direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C2.C42, (C22×C4)⋊5C20, (C22×C20)⋊22C4, (C23×C4).1C10, C22.8(C4×C20), (C23×C20).4C2, C23.53(C5×D4), C23.10(C5×Q8), C22.7(Q8×C10), C10.49(C2×C42), C24.35(C2×C10), (C2×C10).52C42, C23.36(C2×C20), C22.26(D4×C10), (C22×C10).29Q8, (C22×C10).214D4, C23.48(C22×C10), C22.12(C22×C20), (C22×C20).486C22, (C22×C10).439C23, (C23×C10).118C22, C2.1(C2×C4×C20), (C2×C4)⋊9(C2×C20), C2.1(C10×C4⋊C4), (C2×C20)⋊48(C2×C4), C10.80(C2×C4⋊C4), C2.1(C10×C22⋊C4), C22.18(C5×C4⋊C4), (C2×C10).99(C2×Q8), (C2×C10).89(C4⋊C4), (C2×C10).593(C2×D4), C10.130(C2×C22⋊C4), (C22×C4).80(C2×C10), C22.31(C5×C22⋊C4), (C22×C10).213(C2×C4), (C2×C10).312(C22×C4), (C2×C10).197(C22⋊C4), SmallGroup(320,876)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C2.C42
G = < a,b,c,d | a10=b2=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >
Subgroups: 450 in 330 conjugacy classes, 210 normal (12 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C23, C10, C10, C22×C4, C22×C4, C24, C20, C2×C10, C2×C10, C2.C42, C23×C4, C2×C20, C2×C20, C22×C10, C22×C10, C2×C2.C42, C22×C20, C22×C20, C23×C10, C5×C2.C42, C23×C20, C10×C2.C42
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C23, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C20, C2×C10, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C20, C5×D4, C5×Q8, C22×C10, C2×C2.C42, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, C5×C2.C42, C2×C4×C20, C10×C22⋊C4, C10×C4⋊C4, C10×C2.C42
►Regular action on 320 points
Generators in S320
(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)
(1 112)(2 113)(3 114)(4 115)(5 116)(6 117)(7 118)(8 119)(9 120)(10 111)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 21)(18 22)(19 23)(20 24)(31 66)(32 67)(33 68)(34 69)(35 70)(36 61)(37 62)(38 63)(39 64)(40 65)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 311)(52 312)(53 313)(54 314)(55 315)(56 316)(57 317)(58 318)(59 319)(60 320)(81 107)(82 108)(83 109)(84 110)(85 101)(86 102)(87 103)(88 104)(89 105)(90 106)(91 126)(92 127)(93 128)(94 129)(95 130)(96 121)(97 122)(98 123)(99 124)(100 125)(131 145)(132 146)(133 147)(134 148)(135 149)(136 150)(137 141)(138 142)(139 143)(140 144)(151 190)(152 181)(153 182)(154 183)(155 184)(156 185)(157 186)(158 187)(159 188)(160 189)(161 191)(162 192)(163 193)(164 194)(165 195)(166 196)(167 197)(168 198)(169 199)(170 200)(171 206)(172 207)(173 208)(174 209)(175 210)(176 201)(177 202)(178 203)(179 204)(180 205)(211 225)(212 226)(213 227)(214 228)(215 229)(216 230)(217 221)(218 222)(219 223)(220 224)(231 270)(232 261)(233 262)(234 263)(235 264)(236 265)(237 266)(238 267)(239 268)(240 269)(241 271)(242 272)(243 273)(244 274)(245 275)(246 276)(247 277)(248 278)(249 279)(250 280)(251 286)(252 287)(253 288)(254 289)(255 290)(256 281)(257 282)(258 283)(259 284)(260 285)(291 305)(292 306)(293 307)(294 308)(295 309)(296 310)(297 301)(298 302)(299 303)(300 304)
