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## G = C42×Dic5order 320 = 26·5

### Direct product of C42 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C42×Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C4×Dic5 — C42×Dic5
 Lower central C5 — C42×Dic5
 Upper central C1 — C2×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, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C42, C42, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×C42, C2×C42, C2×Dic5, C2×C20, C22×C10, C43, C4×Dic5, C4×C20, C22×Dic5, C22×C20, C2×C4×Dic5, C2×C4×C20, C42×Dic5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C42, C22×C4, Dic5, D10, C2×C42, C4×D5, C2×Dic5, C22×D5, C43, C4×Dic5, C2×C4×D5, C22×Dic5, D5×C42, C2×C4×Dic5, C42×Dic5

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

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

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

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

128 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4X 4Y ··· 4BD 5A 5B 10A ··· 10N 20A ··· 20AV order 1 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 1 ··· 1 5 ··· 5 2 2 2 ··· 2 2 ··· 2

128 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 D5 Dic5 D10 C4×D5 kernel C42×Dic5 C2×C4×Dic5 C2×C4×C20 C4×Dic5 C4×C20 C2×C42 C42 C22×C4 C2×C4 # reps 1 6 1 48 8 2 8 6 48

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

 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 40 0 0 8 34
,
 40 0 0 0 0 32 0 0 0 0 18 37 0 0 30 23
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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