direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C42.C2, C4.9(Q8×C10), (C2×C20).80Q8, C20.98(C2×Q8), (C2×C42).18C10, C42.88(C2×C10), C10.58(C22×Q8), C22.18(Q8×C10), (C2×C10).347C24, (C2×C20).660C23, (C4×C20).372C22, C22.21(C23×C10), C23.71(C22×C10), (C22×C10).469C23, (C22×C20).509C22, C2.4(Q8×C2×C10), (C2×C4×C20).41C2, (C10×C4⋊C4).47C2, (C2×C4⋊C4).18C10, (C2×C4).22(C5×Q8), C4⋊C4.63(C2×C10), C2.10(C10×C4○D4), C10.229(C2×C4○D4), (C2×C10).116(C2×Q8), C22.33(C5×C4○D4), (C5×C4⋊C4).386C22, (C2×C4).15(C22×C10), (C2×C10).233(C4○D4), (C22×C4).101(C2×C10), SmallGroup(320,1529)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C42.C2
G = < a,b,c,d | a10=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >
Subgroups: 274 in 226 conjugacy classes, 178 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C42, C4⋊C4, C22×C4, C22×C4, C20, C20, C2×C10, C2×C10, C2×C42, C2×C4⋊C4, C42.C2, C2×C20, C2×C20, C22×C10, C2×C42.C2, C4×C20, C5×C4⋊C4, C22×C20, C22×C20, C2×C4×C20, C10×C4⋊C4, C5×C42.C2, C10×C42.C2
Quotients: C1, C2, C22, C5, Q8, C23, C10, C2×Q8, C4○D4, C24, C2×C10, C42.C2, C22×Q8, C2×C4○D4, C5×Q8, C22×C10, C2×C42.C2, Q8×C10, C5×C4○D4, C23×C10, C5×C42.C2, Q8×C2×C10, C10×C4○D4, C10×C42.C2
►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)
(1 131 65 141)(2 132 66 142)(3 133 67 143)(4 134 68 144)(5 135 69 145)(6 136 70 146)(7 137 61 147)(8 138 62 148)(9 139 63 149)(10 140 64 150)(11 270 21 260)(12 261 22 251)(13 262 23 252)(14 263 24 253)(15 264 25 254)(16 265 26 255)(17 266 27 256)(18 267 28 257)(19 268 29 258)(20 269 30 259)(31 235 320 249)(32 236 311 250)(33 237 312 241)(34 238 313 242)(35 239 314 243)(36 240 315 244)(37 231 316 245)(38 232 317 246)(39 233 318 247)(40 234 319 248)(41 112 59 126)(42 113 60 127)(43 114 51 128)(44 115 52 129)(45 116 53 130)(46 117 54 121)(47 118 55 122)(48 119 56 123)(49 120 57 124)(50 111 58 125)(71 188 85 178)(72 189 86 179)(73 190 87 180)(74 181 88 171)(75 182 89 172)(76 183 90 173)(77 184 81 174)(78 185 82 175)(79 186 83 176)(80 187 84 177)(91 168 101 154)(92 169 102 155)(93 170 103 156)(94 161 104 157)(95 162 105 158)(96 163 106 159)(97 164 107 160)(98 165 108 151)(99 166 109 152)(100 167 110 153)(191 276 205 290)(192 277 206 281)(193 278 207 282)(194 279 208 283)(195 280 209 284)(196 271 210 285)(197 272 201 286)(198 273 202 287)(199 274 203 288)(200 275 204 289)(211 296 221 306)(212 297 222 307)(213 298 223 308)(214 299 224 309)(215 300 225 310)(216 291 226 301)(217 292 227 302)(218 293 228 303)(219 294 229 304)(220 295 230 305)
(1 79 56 101)(2 80 57 102)(3 71 58 103)(4 72 59 104)(5 73 60 105)(6 74 51 106)(7 75 52 107)(8 76 53 108)(9 77 54 109)(10 78 55 110)(11 282 34 300)(12 283 35 291)(13 284 36 292)(14 285 37 293)(15 286 38 294)(16 287 39 295)(17 288 40 296)(18 289 31 297)(19 290 32 298)(20 281 33 299)(21 278 313 310)(22 279 314 301)(23 280 315 302)(24 271 316 303)(25 272 317 304)(26 273 318 305)(27 274 319 306)(28 275 320 307)(29 276 311 308)(30 277 312 309)(41 94 68 86)(42 95 69 87)(43 96 70 88)(44 97 61 89)(45 98 62 