,286,317,277)(30,287,318,278)(31,87,47,71)(32,88,48,72)(33,89,49,73)(34,90,50,74)(35,81,41,75)(36,82,42,76)(37,83,43,77)(38,84,44,78)(39,85,45,79)(40,86,46,80)(91,150,110,131)(92,141,101,132)(93,142,102,133)(94,143,103,134)(95,144,104,135)(96,145,105,136)(97,146,106,137)(98,147,107,138)(99,148,108,139)(100,149,109,140)(111,170,127,151)(112,161,128,152)(113,162,129,153)(114,163,130,154)(115,164,121,155)(116,165,122,156)(117,166,123,157)(118,167,124,158)(119,168,125,159)(120,169,126,160)(171,227,187,211)(172,228,188,212)(173,229,189,213)(174,230,190,214)(175,221,181,215)(176,222,182,216)(177,223,183,217)(178,224,184,218)(179,225,185,219)(180,226,186,220)(191,250,210,231)(192,241,201,232)(193,242,202,233)(194,243,203,234)(195,244,204,235)(196,245,205,236)(197,246,206,237)(198,247,207,238)(199,248,208,239)(200,249,209,240)(251,307,267,291)(252,308,268,292)(253,309,269,293)(254,310,270,294)(255,301,261,295)(256,302,262,296)(257,303,263,297)(258,304,264,298)(259,305,265,299)(260,306,266,300), (1,111,31,97)(2,112,32,98)(3,113,33,99)(4,114,34,100)(5,115,35,91)(6,116,36,92)(7,117,37,93)(8,118,38,94)(9,119,39,95)(10,120,40,96)(11,125,45,104)(12,126,46,105)(13,127,47,106)(14,128,48,107)(15,129,49,108)(16,130,50,109)(17,121,41,110)(18,122,42,101)(19,123,43,102)(20,124,44,103)(21,224,304,248)(22,225,305,249)(23,226,306,250)(24,227,307,241)(25,228,308,242)(26,229,309,243)(27,230,310,244)(28,221,301,245)(29,222,302,246)(30,223,303,247)(51,155,75,131)(52,156,76,132)(53,157,77,133)(54,158,78,134)(55,159,79,135)(56,160,80,136)(57,151,71,137)(58,152,72,138)(59,153,73,139)(60,154,74,140)(61,165,82,141)(62,166,83,142)(63,167,84,143)(64,168,85,144)(65,169,86,145)(66,170,87,146)(67,161,88,147)(68,162,89,148)(69,163,90,149)(70,164,81,150)(171,251,192,272)(172,252,193,273)(173,253,194,274)(174,254,195,275)(175,255,196,276)(176,256,197,277)(177,257,198,278)(178,258,199,279)(179,259,200,280)(180,260,191,271)(181,261,205,285)(182,262,206,286)(183,263,207,287)(184,264,208,288)(185,265,209,289)(186,266,210,290)(187,267,201,281)(188,268,202,282)(189,269,203,283)(190,270,204,284)(211,291,232,312)(212,292,233,313)(213,293,234,314)(214,294,235,315)(215,295,236,316)(216,296,237,317)(217,297,238,318)(218,298,239,319)(219,299,240,320)(220,300,231,311), (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), (1,173,6,178)(2,172,7,177)(3,171,8,176)(4,180,9,175)(5,179,10,174)(11,181,16,186)(12,190,17,185)(13,189,18,184)(14,188,19,183)(15,187,20,182)(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( (1,66,13,57)(2,67,14,58)(3,68,15,59)(4,69,16,60)(5,70,17,51)(6,61,18,52)(7,62,19,53)(8,63,20,54)(9,64,11,55)(10,65,12,56)(21,288,319,279)(22,289,320,280)(23,290,311,271)(24,281,312,272)(25,282,313,273)(26,283,314,274)(27,284,315,275)(28,285,316,276)(29,286,317,277)(30,287,318,278)(31,87,47,71)(32,88,48,72)(33,89,49,73)(34,90,50,74)(35,81,41,75)(36,82,42,76)(37,83,43,77)(38,84,44,78)(39,85,45,79)(40,86,46,80)(91,150,110,131)(92,141,101,132)(93,142,102,133)(94,143,103,134)(95,144,104,135)(96,145,105,136)(97,146,106,137)(98,147,107,138)(99,148,108,139)(100,149,