direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4.Dic10, C20.66(C2×Q8), (C2×C20).30Q8, C4⋊C4.260D10, C10.8(C22×Q8), (C2×C10).43C24, C10⋊3(C42.C2), (C2×C4).37Dic10, C4.31(C2×Dic10), (C2×C20).134C23, (C22×C4).358D10, C22.81(C23×D5), C4⋊Dic5.357C22, (C22×C20).73C22, C22.37(C2×Dic10), C2.10(C22×Dic10), C23.323(C22×D5), C22.72(D4⋊2D5), (C22×C10).392C23, C22.32(Q8⋊2D5), (C4×Dic5).279C22, (C2×Dic5).193C23, C10.D4.104C22, (C22×Dic5).231C22, C5⋊3(C2×C42.C2), (C2×C4⋊C4).26D5, (C10×C4⋊C4).19C2, C10.70(C2×C4○D4), (C2×C10).51(C2×Q8), C2.5(C2×Q8⋊2D5), (C2×C4×Dic5).13C2, C2.14(C2×D4⋊2D5), (C2×C4⋊Dic5).44C2, (C5×C4⋊C4).292C22, (C2×C4).139(C22×D5), (C2×C10).170(C4○D4), (C2×C10.D4).24C2, SmallGroup(320,1171)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4.Dic10
G = < a,b,c,d | a2=b4=c20=1, d2=c10, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b2c-1 >
Subgroups: 606 in 226 conjugacy classes, 127 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C2×C42.C2, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C4.Dic10, C2×C4×Dic5, C2×C10.D4, C2×C4⋊Dic5, C2×C4⋊Dic5, C10×C4⋊C4, C2×C4.Dic10
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, C24, D10, C42.C2, C22×Q8, C2×C4○D4, Dic10, C22×D5, C2×C42.C2, C2×Dic10, D4⋊2D5, Q8⋊2D5, C23×D5, C4.Dic10, C22×Dic10, C2×D4⋊2D5, C2×Q8⋊2D5, C2×C4.Dic10
►Regular action on 320 points
Generators in S320
(1 108)(2 109)(3 110)(4 111)(5 112)(6 113)(7 114)(8 115)(9 116)(10 117)(11 118)(12 119)(13 120)(14 101)(15 102)(16 103)(17 104)(18 105)(19 106)(20 107)(21 220)(22 201)(23 202)(24 203)(25 204)(26 205)(27 206)(28 207)(29 208)(30 209)(31 210)(32 211)(33 212)(34 213)(35 214)(36 215)(37 216)(38 217)(39 218)(40 219)(41 185)(42 186)(43 187)(44 188)(45 189)(46 190)(47 191)(48 192)(49 193)(50 194)(51 195)(52 196)(53 197)(54 198)(55 199)(56 200)(57 181)(58 182)(59 183)(60 184)(61 167)(62 168)(63 169)(64 170)(65 171)(66 172)(67 173)(68 174)(69 175)(70 176)(71 177)(72 178)(73 179)(74 180)(75 161)(76 162)(77 163)(78 164)(79 165)(80 166)(81 129)(82 130)(83 131)(84 132)(85 133)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 121)(94 122)(95 123)(96 124)(97 125)(98 126)(99 127)(100 128)(141 298)(142 299)(143 300)(144 281)(145 282)(146 283)(147 284)(148 285)(149 286)(150 287)(151 288)(152 289)(153 290)(154 291)(155 292)(156 293)(157 294)(158 295)(159 296)(160 297)(221 313)(222 314)(223 315)(224 316)(225 317)(226 318)(227 319)(228 320)(229 301)(230 302)(231 303)(232 304)(233 305)(234 306)(235 307)(236 308)(237 309)(238 310)(239 311)(240 312)(241 276)(242 277)(243 278)(244 279)(245 280)(246 261)(247 262)(248 263)(249 264)(250 265)(251 266)(252 267)(253 268)(254 269)(255 270)(256 271)(257 272)(258 273)(259 274)(260 275)
