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G = C42×Dic5order 320 = 26·5

Direct product of C42 and Dic5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C42×Dic5, C52C43, C206C42, (C4×C20)⋊24C4, C2.2(D5×C42), (C2×C42).23D5, C10.36(C2×C42), (C22×C4).456D10, C23.263(C22×D5), (C22×C10).305C23, (C22×C20).555C22, C22.16(C22×Dic5), (C22×Dic5).286C22, (C2×C4×C20).27C2, C2.2(C2×C4×Dic5), C22.49(C2×C4×D5), (C2×C4).179(C4×D5), (C2×C4×Dic5).53C2, (C2×C20).423(C2×C4), (C2×C4).99(C2×Dic5), (C2×C10).196(C22×C4), (C2×Dic5).205(C2×C4), SmallGroup(320,557)

Series: Derived Chief Lower central Upper central

C1C5 — C42×Dic5
C1C5C10C2×C10C22×C10C22×Dic5C2×C4×Dic5 — C42×Dic5
C5 — C42×Dic5
C1C2×C42

Generators and relations for C42×Dic5
 G = < a,b,c,d | a4=b4=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 558 in 258 conjugacy classes, 183 normal (9 characteristic)
C1, C2 [×7], C4 [×12], C4 [×16], C22, C22 [×6], C5, C2×C4 [×18], C2×C4 [×24], C23, C10 [×7], C42 [×4], C42 [×24], C22×C4 [×3], C22×C4 [×4], Dic5 [×16], C20 [×12], C2×C10, C2×C10 [×6], C2×C42, C2×C42 [×6], C2×Dic5 [×24], C2×C20 [×18], C22×C10, C43, C4×Dic5 [×24], C4×C20 [×4], C22×Dic5 [×4], C22×C20 [×3], C2×C4×Dic5 [×6], C2×C4×C20, C42×Dic5
Quotients: C1, C2 [×7], C4 [×28], C22 [×7], C2×C4 [×42], C23, D5, C42 [×28], C22×C4 [×7], Dic5 [×4], D10 [×3], C2×C42 [×7], C4×D5 [×12], C2×Dic5 [×6], C22×D5, C43, C4×Dic5 [×12], C2×C4×D5 [×6], C22×Dic5, D5×C42 [×4], C2×C4×Dic5 [×3], C42×Dic5