90)(46 99 63 81)(47 100 64 82)(48 91 65 83)(49 92 66 84)(50 93 67 85)(111 170 143 178)(112 161 144 179)(113 162 145 180)(114 163 146 171)(115 164 147 172)(116 165 148 173)(117 166 149 174)(118 167 150 175)(119 168 141 176)(120 169 142 177)(121 152 139 184)(122 153 140 185)(123 154 131 186)(124 155 132 187)(125 156 133 188)(126 157 134 189)(127 158 135 190)(128 159 136 181)(129 160 137 182)(130 151 138 183)(191 236 223 268)(192 237 224 269)(193 238 225 270)(194 239 226 261)(195 240 227 262)(196 231 228 263)(197 232 229 264)(198 233 230 265)(199 234 221 266)(200 235 222 267)(201 246 219 254)(202 247 220 255)(203 248 211 256)(204 249 212 257)(205 250 213 258)(206 241 214 259)(207 242 215 260)(208 243 216 251)(209 244 217 252)(210 245 218 253)
(1 199 56 221)(2 200 57 222)(3 191 58 223)(4 192 59 224)(5 193 60 225)(6 194 51 226)(7 195 52 227)(8 196 53 228)(9 197 54 229)(10 198 55 230)(11 158 34 190)(12 159 35 181)(13 160 36 182)(14 151 37 183)(15 152 38 184)(16 153 39 185)(17 154 40 186)(18 155 31 187)(19 156 32 188)(20 157 33 189)(21 162 313 180)(22 163 314 171)(23 164 315 172)(24 165 316 173)(25 166 317 174)(26 167 318 175)(27 168 319 176)(28 169 320 177)(29 170 311 178)(30 161 312 179)(41 214 68 206)(42 215 69 207)(43 216 70 208)(44 217 61 209)(45 218 62 210)(46 219 63 201)(47 220 64 202)(48 211 65 203)(49 212 66 204)(50 213 67 205)(71 250 103 258)(72 241 104 259)(73 242 105 260)(74 243 106 251)(75 244 107 252)(76 245 108 253)(77 246 109 254)(78 247 110 255)(79 248 101 256)(80 249 102 257)(81 232 99 264)(82 233 100 265)(83 234 91 266)(84 235 92 267)(85 236 93 268)(86 237 94 269)(87 238 95 270)(88 239 96 261)(89 240 97 262)(90 231 98 263)(111 290 143 298)(112 281 144 299)(113 282 145 300)(114 283 146 291)(115 284 147 292)(116 285 148 293)(117 286 149 294)(118 287 150 295)(119 288 141 296)(120 289 142 297)(121 272 139 304)(122 273 140 305)(123 274 131 306)(124 275 132 307)(125 276 133 308)(126 277 134 309)(127 278 135 310)(128 279 136 301)(129 280 137 302)(130 271 138 303)
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), (1,131,65,141)(2,132,66,142)(3,133,67,143)(4,134,68,144)(5,135,69,145)(6,136,70,146)(7,137,61,147)(8,138,62,148)(9,139,63,149)(10,140,64,150)(11,270,21,260)(12,261,22,251)(13,262,23,252)(14,263,24,253)(15,264,25,254)(16,265,26,255)(17,266,27,256)(18,267,28,257)(19,268,29,258)(20,269,30,259)(31,235,320,249)(32,236,311,250)(33,237,312,241)(34,238,313,242)(35,239,314,243)(36,240,315,244)(37,231,316,245)(38,232,317,246)(39,233,318,247)(40,234,319,248)(41,112,59,126)(42,113,60,127)(43,114,51,128)(44,115,52,129)(45,116,53,130)(46,117,54,121)(47,118,55,122)(48,119,56,123)(49,120,57,124)(50,111,58,125)(71,188,85,178)(72,189,86,179)(73,190,87,180)(74,181,88,171)(75,182,89,172)(76,183,90,173)(77,184,81,174)(78,185,82,175)(79,186,83,176)(80,187,84,177)(91,168,101,154)(92,169,102,155)(93,170,103,156)(94,161,104,157)(95,162,105,158)(96,163,106,159)(97,164,107,160)(98,165,108,151)(99,166,109,152)(100,167,110,153)(191,276,205,290)(192,277,206,281)(193,278,207,282