109,140)(111,170,127,151)(112,161,128,152)(113,162,129,153)(114,163,130,154)(115,164,121,155)(116,165,122,156)(117,166,123,157)(118,167,124,158)(119,168,125,159)(120,169,126,160)(171,227,187,211)(172,228,188,212)(173,229,189,213)(174,230,190,214)(175,221,181,215)(176,222,182,216)(177,223,183,217)(178,224,184,218)(179,225,185,219)(180,226,186,220)(191,250,210,231)(192,241,201,232)(193,242,202,233)(194,243,203,234)(195,244,204,235)(196,245,205,236)(197,246,206,237)(198,247,207,238)(199,248,208,239)(200,249,209,240)(251,307,267,291)(252,308,268,292)(253,309,269,293)(254,310,270,294)(255,301,261,295)(256,302,262,296)(257,303,263,297)(258,304,264,298)(259,305,265,299)(260,306,266,300), (1,111,31,97)(2,112,32,98)(3,113,33,99)(4,114,34,100)(5,115,35,91)(6,116,36,92)(7,117,37,93)(8,118,38,94)(9,119,39,95)(10,120,40,96)(11,125,45,104)(12,126,46,105)(13,127,47,106)(14,128,48,107)(15,129,49,108)(16,130,50,109)(17,121,41,110)(18,122,42,101)(19,123,43,102)(20,124,44,103)(21,224,304,248)(22,225,305,249)(23,226,306,250)(24,227,307,241)(25,228,308,242)(26,229,309,243)(27,230,310,244)(28,221,301,245)(29,222,302,246)(30,223,303,247)(51,155,75,131)(52,156,76,132)(53,157,77,133)(54,158,78,134)(55,159,79,135)(56,160,80,136)(57,151,71,137)(58,152,72,138)(59,153,73,139)(60,154,74,140)(61,165,82,141)(62,166,83,142)(63,167,84,143)(64,168,85,144)(65,169,86,145)(66,170,87,146)(67,161,88,147)(68,162,89,148)(69,163,90,149)(70,164,81,150)(171,251,192,272)(172,252,193,273)(173,253,194,274)(174,254,195,275)(175,255,196,276)(176,256,197,277)(177,257,198,278)(178,258,199,279)(179,259,200,280)(180,260,191,271)(181,261,205,285)(182,262,206,286)(183,263,207,287)(184,264,208,288)(185,265,209,289)(186,266,210,290)(187,267,201,281)(188,268,202,282)(189,269,203,283)(190,270,204,284)(211,291,232,312)(212,292,233,313)(213,293,234,314)(214,294,235,315)(215,295,236,316)(216,296,237,317)(217,297,238,318)(218,298,239,319)(219,299,240,320)(220,300,231,311), (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), (1,173,6,178)(2,172,7,177)(3,171,8,176)(4,180,9,175)(5,179,10,174)(11,181,16,186)(12,190,17,185)(13,189,18,184)(14,188,19,183)(15,187,20,182)(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([[(1,66,13,57),(2,67,14,58),(3,68,15,59),(4,69,16,60),(5,70,17,51),(6,61,18,52),(7,62,19,53),(8,63,20,54),(9,64,11,55),(10,65,12,56),(21,288,319,279),(22,289,320,280),(23,290,311,271),(24,281,312,272),(25,282,313,273),(26,283,314,274),(27,284,315,275),(28,285,316,276),(29,286,317,277),(30,287,318,278),(31,87,47,71),(32,88,48,72),(33,89,49,73),(34,90,50,74),(35,81,41,75),(36,82,42,76),(37,83,43,77),(38,84,44,78),(39,85,45,79),(40,86,46,80),(91,150,110,131),(92,141,101,132),(93,142,102,133),(94,143,103,134),(95,144,104,135),(96,145,105,136),(97,146,106,137),(98,147,107,138),(99,148,108,139),(100,149,109,140),(111,170,127,151),(112,161,128,152),(113,162,129,153),(114,163,130,154),(115,164,121,155),(116,165,122,156),(117,166,123,157),(118,167,124,158),(119,168,125,159),(120,169,126,160),(171,227,187,211),(172,228,188,212),(173,229,189,213),(174,230,190,214),(175,221,181