direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C10×C2.D8, C8⋊7(C2×C20), (C2×C8)⋊5C20, C40⋊43(C2×C4), (C2×C40)⋊25C4, C2.2(C10×D8), C4.2(Q8×C10), (C2×C10).54D8, C10.74(C2×D8), C20.94(C4⋊C4), C2.2(C10×Q16), C20.91(C2×Q8), (C2×C20).77Q8, (C2×C20).418D4, (C22×C8).8C10, C10.49(C2×Q16), (C2×C10).21Q16, C22.13(C5×D8), C23.57(C5×D4), C22.6(C5×Q16), (C22×C40).26C2, C4.25(C22×C20), C22.48(D4×C10), (C2×C20).899C23, (C2×C40).420C22, C20.242(C22×C4), (C22×C10).218D4, (C22×C20).588C22, C4.14(C5×C4⋊C4), C2.12(C10×C4⋊C4), C10.91(C2×C4⋊C4), (C2×C4).73(C5×D4), (C10×C4⋊C4).43C2, (C2×C4⋊C4).14C10, (C2×C4).19(C5×Q8), C4⋊C4.42(C2×C10), (C2×C8).75(C2×C10), (C2×C4).75(C2×C20), C22.21(C5×C4⋊C4), (C2×C10).92(C4⋊C4), (C2×C20).509(C2×C4), (C2×C10).624(C2×D4), (C5×C4⋊C4).363C22, (C2×C4).74(C22×C10), (C22×C4).117(C2×C10), SmallGroup(320,927)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C2.D8
G = < a,b,c,d | a10=b2=c8=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 194 in 130 conjugacy classes, 98 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C20, C20, C20, C2×C10, C2×C10, C2.D8, C2×C4⋊C4, C22×C8, C40, C2×C20, C2×C20, C2×C20, C22×C10, C2×C2.D8, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C22×C20, C22×C20, C5×C2.D8, C10×C4⋊C4, C22×C40, C10×C2.D8
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C23, C10, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C20, C2×C10, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, C2×C20, C5×D4, C5×Q8, C22×C10, C2×C2.D8, C5×C4⋊C4, C5×D8, C5×Q16, C22×C20, D4×C10, Q8×C10, C5×C2.D8, C10×C4⋊C4, C10×D8, C10×Q16, C10×C2.D8
►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)
(1 133)(2 134)(3 135)(4 136)(5 137)(6 138)(7 139)(8 140)(9 131)(10 132)(11 39)(12 40)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 71)(29 72)(30 73)(41 315)(42 316)(43 317)(44 318)(45 319)(46 320)(47 311)(48 312)(49 313)(50 314)(51 67)(52 68)(53 69)(54 70)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(91 124)(92 125)(93 126)(94 127)(95 128)(96 129)(97 130)(98 121)(99 122)(100 123)(101 147)(102 148)(103 149)(104 150)(105 141)(106 142)(107 143)(108 144)(109 145)(110 146)(151 200)(152 191)(153 192)(154 193)(155 194)(156 195)(157 196)(158 197)(159 198)(160 199)(161 228)(162 229)(163 230)(164 221)(165 222)(166 223)(167 224)(168 225)(169 226)(170 227)(171 204)(172 205)(173 206)(174 207)(175 208)(176 209)(177 210)(178 201)(179 202)(180 203)(181 214)(182 215)(183 216)(184 217)(185 218)(186 219)(187 220)(188 211)(189 212)(190 213)(231 280)(232 271)(233 272)(234 273)(235 274)(236 275)(237 276)(238 277)(239 278)(240 279)(241 308)(242 309)(243 310)(244 301)(245 302)(246 303)(247 304)(248 305)(249 306)(250 307)(251 284)(252 285)(253 286)(254 287)(255 288)(256 289)(257 290)(258 281)(259 282)(260 283)(261 294)(262 295)(263 296)(264 297)(265 298)(266 299)(267 300)(268 291)(269 292)(270 293)
(1 168 96 218 120 204 150 196)(2 169 97 219 111 205 141 197)(3 170 98 220 112 206 142 198)(4 161 99 211 113 207 143 199)(5 162 100 212 114 208 144 200)(6 163 91 213 115 209 145 191)(7 164 92 214 116 210 146 192)(8 165 93 215 117 201 147 193)(9 166 94 216 118 202 148 194)(10 167 95 217 119 203 149 195)(11 258 71 262 45 302 66 234)(12 259 72 263 46 303 67 235)(13 260 73 264 47 304 68 236)(14 251 74 265 48 305 69 237)(15 252 75 266 49 306 70 238)(16 253 76 267 50 307 61 239)(17 254 77 268 41 308 62 240)(18 255 78 269 42 309 63 231)(19 256 79 270 43 310 64 232)(20 257 80 261 44 301 65 233)(21 298 312 248 53 276 32 284)(22 299 313 249 54 277 33 285)(23 300 314 250 55 278 34 286)(24 291 315 241 56 279 35 287)(25 292 316 242 57 280 36 288)(26 293 317 243 58 271 37 289)(27 294 318 244 59 272 38 290)(28 295 319 245 60 273 39 281)(29 296 320 246 51 274 40 282)(30 297 311 247 52 275 31 283)(81 172 105 158 134 226 130 186)(82 173 106 159 135 227 121 187)(83 174 107 160 136 228 122 188)(84 175 108 151 137 229 123 189)(85 176 109 152 138 230 124 190)(86 177 110 153 139 221 125 181)(87 178 101 154 140 222 126 182)(88 179 102 155 131 223 127 183)(89 180 103 156 132 224 128 184)(90 171 104 157 133 225 129 185)