Smallest permutation representation of C42×Dic5
Regular action on 320 points
Generators in S320
(1 138 58 106)(2 139 59 107)(3 140 60 108)(4 131 51 109)(5 132 52 110)(6 133 53 101)(7 134 54 102)(8 135 55 103)(9 136 56 104)(10 137 57 105)(11 115 25 81)(12 116 26 82)(13 117 27 83)(14 118 28 84)(15 119 29 85)(16 120 30 86)(17 111 21 87)(18 112 22 88)(19 113 23 89)(20 114 24 90)(31 252 309 229)(32 253 310 230)(33 254 301 221)(34 255 302 222)(35 256 303 223)(36 257 304 224)(37 258 305 225)(38 259 306 226)(39 260 307 227)(40 251 308 228)(41 155 75 121)(42 156 76 122)(43 157 77 123)(44 158 78 124)(45 159 79 125)(46 160 80 126)(47 151 71 127)(48 152 72 128)(49 153 73 129)(50 154 74 130)(61 180 93 141)(62 171 94 142)(63 172 95 143)(64 173 96 144)(65 174 97 145)(66 175 98 146)(67 176 99 147)(68 177 100 148)(69 178 91 149)(70 179 92 150)(161 272 192 241)(162 273 193 242)(163 274 194 243)(164 275 195 244)(165 276 196 245)(166 277 197 246)(167 278 198 247)(168 279 199 248)(169 280 200 249)(170 271 191 250)(181 294 214 261)(182 295 215 262)(183 296 216 263)(184 297 217 264)(185 298 218 265)(186 299 219 266)(187 300 220 267)(188 291 211 268)(189 292 212 269)(190 293 213 270)(201 312 232 281)(202 313 233 282)(203 314 234 283)(204 315 235 284)(205 316 236 285)(206 317 237 286)(207 318 238 287)(208 319 239 288)(209 320 240 289)(210 311 231 290)
(1 61 15 50)(2 62 16 41)(3 63 17 42)(4 64 18 43)(5 65 19 44)(6 66 20 45)(7 67 11 46)(8 68 12 47)(9 69 13 48)(10 70 14 49)(21 76 60 95)(22 77 51 96)(23 78 52 97)(24 79 53 98)(25 80 54 99)(26 71 55 100)(27 72 56 91)(28 73 57 92)(29 74 58 93)(30 75 59 94)(31 271 316 297)(32 272 317 298)(33 273 318 299)(34 274 319 300)(35 275 320 291)(36 276 311 292)(37 277 312 293)(38 278 313 294)(39 279 314 295)(40 280 315 296)(81 126 102 147)(82 127 103 148)(83 128 104 149)(84 129 105 150)(85 130 106 141)(86 121 107 142)(87 122 108 143)(88 123 109 144)(89 124 110 145)(90 125 101 146)(111 156 140 172)(112 157 131 173)(113 158 132 174)(114 159 133 175)(115 160 134 176)(116 151 135 177)(117 152 136 178)(118 153 137 179)(119 154 138 180)(120 155 139 171)(161 206 185 230)(162 207 186 221)(163 208 187 222)(164 209 188 223)(165 210 189 224)(166 201 190 225)(167 202 181 226)(168 203 182 227)(169 204 183 228)(170 205 184 229)(191 236 217 252)(192 237 218 253)(193 238 219 254)(194 239 220 255)(195 240 211 256)(196 231 212 257)(197 232 213 258)(198 233 214 259)(199 234 215 260)(200 235 216 251)(241 286 265 310)(242 287 266 301)(243 288 267 302)(244 289 268 303)(245 290 269 304)(246 281 270 305)(247 282 261 306)(248 283 262 307)(249 284 263 308)(250 285 264 309)
(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)
(1 236 6 231)(2 235 7 240)(3 234 8 239)(4 233 9 238)(5 232 10 237)(11 256 16 251)(12 255 17 260)(13 254 18 259)(14 253 19 258)(15 252 20 257)(21 227 26 222)(22 226 27 221)(23 225 28 230)(24 224 29 229)(25 223 30 228)(31 90 36 85)(32 89 37 84)(33 88 38 83)(34 87 39 82)(35 86 40 81)(41 200 46 195)(42 199 47 194)(43 198 48 193)(44 197 49 192)(45 196 50 191)(51 202 56 207)(52 201 57 206)(53 210 58 205)(54 209 59 204)(55 208 60 203)(61 217 66 212)(62 216 67 211)(63 215 68 220)(64 214 69 219)(65 213 70 218)(71 163 76 168)(72 162 77 167)(73 161 78 166)(74 170 79 165)(75 169 80 164)(91 186 96 181)(92 185 97 190)(93 184 98 189)(94 183 99 188)(95 182 100 187)(101 311 106 316)(102 320 107 315)(103 319 108 314)(104 318 109 313)(105 317 110 312)(111 307 116 302)(112 306 117 301)(113 305 118 310)(114 304 119 309)(115 303 120 308)(121 280 126 275)(122 279 127 274)(123 278 128 273)(124 277 129 272)(125 276 130 271)(131 282 136 287)(132 281 137 286)(133 290 138 285)(134 289 139 284)(135 288 140 283)(141 297 146 292)(142 296 147 291)(143 295 148 300)(144 294 149 299)(145 293 150 298)(151 243 156 248)(152 242 157 247)(153 241 158 246)(154 250 159 245)(155 249 160 244)(171 263 176 268)(172 262 177 267)(173 261 178 266)(174 270 179 265)(175 269 180 264)