(1 97 52 291)(2 292 53 98)(3 99 54 293)(4 294 55 100)(5 81 56 295)(6 296 57 82)(7 83 58 297)(8 298 59 84)(9 85 60 299)(10 300 41 86)(11 87 42 281)(12 282 43 88)(13 89 44 283)(14 284 45 90)(15 91 46 285)(16 286 47 92)(17 93 48 287)(18 288 49 94)(19 95 50 289)(20 290 51 96)(21 169 226 247)(22 248 227 170)(23 171 228 249)(24 250 229 172)(25 173 230 251)(26 252 231 174)(27 175 232 253)(28 254 233 176)(29 177 234 255)(30 256 235 178)(31 179 236 257)(32 258 237 180)(33 161 238 259)(34 260 239 162)(35 163 240 241)(36 242 221 164)(37 165 222 243)(38 244 223 166)(39 167 224 245)(40 246 225 168)(61 316 280 218)(62 219 261 317)(63 318 262 220)(64 201 263 319)(65 320 264 202)(66 203 265 301)(67 302 266 204)(68 205 267 303)(69 304 268 206)(70 207 269 305)(71 306 270 208)(72 209 271 307)(73 308 272 210)(74 211 273 309)(75 310 274 212)(76 213 275 311)(77 312 276 214)(78 215 277 313)(79 314 278 216)(80 217 279 315)(101 147 189 138)(102 139 190 148)(103 149 191 140)(104 121 192 150)(105 151 193 122)(106 123 194 152)(107 153 195 124)(108 125 196 154)(109 155 197 126)(110 127 198 156)(111 157 199 128)(112 129 200 158)(113 159 181 130)(114 131 182 160)(115 141 183 132)(116 133 184 142)(117 143 185 134)(118 135 186 144)(119 145 187 136)(120 137 188 146)
(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)
(1 162 11 172)(2 259 12 249)(3 180 13 170)(4 257 14 247)(5 178 15 168)(6 255 16 245)(7 176 17 166)(8 253 18 243)(9 174 19 164)(10 251 20 241)(21 294 31 284)(22 99 32 89)(23 292 33 282)(24 97 34 87)(25 290 35 300)(26 95 36 85)(27 288 37 298)(28 93 38 83)(29 286 39 296)(30 91 40 81)(41 173 51 163)(42 250 52 260)(43 171 53 161)(44 248 54 258)(45 169 55 179)(46 246 56 256)(47 167 57 177)(48 244 58 254)(49 165 59 175)(50 242 60 252)(61 181 71 191)(62 112 72 102)(63 199 73 189)(64 110 74 120)(65 197 75 187)(66 108 76 118)(67 195 77 185)(68 106 78 116)(69 193 79 183)(70 104 80 114)(82 234 92 224)(84 232 94 222)(86 230 96 240)(88 228 98 238)(90 226 100 236)(101 262 111 272)(103 280 113 270)(105 278 115 268)(107 276 117 266)(109 274 119 264)(121 217 131 207)(122 314 132 304)(123 215 133 205)(124 312 134 302)(125 213 135 203)(126 310 136 320)(127 211 137 201)(128 308 138 318)(129 209 139 219)(130 306 140 316)(141 206 151 216)(142 303 152 313)(143 204 153 214)(144 301 154 311)(145 202 155 212)(146 319 156 309)(147 220 157 210)(148 317 158 307)(149 218 159 208)(150 315 160 305)(182 269 192 279)(184 267 194 277)(186 265 196 275)(188 263 198 273)(190 261 200 271)(221 299 231 289)(223 297 233 287)(225 295 235 285)(227 293 237 283)(229 291 239 281)
G:=sub<Sym(320)| (1,108)(2,109)(3,110)(4,111)(5,112)(6,113)(7,114)(8,115)(9,116)(10,117)(11,118)(12,119)(13,120)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,220)(22,201)(23,202)(24,203)(25,204)(26,205)(27,206)(28,207)(29,208)(30,209)(31,210)(32,211)(33,212)(34,213)(35,214)(36,215)(37,216)(38,217)(39,218)(40,219)(41,185)(42,186)(43,187)(44,188)(45,189)(46,190)(47,191)(48,192)(49,193)(50,194)(51,195)(52,196)(53,197)(54,198)(55,199)(56,200)(57,181)(58,182)(59,183)(60,184)(61,167)(62,168)(63,169)(64,170)(65,171)(66,172)(67,173)(68,174)(69,175)(70,176)(71,177)(72,178)(73,179)(74,180)(75,161)(76,162)(77,163)(78,164)(79,165)(80,166)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128)(141,298)(142,299)(143,300)(144,281)(145,282)(146,283)(147,284)(148,285)(149,286)(150,287)(151,288)(152,289)(153,290)(154,291)(155,292)(156,293)(157,294)(158,295)(159,296)(160,297)(221,313)(222,314)(223,315)(224