,215),(176,222,182,216),(177,223,183,217),(178,224,184,218),(179,225,185,219),(180,226,186,220),(191,250,210,231),(192,241,201,232),(193,242,202,233),(194,243,203,234),(195,244,204,235),(196,245,205,236),(197,246,206,237),(198,247,207,238),(199,248,208,239),(200,249,209,240),(251,307,267,291),(252,308,268,292),(253,309,269,293),(254,310,270,294),(255,301,261,295),(256,302,262,296),(257,303,263,297),(258,304,264,298),(259,305,265,299),(260,306,266,300)], [(1,111,31,97),(2,112,32,98),(3,113,33,99),(4,114,34,100),(5,115,35,91),(6,116,36,92),(7,117,37,93),(8,118,38,94),(9,119,39,95),(10,120,40,96),(11,125,45,104),(12,126,46,105),(13,127,47,106),(14,128,48,107),(15,129,49,108),(16,130,50,109),(17,121,41,110),(18,122,42,101),(19,123,43,102),(20,124,44,103),(21,224,304,248),(22,225,305,249),(23,226,306,250),(24,227,307,241),(25,228,308,242),(26,229,309,243),(27,230,310,244),(28,221,301,245),(29,222,302,246),(30,223,303,247),(51,155,75,131),(52,156,76,132),(53,157,77,133),(54,158,78,134),(55,159,79,135),(56,160,80,136),(57,151,71,137),(58,152,72,138),(59,153,73,139),(60,154,74,140),(61,165,82,141),(62,166,83,142),(63,167,84,143),(64,168,85,144),(65,169,86,145),(66,170,87,146),(67,161,88,147),(68,162,89,148),(69,163,90,149),(70,164,81,150),(171,251,192,272),(172,252,193,273),(173,253,194,274),(174,254,195,275),(175,255,196,276),(176,256,197,277),(177,257,198,278),(178,258,199,279),(179,259,200,280),(180,260,191,271),(181,261,205,285),(182,262,206,286),(183,263,207,287),(184,264,208,288),(185,265,209,289),(186,266,210,290),(187,267,201,281),(188,268,202,282),(189,269,203,283),(190,270,204,284),(211,291,232,312),(212,292,233,313),(213,293,234,314),(214,294,235,315),(215,295,236,316),(216,296,237,317),(217,297,238,318),(218,298,239,319),(219,299,240,320),(220,300,231,311)], [(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)], [(1,173,6,178),(2,172,7,177),(3,171,8,176),(4,180,9,175),(5,179,10,174),(11,181,16,186),(12,190,17,185),(13,189,18,184),(14,188,19,183),(15,187,20,182),(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)]])
104 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AF | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D5 | C4○D4 | Dic5 | D10 | Dic10 | C4×D5 | D20 | C4○D20 |
| kernel | C4×C4⋊Dic5 | C10.10C42 | C2×C4×Dic5 | C2×C4⋊Dic5 | C2×C4×C20 | C4⋊Dic5 | C4×C20 | C2×C20 | C2×C20 | C2×C42 | C2×C10 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 16 | 8 | 2 | 2 | 2 | 4 | 8 | 6 | 8 | 16 | 8 | 16 |
Matrix representation of C4×C4⋊Dic5 ►in GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 25 |
| 32 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,32],[40,0,0,0,0,1,0,0,0,0,23,0,0,0,0,25],[32,0,0,0,0,1,0,0,0,0,0,40,0,0,1,0] >;
C4×C4⋊Dic5 in GAP, Magma, Sage, TeX
C_4\times C_4\rtimes {\rm Dic}_5 % in TeX
G:=Group("C4xC4:Dic5"); // GroupNames label
G:=SmallGroup(320,561);
// by ID
G=gap.SmallGroup(320,561);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,232,100,12550]);
// 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,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations