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## G = C2×C10.10C42order 320 = 26·5

### Direct product of C2 and C10.10C42

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C10.10C42
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C23×Dic5 — C2×C10.10C42
 Lower central C5 — C10 — C2×C10.10C42
 Upper central C1 — C24 — C23×C4

Generators and relations for C2×C10.10C42
G = < a,b,c,d | a2=b10=c4=d4=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b5c >

Subgroups: 878 in 330 conjugacy classes, 183 normal (21 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C23, C10, C10, C22×C4, C22×C4, C24, Dic5, C20, C2×C10, C2×C10, C2.C42, C23×C4, C23×C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C22×C10, C2×C2.C42, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C23×C10, C10.10C42, C23×Dic5, C23×C20, C2×C10.10C42
Quotients:

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

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

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

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

104 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4H 4I ··· 4X 5A 5B 10A ··· 10AD 20A ··· 20AF order 1 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 ··· 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - + + - + image C1 C2 C2 C2 C4 C4 D4 Q8 D5 Dic5 D10 D10 Dic10 C4×D5 D20 C5⋊D4 kernel C2×C10.10C42 C10.10C42 C23×Dic5 C23×C20 C22×Dic5 C22×C20 C22×C10 C22×C10 C23×C4 C22×C4 C22×C4 C24 C23 C23 C23 C23 # reps 1 4 2 1 16 8 6 2 2 8 4 2 8 16 8 16

Matrix representation of C2×C10.10C42 in GL5(𝔽41)

 40 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 7 0 0 0 34 34
,
 40 0 0 0 0 0 40 0 0 0 0 0 9 0 0 0 0 0 39 21 0 0 0 35 2
,
 40 0 0 0 0 0 9 0 0 0 0 0 1 0 0 0 0 0 24 1 0 0 0 40 17

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,34,0,0,0,7,34],[40,0,0,0,0,0,40,0,0,0,0,0,9,0,0,0,0,0,39,35,0,0,0,21,2],[40,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,24,40,0,0,0,1,17] >;

C2×C10.10C42 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC10.10C4^2");
// GroupNames label

G:=SmallGroup(320,835);
// by ID

G=gap.SmallGroup(320,835);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,100,12550]);
// Polycyclic

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

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