,316)(225,317)(226,318)(227,319)(228,320)(229,301)(230,302)(231,303)(232,304)(233,305)(234,306)(235,307)(236,308)(237,309)(238,310)(239,311)(240,312)(241,276)(242,277)(243,278)(244,279)(245,280)(246,261)(247,262)(248,263)(249,264)(250,265)(251,266)(252,267)(253,268)(254,269)(255,270)(256,271)(257,272)(258,273)(259,274)(260,275), (1,97,52,291)(2,292,53,98)(3,99,54,293)(4,294,55,100)(5,81,56,295)(6,296,57,82)(7,83,58,297)(8,298,59,84)(9,85,60,299)(10,300,41,86)(11,87,42,281)(12,282,43,88)(13,89,44,283)(14,284,45,90)(15,91,46,285)(16,286,47,92)(17,93,48,287)(18,288,49,94)(19,95,50,289)(20,290,51,96)(21,169,226,247)(22,248,227,170)(23,171,228,249)(24,250,229,172)(25,173,230,251)(26,252,231,174)(27,175,232,253)(28,254,233,176)(29,177,234,255)(30,256,235,178)(31,179,236,257)(32,258,237,180)(33,161,238,259)(34,260,239,162)(35,163,240,241)(36,242,221,164)(37,165,222,243)(38,244,223,166)(39,167,224,245)(40,246,225,168)(61,316,280,218)(62,219,261,317)(63,318,262,220)(64,201,263,319)(65,320,264,202)(66,203,265,301)(67,302,266,204)(68,205,267,303)(69,304,268,206)(70,207,269,305)(71,306,270,208)(72,209,271,307)(73,308,272,210)(74,211,273,309)(75,310,274,212)(76,213,275,311)(77,312,276,214)(78,215,277,313)(79,314,278,216)(80,217,279,315)(101,147,189,138)(102,139,190,148)(103,149,191,140)(104,121,192,150)(105,151,193,122)(106,123,194,152)(107,153,195,124)(108,125,196,154)(109,155,197,126)(110,127,198,156)(111,157,199,128)(112,129,200,158)(113,159,181,130)(114,131,182,160)(115,141,183,132)(116,133,184,142)(117,143,185,134)(118,135,186,144)(119,145,187,136)(120,137,188,146), (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), (1,162,11,172)(2,259,12,249)(3,180,13,170)(4,257,14,247)(5,178,15,168)(6,255,16,245)(7,176,17,166)(8,253,18,243)(9,174,19,164)(10,251,20,241)(21,294,31,284)(22,99,32,89)(23,292,33,282)(24,97,34,87)(25,290,35,300)(26,95,36,85)(27,288,37,298)(28,93,38,83)(29,286,39,296)(30,91,40,81)(41,173,51,163)(42,250,52,260)(43,171,53,161)(44,248,54,258)(45,169,55,179)(46,246,56,256)(47,167,57,177)(48,244,58,254)(49,165,59,175)(50,242,60,252)(61,181,71,191)(62,112,72,102)(63,199,73,189)(64,110,74,120)(65,197,75,187)(66,108,76,118)(67,195,77,185)(68,106,78,116)(69,193,79,183)(70,104,80,114)(82,234,92,224)(84,232,94,222)(86,230,96,240)(88,228,98,238)(90,226,100,236)(101,262,111,272)(103,280,113,270)(105,278,115,268)(107,276,117,266)(109,274,119,264)(121,217,131,207)(122,314,132,304)(123,215,133,205)(124,312,134,302)(125,213,135,203)(126,310,136,320)(127,211,137,201)(128,308,138,318)(129,209,139,219)(130,306,140,316)(141,206,151,216)(142,303,152,313)(143,204,153,214)(144,301,154,311)(145,202,155,212)(146,319,156,309)(147,220,157,210)(148,317,158,307)(149,218,159,208)(150,315,160,305)(182,269,192,279)(184,267,194,277)(186,265,196,275)(188,263,198,273)(190,261,200,271)(221,299,231,289)(223,297,233,287)(225,295,235,285)(227,293,237,283)(229,291,239,281)>;