G:=sub<Sym(320)| (1,138,58,106)(2,139,59,107)(3,140,60,108)(4,131,51,109)(5,132,52,110)(6,133,53,101)(7,134,54,102)(8,135,55,103)(9,136,56,104)(10,137,57,105)(11,115,25,81)(12,116,26,82)(13,117,27,83)(14,118,28,84)(15,119,29,85)(16,120,30,86)(17,111,21,87)(18,112,22,88)(19,113,23,89)(20,114,24,90)(31,252,309,229)(32,253,310,230)(33,254,301,221)(34,255,302,222)(35,256,303,223)(36,257,304,224)(37,258,305,225)(38,259,306,226)(39,260,307,227)(40,251,308,228)(41,155,75,121)(42,156,76,122)(43,157,77,123)(44,158,78,124)(45,159,79,125)(46,160,80,126)(47,151,71,127)(48,152,72,128)(49,153,73,129)(50,154,74,130)(61,180,93,141)(62,171,94,142)(63,172,95,143)(64,173,96,144)(65,174,97,145)(66,175,98,146)(67,176,99,147)(68,177,100,148)(69,178,91,149)(70,179,92,150)(161,272,192,241)(162,273,193,242)(163,274,194,243)(164,275,195,244)(165,276,196,245)(166,277,197,246)(167,278,198,247)(168,279,199,248)(169,280,200,249)(170,271,191,250)(181,294,214,261)(182,295,215,262)(183,296,216,263)(184,297,217,264)(185,298,218,265)(186,299,219,266)(187,300,220,267)(188,291,211,268)(189,292,212,269)(190,293,213,270)(201,312,232,281)(202,313,233,282)(203,314,234,283)(204,315,235,284)(205,316,236,285)(206,317,237,286)(207,318,238,287)(208,319,239,288)(209,320,240,289)(210,311,231,290), (1,61,15,50)(2,62,16,41)(3,63,17,42)(4,64,18,43)(5,65,19,44)(6,66,20,45)(7,67,11,46)(8,68,12,47)(9,69,13,48)(10,70,14,49)(21,76,60,95)(22,77,51,96)(23,78,52,97)(24,79,53,98)(25,80,54,99)(26,71,55,100)(27,72,56,91)(28,73,57,92)(29,74,58,93)(30,75,59,94)(31,271,316,297)(32,272,317,298)(33,273,318,299)(34,274,319,300)(35,275,320,291)(36,276,311,292)(37,277,312,293)(38,278,313,294)(39,279,314,295)(40,280,315,296)(81,126,102,147)(82,127,103,148)(83,128,104,149)(84,129,105,150)(85,130,106,141)(86,121,107,142)(87,122,108,143)(88,123,109,144)(89,124,110,145)(90,125,101,146)(111,156,140,172)(112,157,131,173)(113,158,132,174)(114,159,133,175)(115,160,134,176)(116,151,135,177)(117,152,136,178)(118,153,137,179)(119,154,138,180)(120,155,139,171)(161,206,185,230)(162,207,186,221)(163,208,187,222)(164,209,188,223)(165,210,189,224)(166,201,190,225)(167,202,181,226)(168,203,182,227)(169,204,183,228)(170,205,184,229)(191,236,217,252)(192,237,218,253)(193,238,219,254)(194,239,220,255)(195,240,211,256)(196,231,212,257)(197,232,213,258)(198,233,214,259)(199,234,215,260)(200,235,216,251)(241,286,265,310)(242,287,266,301)(243,288,267,302)(244,289,268,303)(245,290,269,304)(246,281,270,305)(247,282,261,306)(248,283,262,307)(249,284,263,308)(250,285,264,309), (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), (1,236,6,231)(2,235,7,240)(3,234,8,239)(4,233,9,238)(5,232,10,237)(11,256,16,251)(12,255,17,260)(13,254,18,259)(14,253,19,258)(15,252,20,257)(21,227,26,222)(22,226,27,221)(23,225,28,230)(24,224,29,229)(25,223,30,228)(31,90,36,85)(32,89,37,84)(33,88,38,83)(34,87,39,82)(35,86,40,81)(41,200,46,195)(42,199,47,194)(43,198,48,193)(44,197,49,192)(45,196,50,191)(51,202,56,207)(52,201,57,206)(53,210,58,205)(54,209,59,204)(55,208,60,203)(61,217,66,212)(62,216,67,211)(63,215,68,220)(64,214,69,219)(65,213,70,218)(71,163,76,168)(72,162,77,167)(73,161,78,166)(74,170,79,165)(75,169,80,164)(91,186,96,181)(92,185,97,190)(93,184,98,189)(94,183,99,188)(95,182,100,187)(101,311,106,316)(102,320,107,315)(103,319,108,314)(104,318,109,313)(105,317,110,312)(111,307,116,302)(112,306,117,301)(113,305,118,310)(114,304,119,309)(115,303,120,308)(121,280,126,275)(122,279,127,274)(123,278,128,273)(124,277,129,272)(125,276,130,271)(131,282,136,287)(132,281,137,286)(133,290,138,285)(134,289,139,284)(135,288,140,283)(141,297,146,292)(142,296,147,291)(143,295,148,300)(144,294,149,299)(145,293,150,298)(151,243,156,248)(152,242,157,247)(153,241,158,246)(154,250,159,245)(155,249,160,244)(171,263,176,268)(172,262,177,267)(173,261,178,266)(174,270,179,265)(175,269,180,264)>;

G:=Group( (1,138,58,106)(2,139,59,107)(3,140,60,108)(4,131,51,109)(5,132,52,110)(6,133,53,101)(7,134,54,102)(8,135,55,103)(9,136,56,104)(10,137,57,105)(11,115,25,81)(12,116,26,82)(13,117,27,83)(14,118,28,84)(15,119,29,85)(16,120,30,86)(17,111,21,87)(18,112,22,88)(19,113,23,89)(20,114,24,90)(31,252,309,229)(32,253,310,230)(33,254,301,221)(34,255,302,222)(35,256,303,223)(36,257,304,224)(37,258,305,225)(38,259,306,226)(39,260,307,227)(40,251,308,228)(41,155,75,121)(42,156,76,122)(43,157,77,123)(44,158,78,124)(45,159,79,125)(46,160,80,126)(47,151,71,127)(48,152,72,128)(49,153,73,129)(50,154,74,130)(61,180,93,141)(62,171,94,142)(63,172,95,143)(64,173,96,144)(65,174,97,145)(66,175,98,146)(67,176,99,147)(68,177,100,148)(69,178,91,149)(70,179,92,150)(161,272,192,241)(162,273,193,242)(163,274,194,243)(164,275,195,244)(165,276,196,245)(166,277,197,246)(167,278,198,247)(168,279,199,248)(169,280,200,249)(170,271,191,250)(181,294,214,261)(182,295,215,262)(183,296,216,263)(184,297,217,264)(185,298,218,265)(186,299,219,266)(187,300,220,267)(188,291,211,268)(189,292,212,269)(190,293,213,270)(201,312,232,281)(202,313,233,282)(203,314,234,283)(204,315,235,284)(205,316,236,285)(206,317,237,286)(207,318,238,287)(208,319,239,288)(209,320,240,289)(210,311,231,290), (1,61,15,50)(2,62,16,41)(3,63,17,42)(4,64,18,43)(5,65,19,44)(6,66,20,45)(7,67,11,46)(8,68,12,47)(9,69,13,48)(10,70,14,49)(21,76,60,95)(22,77,51,96)(23,78,52,97)(24,79,53,98)(25,80,54,99)(26,71,55,100)(27,72,56,91)(28,73,57,92)(29,74,58,93)(30,75,59,94)(31,271,316,297)(32,272,317,298)(33,273,318,299)(34,274,319,300)(35,275,320,291)(36,276,311,292)(37,277,312,293)(38,278,313,294)(39,279,314,295)(40,280,315,296)(81,126,102,147)(82,127,103,148)(83,128,104,149)(84,129,105,150)(85,130,106,141)(86,121,107,142)(87,122,108,143)(88,123,109,144)(89,124,110,145)(90,125,101,146)(111,156,140,172)(112,157,131,173)(113,158,132,174)(114,159,133,175)(115,160,134,176)(116,151,135,177)(117,152,136,178)(118,153,137,179)(119,154,138,180)(120,155,139,171)(161,206,185,230)(162,207,186,221)(163,208,187,222)(164,209,188,223)(165,210,189,224)(166,201,190,225)(167,202,181,226)(168,203,182,227)(169,204,183,228)(170,205,184,229)(191,236,217,252)(192,237,218,253)(193,238,219,254)(194,239,220,255)(195,240,211,256)(196,231,212,257)(197,232,213,258)(198,233,214,259)(199,234,215,260)(200,235,216,251)(241,286,265,310)(242,287,266,301)(243,288,267,302)(244,289,268,303)(245,290,269,304)(246,281,270,305)(247,282,261,306)(248,283,262,307)(249,284,263,308)(250,285,264,309), (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), (1,236,6,231)(2,235,7,240)(3,234,8,239)(4,233,9,238)(5,232,10,237)(11,256,16,251)(12,255,17,260)(13,254,18,259)(14,253,19,258)(15,252,20,257)(21,227,26,222)(22,226,27,221)(23,225,28,230)(24,224,29,229)(25,223,30,228)(31,90,36,85)(32,89,37,84)(33,88,38,83)(34,87,39,82)(35,86,40,81)(41,200,46,195)(42,199,47,194)(43,198,48,193)(44,197,49,192)(45,196,50,191)(51,202,56,207)(52,201,57,206)(53,210,58,205)(54,209,59,204)(55,208,60,203)(61,217,66,212)(62,216,67,211)(63,215,68,220)(64,214,69,219)(65,213,70,218)(71,163,76,168)(72,162,77,167)(73,161,78,166)(74,170,79,165)(75,169,80,164)(91,186,96,181)(92,185,97,190)(93,184,98,189)(94,183,99,188)(95,182,100,187)(101,311,106,316)(102,320,107,315)(103,319,108,314)(104,318,109,313)(105,317,110,312)(111,307,116,302)(112,306,117,301)(113,305,118,310)(114,304,119,309)(115,303,120,308)(121,280,126,275)(122,279,127,274)(123,278,128,273)(124,277,129,272)(125,276,130,271)(131,282,136,287)(132,281,137,286)(133,290,138,285)(134,289,139,284)(135,288,140,283)(141,297,146,292)(142,296,147,291)(143,295,148,300)(144,294,149,299)(145,293,150,298)(151,243,156,248)(152,242,157,247)(153,241,158,246)(154,250,159,245)(155,249,160,244)(171,263,176,268)(172,262,177,267)(173,261,178,266)(174,270,179,265)(175,269,180,264) );

G=PermutationGroup([(1,138,58,106),(2,139,59,107),(3,140,60,108),(4,131,51,109),(5,132,52,110),(6,133,53,101),(7,134,54,102),(8,135,55,103),(9,136,56,104),(10,137,57,105),(11,115,25,81),(12,116,26,82),(13,117,27,83),(14,118,28,84),(15,119,29,85),(16,120,30,86),(17,111,21,87),(18,112,22,88),(19,113,23,89),(20,114,24,90),(31,252,309,229),(32,253,310,230),(33,254,301,221),(34,255,302,222),(35,256,303,223),(36,257,304,224),(37,258,305,225),(38,259,306,226),(39,260,307,227),(40,251,308,228),(41,155,75,121),(42,156,76,122),(43,157,77,123),(44,158,78,124),(45,159,79,125),(46,160,80,126),(47,151,71,127),(48,152,72,128),(49,153,73,129),(50,154,74,130),(61,180,93,141),(62,171,94,142),(63,172,95,143),(64,173,96,144),(65,174,97,145),(66,175,98,146),(67,176,99,147),(68,177,100,148),(69,178,91,149),(70,179,92,150),(161,272,192,241),(162,273,193,242),(163,274,194,243),(164,275,195,244),(165,276,196,245),(166,277,197,246),(167,278,198,247),(168,279,199,248),(169,280,200,249),(170,271,191,250),(181,294,214,261),(182,295,215,262),(183,296,216,263),(184,297,217,264),(185,298,218,265),(186,299,219,266),(187,300,220,267),(188,291,211,268),(189,292,212,269),(190,293,213,270),(201,312,232,281),(202,313,233,282),(203,314,234,283),(204,315,235,284),(205,316,236,285),(206,317,237,286),(207,318,238,287),(208,319,239,288),(209,320,240,289),(210,311,231,290)], [(1,61,15,50),(2,62,16,41),(3,63,17,42),(4,64,18,43),(5,65,19,44),(6,66,20,45),(7,67,11,46),(8,68,12,47),(9,69,13,48),(10,70,14,49),(21,76,60,95),(22,77,51,96),(23,78,52,97),(24,79,53,98),(25,80,54,99),(26,71,55,100),(27,72,56,91),(28,73,57,92),(29,74,58,93),(30,75,59,94),(31,271,316,297),(32,272,317,298),(33,273,318,299),(34,274,319,300),(35,275,320,291),(36,276,311,292),(37,277,312,293),(38,278,313,294),(39,279,314,295),(40,280,315,296),(81,126,102,147),(82,127,103,148),(83,128,104,149),(84,129,105,150),(85,130,106,141),(86,121,107,142),(87,122,108,143),(88,123,109,144),(89,124,110,145),(90,125,101,146),(111,156,140,172),(112,157,131,173),(113,158,132,174),(114,159,133,175),(115,160,134,176),(116,151,135,177),(117,152,136,178),(118,153,137,179),(119,154,138,180),(120,155,139,171),(161,206,185,230),(162,207,186,221),(163,208,187,222),(164,209,188,223),(165,210,189,224),(166,201,190,225),(167,202,181,226),(168,203,182,227),(169,204,183,228),(170,205,184,229),(191,236,217,252),(192,237,218,253),(193,238,219,254),(194,239,220,255),(195,240,211,256),(196,231,212,257),(197,232,213,258),(198,233,214,259),(199,234,215,260),(200,235,216,251),(241,286,265,310),(242,287,266,301),(243,288,