(1 266 129 299)(2 267 130 300)(3 268 121 291)(4 269 122 292)(5 270 123 293)(6 261 124 294)(7 262 125 295)(8 263 126 296)(9 264 127 297)(10 265 128 298)(11 163 58 210)(12 164 59 201)(13 165 60 202)(14 166 51 203)(15 167 52 204)(16 168 53 205)(17 169 54 206)(18 170 55 207)(19 161 56 208)(20 162 57 209)(21 199 314 171)(22 200 315 172)(23 191 316 173)(24 192 317 174)(25 193 318 175)(26 194 319 176)(27 195 320 177)(28 196 311 178)(29 197 312 179)(30 198 313 180)(31 222 78 156)(32 223 79 157)(33 224 80 158)(34 225 71 159)(35 226 72 160)(36 227 73 151)(37 228 74 152)(38 229 75 153)(39 230 76 154)(40 221 77 155)(41 188 69 211)(42 189 70 212)(43 190 61 213)(44 181 62 214)(45 182 63 215)(46 183 64 216)(47 184 65 217)(48 185 66 218)(49 186 67 219)(50 187 68 220)(81 246 133 283)(82 247 134 284)(83 248 135 285)(84 249 136 286)(85 250 137 287)(86 241 138 288)(87 242 139 289)(88 243 140 290)(89 244 131 281)(90 245 132 282)(91 310 119 234)(92 301 120 235)(93 302 111 236)(94 303 112 237)(95 304 113 238)(96 305 114 239)(97 306 115 240)(98 307 116 231)(99 308 117 232)(100 309 118 233)(101 280 141 252)(102 271 142 253)(103 272 143 254)(104 273 144 255)(105 274 145 256)(106 275 146 257)(107 276 147 258)(108 277 148 259)(109 278 149 260)(110 279 150 251)
(1 179 108 219)(2 180 109 220)(3 171 110 211)(4 172 101 212)(5 173 102 213)(6 174 103 214)(7 175 104 215)(8 176 105 216)(9 177 106 217)(10 178 107 218)(11 273 75 262)(12 274 76 263)(13 275 77 264)(14 276 78 265)(15 277 79 266)(16 278 80 267)(17 279 71 268)(18 280 72 269)(19 271 73 270)(20 272 74 261)(21 249 41 239)(22 250 42 240)(23 241 43 231)(24 242 44 232)(25 243 45 233)(26 244 46 234)(27 245 47 235)(28 246 48 236)(29 247 49 237)(30 248 50 238)(31 298 51 258)(32 299 52 259)(33 300 53 260)(34 291 54 251)(35 292 55 252)(36 293 56 253)(37 294 57 254)(38 295 58 255)(39 296 59 256)(40 297 60 257)(61 307 316 288)(62 308 317 289)(63 309 318 290)(64 310 319 281)(65 301 320 282)(66 302 311 283)(67 303 312 284)(68 304 313 285)(69 305 314 286)(70 306 315 287)(81 222 111 203)(82 223 112 204)(83 224 113 205)(84 225 114 206)(85 226 115 207)(86 227 116 208)(87 228 117 209)(88 229 118 210)(89 230 119 201)(90 221 120 202)(91 164 131 154)(92 165 132 155)(93 166 133 156)(94 167 134 157)(95 168 135 158)(96 169 136 159)(97 170 137 160)(98 161 138 151)(99 162 139 152)(100 163 140 153)(121 199 150 188)(122 200 141 189)(123 191 142 190)(124 192 143 181)(125 193 144 182)(126 194 145 183)(127 195 146 184)(128 196 147 185)(129 197 148 186)(130 198 149 187)
G:=sub<Sym(320)| (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), (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,118)(8,119)(9,120)(10,111)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,21)(18,22)(19,23)(20,24)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,311)(52,312)(53,313)(54,314)(55,315)(56,316)(57,317)(58,318)(59,319)(60,320)(81,107)(82,108)(83,109)(84,110)(85,101)(86,102)(87,103)(88,104)(89,105)(90,106)(91,126)(92,127)(93,128)(94,129)(95,130)(96,121)(97,122)(98,123)(99,124)(100,125)(131,145)(132,146)(133,147)(134,148)(135,149)(136,150)(137,141)(138,142)(139,143)(140,144)(151,190)(152,181)(153,182)(154,183)(155,184)(156,185)(157,186)(158,187)(159,188)(160,189)(161,191)(162,192)(163,193)(164,194)(165,195)(166,