G:=Group( (1,108)(2,109)(3,110)(4,111)(5,112)(6,113)(7,114)(8,115)(9,116)(10,117)(11,118)(12,119)(13,120)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,220)(22,201)(23,202)(24,203)(25,204)(26,205)(27,206)(28,207)(29,208)(30,209)(31,210)(32,211)(33,212)(34,213)(35,214)(36,215)(37,216)(38,217)(39,218)(40,219)(41,185)(42,186)(43,187)(44,188)(45,189)(46,190)(47,191)(48,192)(49,193)(50,194)(51,195)(52,196)(53,197)(54,198)(55,199)(56,200)(57,181)(58,182)(59,183)(60,184)(61,167)(62,168)(63,169)(64,170)(65,171)(66,172)(67,173)(68,174)(69,175)(70,176)(71,177)(72,178)(73,179)(74,180)(75,161)(76,162)(77,163)(78,164)(79,165)(80,166)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128)(141,298)(142,299)(143,300)(144,281)(145,282)(146,283)(147,284)(148,285)(149,286)(150,287)(151,288)(152,289)(153,290)(154,291)(155,292)(156,293)(157,294)(158,295)(159,296)(160,297)(221,313)(222,314)(223,315)(224,316)(225,317)(226,318)(227,319)(228,320)(229,301)(230,302)(231,303)(232,304)(233,305)(234,306)(235,307)(236,308)(237,309)(238,310)(239,311)(240,312)(241,276)(242,277)(243,278)(244,279)(245,280)(246,261)(247,262)(248,263)(249,264)(250,265)(251,266)(252,267)(253,268)(254,269)(255,270)(256,271)(257,272)(258,273)(259,274)(260,275), (1,97,52,291)(2,292,53,98)(3,99,54,293)(4,294,55,100)(5,81,56,295)(6,296,57,82)(7,83,58,297)(8,298,59,84)(9,85,60,299)(10,300,41,86)(11,87,42,281)(12,282,43,88)(13,89,44,283)(14,284,45,90)(15,91,46,285)(16,286,47,92)(17,93,48,287)(18,288,49,94)(19,95,50,289)(20,290,51,96)(21,169,226,247)(22,248,227,170)(23,171,228,249)(24,250,229,172)(25,173,230,251)(26,252,231,174)(27,175,232,253)(28,254,233,176)(29,177,234,255)(30,256,235,178)(31,179,236,257)(32,258,237,180)(33,161,238,259)(34,260,239,162)(35,163,240,241)(36,242,221,164)(37,165,222,243)(38,244,223,166)(39,167,224,245)(40,246,225,168)(61,316,280,218)(62,219,261,317)(63,318,262,220)(64,201,263,319)(65,320,264,202)(66,203,265,301)(67,302,266,204)(68,205,267,303)(69,304,268,206)(70,207,269,305)(71,306,270,208)(72,209,271,307)(73,308,272,210)(74,211,273,309)(75,310,274,212)(76,213,275,311)(77,312,276,214)(78,215,277,313)(79,314,278,216)(80,217,279,315)(101,147,189,138)(102,139,190,148)(103,149,191,140)(104,121,192,150)(105,151,193,122)(106,123,194,152)(107,153,195,124)(108,125,196,154)(109,155,197,126)(110,127,198,156)(111,157,199,128)(112,129,200,158)(113,159,181,130)(114,131,182,160)(115,141,183,132)(116,133,184,142)(117,143,185,134)(118,135,186,144)(119,145,187,136)(120,137,188,146), (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), (1,162,11,172)(2,259,12,249)(3,180,13,170)(4,257,14,247)(5,178,15,168)(6,255,16,245)(7,176,17,166)(8,253,18,243)(9,174,19,164)(10,251,20,241)(21,294,31,284)(22,99,32,89)(23,292,33,282)(24,97,34,87)(25,290,35,300)(26,95,36,85)(27,288,37,298)(28,93,38,83)(29,286,39,296)(30,91,40,81)(41,173,51,163)(42,250,52,260)(43,171,53,161)(44,248,54,258)(45,169,55,179)(46,246,56,256)(47,167,57,177)(48,244,58,254)(49,165,59,175)(50,242,60,252)(61,181,71,191)(62,112,72,102)(63,199,73,189)(64,110,74,120)(65,197,75,187)(66,108,76,118)(67,195,77,185)(68,106,78,116)(69,193,79,183)(70,104,80,114)(82,234,92,224)(84,232,94,222)(86,230,96,240)(88,228,98,238)(90,226,100,236)(101,262,111,272)(103,280,113,270)(105,278,115,268)(107,276,117,266)(109,274,119,264)(121,217,131,207)(122,314,132,304)(123,215,133,205)(124,312,134,302)(125,213,135,203)(126,310,136,320)(127,211,137,201)(128,308,138,318)(129,209,139,219)(130,306,140,316)(141,206,151,216)(142,303,152,313)(143,204,153,214)(144,301,154,311)(145,202,155,212)(146,319,156,309)(147,220,157,210)(148,317,158,307)(149,218,159,208)(150,315,160,305)(182,269,192,279)(184,267,194,277)(186,265,196,275)(188,263,198,273)(190,261,200,271)(221,299,231,289)(223,297,233,287)(225,295,235,285)(227,293,237,283)(229,291,239,281) );
G=PermutationGroup([[(1,108),(2,109),(3,110),(4,111),(5,112),(6,113),(7,114),(8,115),(9,116),(10,117),(11,118),(12,119),(13,120),(14,101),(15,102),(16,103),(17,104),(18,105),(19,106),(20,107),(21,220),(22,201),(23,202),(24,203),(25,204),(26,205),(27,206),(28,207),(29,208),(30,209),(31,210),(32,211),(33,212),(34,213),(35,214),(36,215),(37,216),(38,217),(39,218),(40,219),(41,185),(42,186),(43,187),(44,188),(45,189),(46,190),(47,191),(48,192),(49,193),(50,194),(51,195),(52,196),(53,197),(54,198),(55,199),(56,200),(57,181),(58,182),(59,183),(60,184),(61,167),(62,168),(63,169),(64,170),(65,171),(66,172),(67,173),(68,174),(69,175),(70,176),(71,177),(72,178),(73,179),(74,180),(75,161),(76,162),(77,163),(78,164),(79,165),(80,166),(81,129),(82,130),(83,131),(84,132),(85,133),(86,134),(87,135),(88,136),(89,137),(90,138),(91,139),(92,140),(93,121),(94,122),(95,123),(96,124),(97,125),(98,126),(99,127),(100,128),(141,298),(142,299),(143,300),(144,281),(145,282),(146,283),(147,284),(148,285),(149,286),(150,287),(151,288),(152,289),(153,290),(154,291),(155,292),(156,293),(157,294),(158,295),(159,296),(160,297),(221,313),(222,314),(223,315),(224,316),(225,317),(226,318),(227,319),(228,320),(229,301),(230,302),(231,303),(232,304),(233,305),(234,306),(235,307),(236,308),(237,309),(238,310),(239,311),(240,312),(241,276),(242,277),(243,278),(244,279),(245,280),(246,261),(247,262),(248,263),(249,264),(250,265),(251,266),(252,267),(253,268),(254,269),(255,270),(256,271),(257,272),(258,273),(259,274),(260,275)], [(1,97,52,291),(2,292,53,98),(3,99,54,293),(4,294,55,100),(5,81,56,295),(6,296,57,82),(7,83,58,297),(8,298,59,84),(9,85,60,299),(10,300,41,86),(11,87,42,281),(12,282,43,88),(13,89,44,283),(14,284,45,90),(15,91,46,285),(16,286,47,92),(17,93,48,287),(18,288,49,94),(19,95,50,289),(20,290,51,96),(21,169,226,247),(22,248,227,170),(23,171,228,249),(24,250,229,172),(25,173,230,251),(26,252,231,174),(27,175,232,253),(28,254,233,176),(29,177,234,255),(30,256,235,178),(31,179,236,257),(32,258,237,180),(33,161,238,259),(34,260,239,162),(35,163,240,241),(36,242,221,164),(37,165,222,243),(38,244,223,166),(39,167,224,245),(40,246,225,168),(61,316,280,218),(62,219,261,317),(63,318,262,220),(64,201,263,319),(65,320,264,202),(66,203,265,301),(67,302,266,204),(68,205,267,303),(69,304,268,206),(70,207,269,305),(71,306,270,208),(72,209,271,307),(73,308,272,210),(74,211,273,309),(75,310,274,212),(76,213,275,311),(77,312,276,214),(78,215,277,313),(79,314,278,216),(80,217,279,315),(101,147,189,138),(102,139,190,148),(103,149,191,140