267,302),(244,289,268,303),(245,290,269,304),(246,281,270,305),(247,282,261,306),(248,283,262,307),(249,284,263,308),(250,285,264,309)], [(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)], [(1,236,6,231),(2,235,7,240),(3,234,8,239),(4,233,9,238),(5,232,10,237),(11,256,16,251),(12,255,17,260),(13,254,18,259),(14,253,19,258),(15,252,20,257),(21,227,26,222),(22,226,27,221),(23,225,28,230),(24,224,29,229),(25,223,30,228),(31,90,36,85),(32,89,37,84),(33,88,38,83),(34,87,39,82),(35,86,40,81),(41,200,46,195),(42,199,47,194),(43,198,48,193),(44,197,49,192),(45,196,50,191),(51,202,56,207),(52,201,57,206),(53,210,58,205),(54,209,59,204),(55,208,60,203),(61,217,66,212),(62,216,67,211),(63,215,68,220),(64,214,69,219),(65,213,70,218),(71,163,76,168),(72,162,77,167),(73,161,78,166),(74,170,79,165),(75,169,80,164),(91,186,96,181),(92,185,97,190),(93,184,98,189),(94,183,99,188),(95,182,100,187),(101,311,106,316),(102,320,107,315),(103,319,108,314),(104,318,109,313),(105,317,110,312),(111,307,116,302),(112,306,117,301),(113,305,118,310),(114,304,119,309),(115,303,120,308),(121,280,126,275),(122,279,127,274),(123,278,128,273),(124,277,129,272),(125,276,130,271),(131,282,136,287),(132,281,137,286),(133,290,138,285),(134,289,139,284),(135,288,140,283),(141,297,146,292),(142,296,147,291),(143,295,148,300),(144,294,149,299),(145,293,150,298),(151,243,156,248),(152,242,157,247),(153,241,158,246),(154,250,159,245),(155,249,160,244),(171,263,176,268),(172,262,177,267),(173,261,178,266),(174,270,179,265),(175,269,180,264)])

128 conjugacy classes

class 1 2A···2G4A···4X4Y···4BD5A5B10A···10N20A···20AV
order12···24···44···45510···1020···20
size11···11···15···5222···22···2

128 irreducible representations

dim111112222
type++++-+
imageC1C2C2C4C4D5Dic5D10C4×D5
kernelC42×Dic5C2×C4×Dic5C2×C4×C20C4×Dic5C4×C20C2×C42C42C22×C4C2×C4
# reps16148828648

Matrix representation of C42×Dic5 in GL4(𝔽41) generated by

9000
0900
00320
00032
,
9000
03200
00400
00040
,
1000
04000
00140
00834
,
40000
03200
001837
003023
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,32,0,0,0,0,32],[9,0,0,0,0,32,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,1,8,0,0,40,34],[40,0,0,0,0,32,0,0,0,0,18,30,0,0,37,23] >;

C42×Dic5 in GAP, Magma, Sage, TeX

C_4^2\times {\rm Dic}_5
% in TeX

G:=Group("C4^2xDic5");
// GroupNames label

G:=SmallGroup(320,557);
// by ID

G=gap.SmallGroup(320,557);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,100,12550]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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