196)(167,197)(168,198)(169,199)(170,200)(171,206)(172,207)(173,208)(174,209)(175,210)(176,201)(177,202)(178,203)(179,204)(180,205)(211,225)(212,226)(213,227)(214,228)(215,229)(216,230)(217,221)(218,222)(219,223)(220,224)(231,270)(232,261)(233,262)(234,263)(235,264)(236,265)(237,266)(238,267)(239,268)(240,269)(241,271)(242,272)(243,273)(244,274)(245,275)(246,276)(247,277)(248,278)(249,279)(250,280)(251,286)(252,287)(253,288)(254,289)(255,290)(256,281)(257,282)(258,283)(259,284)(260,285)(291,305)(292,306)(293,307)(294,308)(295,309)(296,310)(297,301)(298,302)(299,303)(300,304), (1,266,129,299)(2,267,130,300)(3,268,121,291)(4,269,122,292)(5,270,123,293)(6,261,124,294)(7,262,125,295)(8,263,126,296)(9,264,127,297)(10,265,128,298)(11,163,58,210)(12,164,59,201)(13,165,60,202)(14,166,51,203)(15,167,52,204)(16,168,53,205)(17,169,54,206)(18,170,55,207)(19,161,56,208)(20,162,57,209)(21,199,314,171)(22,200,315,172)(23,191,316,173)(24,192,317,174)(25,193,318,175)(26,194,319,176)(27,195,320,177)(28,196,311,178)(29,197,312,179)(30,198,313,180)(31,222,78,156)(32,223,79,157)(33,224,80,158)(34,225,71,159)(35,226,72,160)(36,227,73,151)(37,228,74,152)(38,229,75,153)(39,230,76,154)(40,221,77,155)(41,188,69,211)(42,189,70,212)(43,190,61,213)(44,181,62,214)(45,182,63,215)(46,183,64,216)(47,184,65,217)(48,185,66,218)(49,186,67,219)(50,187,68,220)(81,246,133,283)(82,247,134,284)(83,248,135,285)(84,249,136,286)(85,250,137,287)(86,241,138,288)(87,242,139,289)(88,243,140,290)(89,244,131,281)(90,245,132,282)(91,310,119,234)(92,301,120,235)(93,302,111,236)(94,303,112,237)(95,304,113,238)(96,305,114,239)(97,306,115,240)(98,307,116,231)(99,308,117,232)(100,309,118,233)(101,280,141,252)(102,271,142,253)(103,272,143,254)(104,273,144,255)(105,274,145,256)(106,275,146,257)(107,276,147,258)(108,277,148,259)(109,278,149,260)(110,279,150,251), (1,179,108,219)(2,180,109,220)(3,171,110,211)(4,172,101,212)(5,173,102,213)(6,174,103,214)(7,175,104,215)(8,176,105,216)(9,177,106,217)(10,178,107,218)(11,273,75,262)(12,274,76,263)(13,275,77,264)(14,276,78,265)(15,277,79,266)(16,278,80,267)(17,279,71,268)(18,280,72,269)(19,271,73,270)(20,272,74,261)(21,249,41,239)(22,250,42,240)(23,241,43,231)(24,242,44,232)(25,243,45,233)(26,244,46,234)(27,245,47,235)(28,246,48,236)(29,247,49,237)(30,248,50,238)(31,298,51,258)(32,299,52,259)(33,300,53,260)(34,291,54,251)(35,292,55,252)(36,293,56,253)(37,294,57,254)(38,295,58,255)(39,296,59,256)(40,297,60,257)(61,307,316,288)(62,308,317,289)(63,309,318,290)(64,310,319,281)(65,301,320,282)(66,302,311,283)(67,303,312,284)(68,304,313,285)(69,305,314,286)(70,306,315,287)(81,222,111,203)(82,223,112,204)(83,224,113,205)(84,225,114,206)(85,226,115,207)(86,227,116,208)(87,228,117,209)(88,229,118,210)(89,230,119,201)(90,221,120,202)(91,164,131,154)(92,165,132,155)(93,166,133,156)(94,167,134,157)(95,168,135,158)(96,169,136,159)(97,170,137,160)(98,161,138,151)(99,162,139,152)(100,163,140,153)(121,199,150,188)(122,200,141,189)(123,191,142,190)(124,192,143,181)(125,193,144,182)(126,194,145,183)(127,195,146,184)(128,196,147,185)(129,197,148,186)(130,198,149,187)>;