),(104,121,192,150),(105,151,193,122),(106,123,194,152),(107,153,195,124),(108,125,196,154),(109,155,197,126),(110,127,198,156),(111,157,199,128),(112,129,200,158),(113,159,181,130),(114,131,182,160),(115,141,183,132),(116,133,184,142),(117,143,185,134),(118,135,186,144),(119,145,187,136),(120,137,188,146)], [(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)], [(1,162,11,172),(2,259,12,249),(3,180,13,170),(4,257,14,247),(5,178,15,168),(6,255,16,245),(7,176,17,166),(8,253,18,243),(9,174,19,164),(10,251,20,241),(21,294,31,284),(22,99,32,89),(23,292,33,282),(24,97,34,87),(25,290,35,300),(26,95,36,85),(27,288,37,298),(28,93,38,83),(29,286,39,296),(30,91,40,81),(41,173,51,163),(42,250,52,260),(43,171,53,161),(44,248,54,258),(45,169,55,179),(46,246,56,256),(47,167,57,177),(48,244,58,254),(49,165,59,175),(50,242,60,252),(61,181,71,191),(62,112,72,102),(63,199,73,189),(64,110,74,120),(65,197,75,187),(66,108,76,118),(67,195,77,185),(68,106,78,116),(69,193,79,183),(70,104,80,114),(82,234,92,224),(84,232,94,222),(86,230,96,240),(88,228,98,238),(90,226,100,236),(101,262,111,272),(103,280,113,270),(105,278,115,268),(107,276,117,266),(109,274,119,264),(121,217,131,207),(122,314,132,304),(123,215,133,205),(124,312,134,302),(125,213,135,203),(126,310,136,320),(127,211,137,201),(128,308,138,318),(129,209,139,219),(130,306,140,316),(141,206,151,216),(142,303,152,313),(143,204,153,214),(144,301,154,311),(145,202,155,212),(146,319,156,309),(147,220,157,210),(148,317,158,307),(149,218,159,208),(150,315,160,305),(182,269,192,279),(184,267,194,277),(186,265,196,275),(188,263,198,273),(190,261,200,271),(221,299,231,289),(223,297,233,287),(225,295,235,285),(227,293,237,283),(229,291,239,281)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | Dic10 | D4⋊2D5 | Q8⋊2D5 |
| kernel | C2×C4.Dic10 | C4.Dic10 | C2×C4×Dic5 | C2×C10.D4 | C2×C4⋊Dic5 | C10×C4⋊C4 | C2×C20 | C2×C4⋊C4 | C2×C10 | C4⋊C4 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 8 | 1 | 2 | 3 | 1 | 4 | 2 | 8 | 8 | 6 | 16 | 4 | 4 |
Matrix representation of C2×C4.Dic10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 37 |
| 0 | 0 | 0 | 0 | 22 | 28 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 30 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 27 | 40 |
| 2 | 13 | 0 | 0 | 0 | 0 |
| 25 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 12 | 0 | 0 |
| 0 | 0 | 23 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 36 |
| 0 | 0 | 0 | 0 | 7 | 35 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,22,0,0,0,0,37,28],[35,40,0,0,0,0,1,0,0,0,0,0,0,0,14,11,0,0,0,0,30,9,0,0,0,0,0,0,1,27,0,0,0,0,0,40],[2,25,0,0,0,0,13,39,0,0,0,0,0,0,16,23,0,0,0,0,12,25,0,0,0,0,0,0,6,7,0,0,0,0,36,35] >;
C2×C4.Dic10 in GAP, Magma, Sage, TeX
C_2\times C_4.{\rm Dic}_{10} % in TeX
G:=Group("C2xC4.Dic10"); // GroupNames label
G:=SmallGroup(320,1171);
// by ID
G=gap.SmallGroup(320,1171);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,1571,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=1,d^2=c^10,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^2*c^-1>;
// generators/relations