)(194,279,208,283)(195,280,209,284)(196,271,210,285)(197,272,201,286)(198,273,202,287)(199,274,203,288)(200,275,204,289)(211,296,221,306)(212,297,222,307)(213,298,223,308)(214,299,224,309)(215,300,225,310)(216,291,226,301)(217,292,227,302)(218,293,228,303)(219,294,229,304)(220,295,230,305), (1,79,56,101)(2,80,57,102)(3,71,58,103)(4,72,59,104)(5,73,60,105)(6,74,51,106)(7,75,52,107)(8,76,53,108)(9,77,54,109)(10,78,55,110)(11,282,34,300)(12,283,35,291)(13,284,36,292)(14,285,37,293)(15,286,38,294)(16,287,39,295)(17,288,40,296)(18,289,31,297)(19,290,32,298)(20,281,33,299)(21,278,313,310)(22,279,314,301)(23,280,315,302)(24,271,316,303)(25,272,317,304)(26,273,318,305)(27,274,319,306)(28,275,320,307)(29,276,311,308)(30,277,312,309)(41,94,68,86)(42,95,69,87)(43,96,70,88)(44,97,61,89)(45,98,62,90)(46,99,63,81)(47,100,64,82)(48,91,65,83)(49,92,66,84)(50,93,67,85)(111,170,143,178)(112,161,144,179)(113,162,145,180)(114,163,146,171)(115,164,147,172)(116,165,148,173)(117,166,149,174)(118,167,150,175)(119,168,141,176)(120,169,142,177)(121,152,139,184)(122,153,140,185)(123,154,131,186)(124,155,132,187)(125,156,133,188)(126,157,134,189)(127,158,135,190)(128,159,136,181)(129,160,137,182)(130,151,138,183)(191,236,223,268)(192,237,224,269)(193,238,225,270)(194,239,226,261)(195,240,227,262)(196,231,228,263)(197,232,229,264)(198,233,230,265)(199,234,221,266)(200,235,222,267)(201,246,219,254)(202,247,220,255)(203,248,211,256)(204,249,212,257)(205,250,213,258)(206,241,214,259)(207,242,215,260)(208,243,216,251)(209,244,217,252)(210,245,218,253), (1,199,56,221)(2,200,57,222)(3,191,58,223)(4,192,59,224)(5,193,60,225)(6,194,51,226)(7,195,52,227)(8,196,53,228)(9,197,54,229)(10,198,55,230)(11,158,34,190)(12,159,35,181)(13,160,36,182)(14,151,37,183)(15,152,38,184)(16,153,39,185)(17,154,40,186)(18,155,31,187)(19,156,32,188)(20,157,33,189)(21,162,313,180)(22,163,314,171)(23,164,315,172)(24,165,316,173)(25,166,317,174)(26,167,318,175)(27,168,319,176)(28,169,320,177)(29,170,311,178)(30,161,312,179)(41,214,68,206)(42,215,69,207)(43,216,70,208)(44,217,61,209)(45,218,62,210)(46,219,63,201)(47,220,64,202)(48,211,65,203)(49,212,66,204)(50,213,67,205)(71,250,103,258)(72,241,104,259)(73,242,105,260)(74,243,106,251)(75,244,107,252)(76,245,108,253)(77,246,109,254)(78,247,110,255)(79,248,101,256)(80,249,102,257)(81,232,99,264)(82,233,100,265)(83,234,91,266)(84,235,92,267)(85,236,93,268)(86,237,94,269)(87,238,95,270)(88,239,96,261)(89,240,97,262)(90,231,98,263)(111,290,143,298)(112,281,144,299)(113,282,145,300)(114,283,146,291)(115,284,147,292)(116,285,148,293)(117,286,149,294)(118,287,150,295)(119,288,141,296)(120,289,142,297)(121,272,139,304)(122,273,140,305)(123,274,131,306)(124,275,132,307)(125,276,133,308)(126,277,134,309)(127,278,135,310)(128,279,136,301)(129,280,137,302)(130,271,138,303)>;
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), (1,131,65,141)(2,132,66,142)(3,133,67,143)(4,134,68,144)(5,135,69,145)(6,136,70,146)(7,137,61,147)(8,138,62,148)(9,139,63,149)(10,140,64,150)(11,270,21,260)(12,261,22,251)(13,262,23,252)(14,263,24,253)(15,264,25,254)(16,265,26,255)(17,266,27,256)(18,267,28,257)(19,268,29,258)(20,269,30,259)(31,235,320,249)(32,236,311,250)(33,237,312,241)(34,238,313,242)(35,239,314,243)(36,240,315,244)(37,231,316,245)(38,232,317,246)(39,233,318,247)(40,234,319,248)(41,112,59,126)(42,113,60,127)(43,114,51,128)(44,115,52,129)(45,116,53,130)(46,117,54,121)(47,118,55,122)(48,119,56,123)(49,120,57,124)(50,111,58,125)(71,188,85,178)(72,189,86,179)(73,190,87,180)(74,181,88,171)(75,182,89,172)(76,183,90,173)(77,184,81,174)(78,185,82,175)(79,186,83,176)(80,187,84,177)(91,168,101,154)(92,169,102,155)(93,170,103,156)(94,161,104,157)(95,162,105,158)(96,163,106,159)(97,164,107,160)(98,165,108,151)(99,166,109,152)(100,167,110,153)(191,276,205,290)(192,277,206,281)(193,278,207,282)(194,279,208,283)(195,280,209,284)(196,271,210,285)(197,272,201,286)(198,273,202,287)(199,274,203,288)(200,275,204,289)(211,296,221,306)(212,297,222,307)(213,298,223,308)(214,299,224,309)(215,300,225,310)(216,291,226,301)(217,292,227,302)(218,293,228,303)(219,294,229,304)(220,295,230,305), (1,79,56,101)(2,80,57,102)(3,71,58,103)(4,72,59,104)(5,73,60,105)(6,74,51,106)(7,75,52,107)(8,76,53,108)(9,77,54,109)(10,78,55,110)(11,282,34,300)(12,283,35,291)(13,284,36,292)(14,285,37,293)(15,286,38,294)(16,287,39,295)(17,288,40,296)(18,289,31,297)(19,290,32,298)(20,281,33,299)(21,278,313,310)(22,279,314,301)(23,280,315,302)(24,271,316,303)(25,272,317,304)(26,273,318,305)(27,274,319,306)(28,275,320,307)(29,276,311,308)(30,277,312,309)(41,94,68,86)(42,95,69,87)(43,96,70,88)(44,97,61,89)(45,98,62,90)(46,99,63,81)(47,100,64,82)(48,91,65,83)(49,92,66,84)(50,93,67,85)(111,170,143,178)(112,161,144,179)(113,162,145,180)(114,163,146,171)(115,164,147,172)(116,165,148,173)(117,166,149,174)(118,167,150,175)(119,168,141,176)(120,169,142,177)(121,152,139,184)(122,153,140,185)(123,154,131,186)(124,155,132,187)(125,156,133,188)(126,157,134,189)(127,158,135,190)(128,159,136,181)(129,160,137,182)(130,151,138,183)(191,236,223,268)(192,237,224,269)(193,238,225,270)(194,239,226,261)(195,240,227,262)(196,231,228,263)(197,232,229,264)(198,233,230,265)(199,234,221,266)(200,235,222,267)(201,246,219,254)(202,247,220,255)(203,248,211,256)(204,249,212,257)(205,250,213,258)(206,241,214,259)(207,242,215,260)(208,243,216,251)(209,244,217,252)(210,245,218,253), (1,199,56,221)(2,200,57,222)(3,191,58,223)(4,192,59,224)(5,193,60,225)(6,194,51,226)(7,195,52,227)(8,196,53,228)(9,197,54,229)(10,198,55,230)(11,158,34,190)(12,159,35,181)(13,160,36,182)(14,151,37,183)(15,152,38,184)(16,153,39,185)(17,154,40,186)(18,155,31,187)(19,156,32,188)(20,157,33,189)(21,162,313,180)(22,163,314,171)(23,164,315,172)(24,165,316,173)(25,166,317,174)(26,167,318,175)(27,168,319,176)(28,169,320,177)(29,170,311,178)(30,161,312,179)(41,214,68,206)(42,215,69,207)(43,216,70,208)(44,217,61,209)(45,218,62,210)(46,219,63,201)(47,220,64,202)(48,211,65,203)(49,212,66,204)(50,213,67,205)(71,250,103,258)(72,241,104,259)(73,242,105,260)(74,243,106,251)(75,244,107,252)(76,245,108,253)(77,246,109,254)(78,247,110,255)(79,248,101,256)(80,249,102,257)(81,232,99,264)(82,233,100,265)(83,234,91,266)(84,235,92,267)(85,236,93,268)(86,237,94,269)(87,238,95,270)(88,239,96,261)(89,240,97,262)(90,231,98,263)(111,290,143,298)(112,281,144,299)(113,282,145,300)(114,283,146,291)(115,284,147,292)(116,285,148,293)(117,286,149,294)(118,287,150,295)(119,288,141,296)(120,289,142,297)(121,272,139,304)(122,273,140,305)(123,274,131,306)(124,275,132,307)(125,276,133,308)(126,277,134,309)(127,278,135,310)(128,279,136,301)(129,280,137,302)(130,271,138,303) );
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)], [(1,131,65,141),(2,132,66,142),(3,133,67,143),(4,134,68,144),(5,135,69,145),(6,136,70,146),(7,137,61,147),(8,138,62,148),(9,139,63,149),(10,140,64,150),(11,270,21,260),(12,261,22,251),(13,262,23,252),(14,263,24,253),(15,264,25,254),(16,265,26,255),(17,266,27,256),(18,267,28,257),(19,268,29,258),(20,269,30,259),(31,235,320,249),(32,236,311,250),(33,237,312,241),(34,238,313,242),(35,239,314,243),(36,240,315,244),(37,231,316,245),(38,232,317,246),(39,233,318,247),(40,234,319,248),(41,112,59,126),(42,113,60,127),(43,114,51,128),(44,115,52,129),(45,116,53,130),(46,117,54,121),(47,118,55,122),(48,119,56,123),(49,120,57,124),(50,111,58,125),(71,188,85,178),(72,189,86,179),(73,190,87,180),(74,181,88,171),(75,182,89,172),(76,183,90,173),(77,184,81,174),(78,185,82,175),(79,186,83,176),(80,187,84,177),(91,168,101,154),(92,169,102,155),(93,170,103,156),(94,161,104,157),(95,162,105,158),(96,163,106,159),(97,164,107,160),(98,165,108,151),(99,166,109,152),(100,167,110,153),(191,276,205,290),(192,277,206,281),(193,278,207,282),(194,279,208,283),(195,280,209,284),(196,271,210,285),(197,272,201,286),(198,273,202,287),(199,274,203,288),(200,275,204,289),(211,296,221,306),(212,297,222,307),(213,298,223,308),(214,299,224,309),(215,300,225,310),(216,291,226,301),(217,292,227,302),(218,293,228,303),(219,294,229,304),(220,295,230,305)], [(1,79,56,101),(2,80,57,102),(3,71,58,103),(4,72,59,104),(5,73,60,105),(6,74,51,106),(7,75,52,107),(8,76,53,108),(9,77,54,109),(10,78,55,110),(11,282,34,300),(12,283,35,291),(13,284,36,292),(14,285,37,293),(15,286,38,294),(16,287,39,295),(17,288,40,296),(18,289,31,297),(19,290,32,298),(20,281,33,299),(21,278,313,310),(22,279,314,301),(23,280,315,302),(24,271,316,303),(25,272,317,304),(26,273,318,305),(27,274,319,306),(28,275,320,307),(29,276,311,308),(30,277,312,309),(41,94,68,86),(42,95,69,87),(43,96,70,88),(44,97,61,89),(45,98,62,90),(46,99,63,81),(47,100,64,82),(48,91,65,83),(49,92,66,84),(50,93,67,85),(111,170,143,178),(112,161,144,179),(113,162,145,180),(114,163,146,171),(115,164,147,172),(116,165,148,173),(117,166,149,174),(118,167,150,175),(119,168,141,176),(120,169,142,177),(121,152,139,184),(122,153,140,185),(123,154,131,186),(124,155,132,187),(125,156,133,188),(126,157,134,189),(127,158,135,190),(128,159,136,181),(129,160,137,182),(130,151,138,183),(191,236,223,268),(192,237,224,269),(193,238,225,270),(194,239,226,