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G = C2×C10.C42order 320 = 26·5

Direct product of C2 and C10.C42

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C10.C42
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C22×C5⋊C8 — C2×C10.C42
 Lower central C5 — C10 — C2×C10.C42
 Upper central C1 — C23 — C22×C4

Generators and relations for C2×C10.C42
G = < a,b,c,d | a2=b10=d4=1, c4=b5, ab=ba, ac=ca, ad=da, cbc-1=b3, bd=db, dcd-1=b5c >

Subgroups: 378 in 146 conjugacy classes, 84 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C22, C22 [×6], C5, C8 [×8], C2×C4 [×2], C2×C4 [×12], C23, C10 [×3], C10 [×4], C42 [×4], C2×C8 [×12], C22×C4, C22×C4 [×2], Dic5 [×4], Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×6], C8⋊C4 [×4], C2×C42, C22×C8 [×2], C5⋊C8 [×8], C2×Dic5 [×2], C2×Dic5 [×6], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×C10, C2×C8⋊C4, C4×Dic5 [×4], C2×C5⋊C8 [×12], C22×Dic5 [×2], C22×C20, C10.C42 [×4], C2×C4×Dic5, C22×C5⋊C8 [×2], C2×C10.C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×4], C22×C4 [×3], F5, C8⋊C4 [×4], C2×C42, C2×M4(2) [×2], C2×F5 [×3], C2×C8⋊C4, C4.F5 [×2], C4×F5 [×2], C22.F5 [×2], C22×F5, C10.C42 [×4], C2×C4.F5, C2×C4×F5, C2×C22.F5, C2×C10.C42