G:=Group( (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), (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,118)(8,119)(9,120)(10,111)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,21)(18,22)(19,23)(20,24)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,311)(52,312)(53,313)(54,314)(55,315)(56,316)(57,317)(58,318)(59,319)(60,320)(81,107)(82,108)(83,109)(84,110)(85,101)(86,102)(87,103)(88,104)(89,105)(90,106)(91,126)(92,127)(93,128)(94,129)(95,130)(96,121)(97,122)(98,123)(99,124)(100,125)(131,145)(132,146)(133,147)(134,148)(135,149)(136,150)(137,141)(138,142)(139,143)(140,144)(151,190)(152,181)(153,182)(154,183)(155,184)(156,185)(157,186)(158,187)(159,188)(160,189)(161,191)(162,192)(163,193)(164,194)(165,195)(166,196)(167,197)(168,198)(169,199)(170,200)(171,206)(172,207)(173,208)(174,209)(175,210)(176,201)(177,202)(178,203)(179,204)(180,205)(211,225)(212,226)(213,227)(214,228)(215,229)(216,230)(217,221)(218,222)(219,223)(220,224)(231,270)(232,261)(233,262)(234,263)(235,264)(236,265)(237,266)(238,267)(239,268)(240,269)(241,271)(242,272)(243,273)(244,274)(245,275)(246,276)(247,277)(248,278)(249,279)(250,280)(251,286)(252,287)(253,288)(254,289)(255,290)(256,281)(257,282)(258,283)(259,284)(260,285)(291,305)(292,306)(293,307)(294,308)(295,309)(296,310)(297,301)(298,302)(299,303)(300,304), (1,266,129,299)(2,267,130,300)(3,268,121,291)(4,269,122,292)(5,270,123,293)(6,261,124,294)(7,262,125,295)(8,263,126,296)(9,264,127,297)(10,265,128,298)(11,163,58,210)(12,164,59,201)(13,165,60,202)(14,166,51,203)(15,167,52,204)(16,168,53,205)(17,169,54,206)(18,170,55,207)(19,161,56,208)(20,162,57,209)(21,199,314,171)(22,200,315,172)(23,191,316,173)(24,192,317,174)(25,193,318,175)(26,194,319,176)(27,195,320,177)(28,196,311,178)(29,197,312,179)(30,198,313,180)(31,222,78,156)(32,223,79,157)(33,224,80,158)(34,225,71,159)(35,226,72,160)(36,227,73,151)(37,228,74,152)(38,229,75,153)(39,230,76,154)(40,221,77,155)(41,188,69,211)(42,189,70,212)(43,190,61,213)(44,181,62,214)(45,182,63,215)(46,183,64,216)(47,184,65,217)(48,185,66,218)(49,186,67,219)(50,187,68,220)(81,246,133,283)(82,247,134,284)(83,248,135,285)(84,249,136,286)(85,250,137,287)(86,241,138,288)(87,242,139,289)(88,243,140,290)(89,244,131,281)(90,245,132,282)(91,310,119,234)(92,301,120,235)(93,302,111,236)(94,303,112,237)(95,304,113,238)(96,305,114,239)(97,306,115,240)(98,307,116,231)(99,308,117,232)(100,309,118,233)(101,280,141,252)(102,271,142,253)(103,272,143,254)(104,273,144,255)(105,274,145,256)(106,275,146,257)(107,276,147,258)(108,277,148,259)(109,278,149,260)(110,279,150,251), (1,179,108,219)(2,180,109,220)(3,171,110,211)(4,172,101,212)(5,173,102,213)(6,174,103,214)(7,175,104,215)(8,176,105,216)(9,177,106,217)(10,178,107,218)(11,273,75,262)(12,274,76,263)(13,275,77,264)(14,276,78,265)(15,277,79,266)(16,278,80,267)(17,279,71,268)(18,280,72,269)(19,271,73,270)(20,272,74,261)(21,249,41,239)(22,250,42,240)(23,241,43,231)(24,242,44,232)(25,243,45,233)(26,244,46,234)(27,245,47,235)(28,246,48,236)(29,247,49,237)(30,248,50,238)(31,298,51,258)(32,299,52,259