261),(195,240,227,262),(196,231,228,263),(197,232,229,264),(198,233,230,265),(199,234,221,266),(200,235,222,267),(201,246,219,254),(202,247,220,255),(203,248,211,256),(204,249,212,257),(205,250,213,258),(206,241,214,259),(207,242,215,260),(208,243,216,251),(209,244,217,252),(210,245,218,253)], [(1,199,56,221),(2,200,57,222),(3,191,58,223),(4,192,59,224),(5,193,60,225),(6,194,51,226),(7,195,52,227),(8,196,53,228),(9,197,54,229),(10,198,55,230),(11,158,34,190),(12,159,35,181),(13,160,36,182),(14,151,37,183),(15,152,38,184),(16,153,39,185),(17,154,40,186),(18,155,31,187),(19,156,32,188),(20,157,33,189),(21,162,313,180),(22,163,314,171),(23,164,315,172),(24,165,316,173),(25,166,317,174),(26,167,318,175),(27,168,319,176),(28,169,320,177),(29,170,311,178),(30,161,312,179),(41,214,68,206),(42,215,69,207),(43,216,70,208),(44,217,61,209),(45,218,62,210),(46,219,63,201),(47,220,64,202),(48,211,65,203),(49,212,66,204),(50,213,67,205),(71,250,103,258),(72,241,104,259),(73,242,105,260),(74,243,106,251),(75,244,107,252),(76,245,108,253),(77,246,109,254),(78,247,110,255),(79,248,101,256),(80,249,102,257),(81,232,99,264),(82,233,100,265),(83,234,91,266),(84,235,92,267),(85,236,93,268),(86,237,94,269),(87,238,95,270),(88,239,96,261),(89,240,97,262),(90,231,98,263),(111,290,143,298),(112,281,144,299),(113,282,145,300),(114,283,146,291),(115,284,147,292),(116,285,148,293),(117,286,149,294),(118,287,150,295),(119,288,141,296),(120,289,142,297),(121,272,139,304),(122,273,140,305),(123,274,131,306),(124,275,132,307),(125,276,133,308),(126,277,134,309),(127,278,135,310),(128,279,136,301),(129,280,137,302),(130,271,138,303)]])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 20A | ··· | 20AV | 20AW | ··· | 20CB |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | Q8 | C4○D4 | C5×Q8 | C5×C4○D4 |
| kernel | C10×C42.C2 | C2×C4×C20 | C10×C4⋊C4 | C5×C42.C2 | C2×C42.C2 | C2×C42 | C2×C4⋊C4 | C42.C2 | C2×C20 | C2×C10 | C2×C4 | C22 |
| # reps | 1 | 1 | 6 | 8 | 4 | 4 | 24 | 32 | 4 | 8 | 16 | 32 |
Matrix representation of C10×C42.C2 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 10 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 39 |
| 0 | 0 | 0 | 0 | 9 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 32 | 39 |
| 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 39 | 28 | 0 | 0 |
| 0 | 13 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 23 |
| 0 | 0 | 0 | 32 | 40 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,10,0,0,0,0,0,10],[40,0,0,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,39,9],[40,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,32,0,0,0,0,39,9],[1,0,0,0,0,0,39,13,0,0,0,28,2,0,0,0,0,0,1,32,0,0,0,23,40] >;
C10×C42.C2 in GAP, Magma, Sage, TeX
C_{10}\times C_4^2.C_2 % in TeX
G:=Group("C10xC4^2.C2"); // GroupNames label
G:=SmallGroup(320,1529);
// by ID
G=gap.SmallGroup(320,1529);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,1128,3446,436]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^2,d*c*d^-1=b^2*c>;
// generators/relations