Smallest permutation representation of C2×C10.C42
Regular action on 320 points
Generators in S320
(1 97)(2 98)(3 99)(4 100)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 111)(22 112)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(31 101)(32 102)(33 103)(34 104)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 131)(42 132)(43 133)(44 134)(45 135)(46 136)(47 137)(48 138)(49 139)(50 140)(51 121)(52 122)(53 123)(54 124)(55 125)(56 126)(57 127)(58 128)(59 129)(60 130)(61 151)(62 152)(63 153)(64 154)(65 155)(66 156)(67 157)(68 158)(69 159)(70 160)(71 141)(72 142)(73 143)(74 144)(75 145)(76 146)(77 147)(78 148)(79 149)(80 150)(161 251)(162 252)(163 253)(164 254)(165 255)(166 256)(167 257)(168 258)(169 259)(170 260)(171 241)(172 242)(173 243)(174 244)(175 245)(176 246)(177 247)(178 248)(179 249)(180 250)(181 271)(182 272)(183 273)(184 274)(185 275)(186 276)(187 277)(188 278)(189 279)(190 280)(191 261)(192 262)(193 263)(194 264)(195 265)(196 266)(197 267)(198 268)(199 269)(200 270)(201 291)(202 292)(203 293)(204 294)(205 295)(206 296)(207 297)(208 298)(209 299)(210 300)(211 281)(212 282)(213 283)(214 284)(215 285)(216 286)(217 287)(218 288)(219 289)(220 290)(221 311)(222 312)(223 313)(224 314)(225 315)(226 316)(227 317)(228 318)(229 319)(230 320)(231 301)(232 302)(233 303)(234 304)(235 305)(236 306)(237 307)(238 308)(239 309)(240 310)
(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)
(1 264 28 246 6 269 23 241)(2 261 27 249 7 266 22 244)(3 268 26 242 8 263 21 247)(4 265 25 245 9 270 30 250)(5 262 24 248 10 267 29 243)(11 272 34 258 16 277 39 253)(12 279 33 251 17 274 38 256)(13 276 32 254 18 271 37 259)(14 273 31 257 19 278 36 252)(15 280 40 260 20 275 35 255)(41 302 64 288 46 307 69 283)(42 309 63 281 47 304 68 286)(43 306 62 284 48 301 67 289)(44 303 61 287 49 308 66 282)(45 310 70 290 50 305 65 285)(51 312 74 298 56 317 79 293)(52 319 73 291 57 314 78 296)(53 316 72 294 58 311 77 299)(54 313 71 297 59 318 76 292)(55 320 80 300 60 315 75 295)(81 182 104 168 86 187 109 163)(82 189 103 161 87 184 108 166)(83 186 102 164 88 181 107 169)(84 183 101 167 89 188 106 162)(85 190 110 170 90 185 105 165)(91 192 114 178 96 197 119 173)(92 199 113 171 97 194 118 176)(93 196 112 174 98 191 117 179)(94 193 111 177 99 198 116 172)(95 200 120 180 100 195 115 175)(121 222 144 208 126 227 149 203)(122 229 143 201 127 224 148 206)(123 226 142 204 128 221 147 209)(124 223 141 207 129 228 146 202)(125 230 150 210 130 225 145 205)(131 232 154 218 136 237 159 213)(132 239 153 211 137 234 158 216)(133 236 152 214 138 231 157 219)(134 233 151 217 139 238 156 212)(135 240 160 220 140 235 155 215)
(1 57 17 47)(2 58 18 48)(3 59 19 49)(4 60 20 50)(5 51 11 41)(6 52 12 42)(7 53 13 43)(8 54 14 44)(9 55 15 45)(10 56 16 46)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)(81 131 91 121)(82 132 92 122)(83 133 93 123)(84 134 94 124)(85 135 95 125)(86 136 96 126)(87 137 97 127)(88 138 98 128)(89 139 99 129)(90 140 100 130)(101 151 111 141)(102 152 112 142)(103 153 113 143)(104 154 114 144)(105 155 115 145)(106 156 116 146)(107 157 117 147)(108 158 118 148)(109 159 119 149)(110 160 120 150)(161 216 171 206)(162 217 172 207)(163 218 173 208)(164 219 174 209)(165 220 175 210)(166 211 176 201)(167 212 177 202)(168 213 178 203)(169 214 179 204)(170 215 180 205)(181 236 191 226)(182 237 192 227)(183 238 193 228)(184 239 194 229)(185 240 195 230)(186 231 196 221)(187 232 197 222)(188 233 198 223)(189 234 199 224)(190 235 200 225)(241 296 251 286)(242 297 252 287)(243 298 253 288)(244 299 254 289)(245 300 255 290)(246 291 256 281)(247 292 257 282)(248 293 258 283)(249 294 259 284)(250 295 260 285)(261 316 271 306)(262 317 272 307)(263 318 273 308)(264 319 274 309)(265 320 275 310)(266 311 276 301)(267 312 277 302)(268 313 278 303)(269 314 279 304)(270 315 280 305)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E ··· 4L 4M 4N 4O 4P 5 8A ··· 8P 10A ··· 10G 20A ··· 20H order 1 2 ··· 2 4 4 4 4 4 ··· 4 4 4 4 4 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 5 ··· 5 10 10 10 10 4 10 ··· 10 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 4 4 4 4 4 4 type + + + + + + + - image C1 C2 C2 C2 C4 C4 C4 C4 M4(2) F5 C2×F5 C2×F5 C4.F5 C4×F5 C22.F5 kernel C2×C10.C42 C10.C42 C2×C4×Dic5 C22×C5⋊C8 C4×Dic5 C2×C5⋊C8 C22×Dic5 C22×C20 C2×C10 C22×C4 C2×C4 C23 C22 C22 C22 # reps 1 4 1 2 4 16 2 2 8 1 2 1 4 4 4

Matrix representation of C2×C10.C42 in GL8(𝔽41)

 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 31 0 0 0 0 0 0 0 0 25 0 0 0 0 0 0 0 0 23
,
 14 0 0 0 0 0 0 0 36 27 0 0 0 0 0 0 0 0 4 29 0 0 0 0 0 0 4 37 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0
,
 9 34 0 0 0 0 0 0 29 32 0 0 0 0 0 0 0 0 1 39 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 32

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,23],[14,36,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,29,37,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0],[9,29,0,0,0,0,0,0,34,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,39,40,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32] >;

C2×C10.C42 in GAP, Magma, Sage, TeX

C_2\times C_{10}.C_4^2
% in TeX

G:=Group("C2xC10.C4^2");
// GroupNames label

G:=SmallGroup(320,1087);
// by ID

G=gap.SmallGroup(320,1087);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,100,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=d^4=1,c^4=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^3,b*d=d*b,d*c*d^-1=b^5*c>;
// generators/relations

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