)(33,300,53,260)(34,291,54,251)(35,292,55,252)(36,293,56,253)(37,294,57,254)(38,295,58,255)(39,296,59,256)(40,297,60,257)(61,307,316,288)(62,308,317,289)(63,309,318,290)(64,310,319,281)(65,301,320,282)(66,302,311,283)(67,303,312,284)(68,304,313,285)(69,305,314,286)(70,306,315,287)(81,222,111,203)(82,223,112,204)(83,224,113,205)(84,225,114,206)(85,226,115,207)(86,227,116,208)(87,228,117,209)(88,229,118,210)(89,230,119,201)(90,221,120,202)(91,164,131,154)(92,165,132,155)(93,166,133,156)(94,167,134,157)(95,168,135,158)(96,169,136,159)(97,170,137,160)(98,161,138,151)(99,162,139,152)(100,163,140,153)(121,199,150,188)(122,200,141,189)(123,191,142,190)(124,192,143,181)(125,193,144,182)(126,194,145,183)(127,195,146,184)(128,196,147,185)(129,197,148,186)(130,198,149,187) );
G=PermutationGroup([[(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)], [(1,112),(2,113),(3,114),(4,115),(5,116),(6,117),(7,118),(8,119),(9,120),(10,111),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(17,21),(18,22),(19,23),(20,24),(31,66),(32,67),(33,68),(34,69),(35,70),(36,61),(37,62),(38,63),(39,64),(40,65),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,311),(52,312),(53,313),(54,314),(55,315),(56,316),(57,317),(58,318),(59,319),(60,320),(81,107),(82,108),(83,109),(84,110),(85,101),(86,102),(87,103),(88,104),(89,105),(90,106),(91,126),(92,127),(93,128),(94,129),(95,130),(96,121),(97,122),(98,123),(99,124),(100,125),(131,145),(132,146),(133,147),(134,148),(135,149),(136,150),(137,141),(138,142),(139,143),(140,144),(151,190),(152,181),(153,182),(154,183),(155,184),(156,185),(157,186),(158,187),(159,188),(160,189),(161,191),(162,192),(163,193),(164,194),(165,195),(166,196),(167,197),(168,198),(169,199),(170,200),(171,206),(172,207),(173,208),(174,209),(175,210),(176,201),(177,202),(178,203),(179,204),(180,205),(211,225),(212,226),(213,227),(214,228),(215,229),(216,230),(217,221),(218,222),(219,223),(220,224),(231,270),(232,261),(233,262),(234,263),(235,264),(236,265),(237,266),(238,267),(239,268),(240,269),(241,271),(242,272),(243,273),(244,274),(245,275),(246,276),(247,277),(248,278),(249,279),(250,280),(251,286),(252,287),(253,288),(254,289),(255,290),(256,281),(257,282),(258,283),(259,284),(260,285),(291,305),(292,306),(293,307),(294,308),(295,309),(296,310),(297,301),(298,302),(299,303),(300,304)], [(1,266,129,299),(2,267,130,300),(3,268,121,291),(4,269,122,292),(5,270,123,293),(6,261,124,294),(7,262,125,295),(8,263,126,296),(9,264,127,297),(10,265,128,298),(11,163,58,210),(12,164,59,201),(13,165,60,202),(14,166,51,203),(15,167,52,204),(16,168,53,205),(17,169,54,206),(18,170,55,207),(19,161,56,208),(20,162,57,209),(21,199,314,171),(22,200,315,172),(23,191,316,173),(24,192,317,174),(25,193,318,175),(26,194,319,176),(27,195,320,177),(28,196,311,178),(29,197,312,179),(30,198,313,180),(31,222,78,156),(32,223,79,157),(33,224,80,158),(34,225,71,159),(35,226,72,160),(36,227,73,151),(37,228,74,152),(38,229,75,153),(39,230,76,154),(40,221,77,155),(41,188,69,211),(42,189,70,212),(43,190,61,213),(44,181,62,214),(45,182,63,215),(46,183,64,216),(47,184,65,217),(48,185,66,218),(49,186,67,219),(50,187,68,220),(81,246,133,283),(82,247,134,284),(83,248,135,285),(84,249,136,286),(85,250,137,287),(86,241,138,288),(87,242,139,289),(88,243,140,290),(89,244,131,281),(90,245,132,282),(91,310,119,234),(92,301,120,235),(93,302,111,236),(94,303,112,237),(95,304,113,238),(96,305,114,239),(97,306,115,240),(98,307,116,231),(99,308,117,232),(100,309,118,233),(101,280,141,252),(102,271,142,253),(103,272,143,254),(104,273,144,255),(105,274,145,256),(106,275,146,257),(107,276,147,258),(108,277,148,259),(109,278,149,260),(110,279,150,251)], [(1,179,108,219),(2,180,109,220),(3,171,110,211),(4,172,101,212),(5,173,102,213),(6,174,103,214),(7,175,104,215),(8,176,105,216),(9,177,106,217),(10,178,107,218),(11,273,75,262),(12,274,76,263),(13,275,77,264),(14,276,78,265),(15,277,79,266),(16,278,80,267),(17,279,71,268),(18,280,72,269),(19,271,73,270),(20,272,74,261),(21,249,41,239),(22,250,42,240),(23,241,43,231),(24,242,44,232),(25,243,45,233),(26,244,46,234),(27,245,47,235),(28,246,48,236),(29,247,49,237),(30,248,50,238),(31,298,51,258),(32,299,52,259),(33,300,53,260),(34,291,54,251),(35,292,55,252),(36,293,56,253),(37,294,57,254),(38,295,58,255),(39,296,59,256),(40,297,60,257),(61,307,316,288),(62,308,317,289),(63,309,318,290),(64,310,319,281),(65,301,320,282),(66,302,311,283),(67,303,312,284),(68,304,313,285),(69,305,314,286),(70,306,315,287),(81,222,111,203),(82,223,112,204),(83,224,113,205),(84,225,114,206),(85,226,115,207),(86,227,116,208),(87,228,117,209),(88,229,118,210),(89,230,119,201),(90,221,120,202),(91,164,131,154),(92,165,132,155),(93,166,133,156),(94,167,134,157),(95,168,135,158),(96,169,136,159),(97,170,137,160),(98,161,138,151),(99,162,139,152),(100,163,140,153),(121,199,150,188),(122,200,141,189),(123,191,142,190),(124,192,143,181),(125,193,144,182),(126,194,145,183),(127,195,146,184),(128,196,147,185),(129,197,148,186),(130,198,149,187)]])
200 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4X | 5A | 5B | 5C | 5D | 10A | ··· | 10BH | 20A | ··· | 20CR |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | D4 | Q8 | C5×D4 | C5×Q8 |
| kernel | C10×C2.C42 | C5×C2.C42 | C23×C20 | C22×C20 | C2×C2.C42 | C2.C42 | C23×C4 | C22×C4 | C22×C10 | C22×C10 | C23 | C23 |
| # reps | 1 | 4 | 3 | 24 | 4 | 16 | 12 | 96 | 6 | 2 | 24 | 8 |
Matrix representation of C10×C2.C42 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 0 | 23 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 14 | 16 |
| 0 | 0 | 0 | 16 | 27 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,23,0,0,0,0,0,23],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[9,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,14,16,0,0,0,16,27],[1,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,40,0] >;
C10×C2.C42 in GAP, Magma, Sage, TeX
C_{10}\times C_2.C_4^2 % in TeX
G:=Group("C10xC2.C4^2"); // GroupNames label
G:=SmallGroup(320,876);
// by ID
G=gap.SmallGroup(320,876);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1128]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,b*d=d*b>;
// generators/relations