(1 69 133 53)(2 70 134 54)(3 61 135 55)(4 62 136 56)(5 63 137 57)(6 64 138 58)(7 65 139 59)(8 66 140 60)(9 67 131 51)(10 68 132 52)(11 101 39 147)(12 102 40 148)(13 103 31 149)(14 104 32 150)(15 105 33 141)(16 106 34 142)(17 107 35 143)(18 108 36 144)(19 109 37 145)(20 110 38 146)(21 120 74 90)(22 111 75 81)(23 112 76 82)(24 113 77 83)(25 114 78 84)(26 115 79 85)(27 116 80 86)(28 117 71 87)(29 118 72 88)(30 119 73 89)(41 122 315 99)(42 123 316 100)(43 124 317 91)(44 125 318 92)(45 126 319 93)(46 127 320 94)(47 128 311 95)(48 129 312 96)(49 130 313 97)(50 121 314 98)(151 280 200 231)(152 271 191 232)(153 272 192 233)(154 273 193 234)(155 274 194 235)(156 275 195 236)(157 276 196 237)(158 277 197 238)(159 278 198 239)(160 279 199 240)(161 308 228 241)(162 309 229 242)(163 310 230 243)(164 301 221 244)(165 302 222 245)(166 303 223 246)(167 304 224 247)(168 305 225 248)(169 306 226 249)(170 307 227 250)(171 284 204 251)(172 285 205 252)(173 286 206 253)(174 287 207 254)(175 288 208 255)(176 289 209 256)(177 290 210 257)(178 281 201 258)(179 282 202 259)(180 283 203 260)(181 294 214 261)(182 295 215 262)(183 296 216 263)(184 297 217 264)(185 298 218 265)(186 299 219 266)(187 300 220 267)(188 291 211 268)(189 292 212 269)(190 293 213 270)
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), (1,133)(2,134)(3,135)(4,136)(5,137)(6,138)(7,139)(8,140)(9,131)(10,132)(11,39)(12,40)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,71)(29,72)(30,73)(41,315)(42,316)(43,317)(44,318)(45,319)(46,320)(47,311)(48,312)(49,313)(50,314)(51,67)(52,68)(53,69)(54,70)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(91,124)(92,125)(93,126)(94,127)(95,128)(96,129)(97,130)(98,121)(99,122)(100,123)(101,147)(102,148)(103,149)(104,150)(105,141)(106,142)(107,143)(108,144)(109,145)(110,146)(151,200)(152,191)(153,192)(154,193)(155,194)(156,195)(157,196)(158,197)(159,198)(160,199)(161,228)(162,229)(163,230)(164,221)(165,222)(166,223)(167,224)(168,225)(169,226)(170,227)(171,204)(172,205)(173,206)(174,207)(175,208)(176,209)(177,210)(178,201)(179,202)(180,203)(181,214)(182,215)(183,216)(184,217)(185,218)(186,219)(187,220)(188,211)(189,212)(190,213)(231,280)(232,271)(233,272)(234,273)(235,274)(236,275)(237,276)(238,277)(239,278)(240,279)(241,308)(242,309)(243,310)(244,301)(245,302)(246,303)(247,304)(248,305)(249,306)(250,307)(251,284)(252,285)(253,286)(254,287)(255,288)(256,289)(257,290)(258,281)(259,282)(260,283)(261,294)(262,295)(263,296)(264,297)(265,298)(266,299)(267,300)(268,291)(269,292)(270,293), (1,168,96,218,120,204,150,196)(2,169,97,219,111,205,141,197)(3,170,98,220,112,206,142,198)(4,161,99,211,113,207,143,199)(5,162,100,212,114,208,144,200)(6,163,91,213,115,209,145,191)(7,164,92,214,116,210,146,192)(8,165,93,215,117,201,147,193)(9,166,94,216,118,202,148,194)(10,167,95,217,119,203,149,195)(11,258,71,262,45,302,66,234)(12,259,72,263,46,303,67,235)(13,260,73,264,47,304,68,236)(14,251,74,265,48,305,69,237)(15,252,75,266,49,306,70,238)(16,253,76,267,50,307,61,239)(17,254,77,268,41,308,62,240)(18,255,78,269,42,309,63,231)(19,256,79,270,43,310,64,232)(20,257,80,261,44,301,65,233)(21,298,312,248,53,276,32,284)(22,299,313,249,54,277,33,285)(23,300,314,250,55,278,34,286)(24,291,315,241,56,279,35,287)(25,292,316,242,57,280,36,288)(26,293,317,243,58,271,37,289)(27,294,318,244,59,272,38,290)(28,295,319,245,60,273,39,281)(29,296,320,246,51,274,40,282)(30,297,311,247,52,275,31,283)(81,172,105,158,134,226,130,186)(82,173,106,159,135,227,121,187)(83,174,107,160,136,228,122,188)(84,175,108,151,137,229,123,189)(85,176,109,152,138,230,124,190)(86,177,110,153,139,221,125,181)(87,178,101,154,140,222,126,182)(88,179,102,155,131,223,127,183)(89,180,103,156,132,224,128,184)(90,171,104,157,133,225,129,185), (1,69,133,53)(2,70,134,54)(3,61,135,55)(4,62,136,56)(5,63,137,57)(6,64,138,58)(7,65,139,59)(8,66,140,60)(9,67,131,51)(10,68,132,52)(11,101,39,147)(12,102,40,148)(13,103,31,149)(14,104,32,150)(15,105,33,141)(16,106,34,142)(17,107,35,143)(18,108,36,144)(19,109,37,145)(20,110,38,146)(21,120,74,90)(22,111,75,81)(23,112,76,82)(24,113,77,83)(25,114,78,84)(26,115,79,85)(27,116,80,86)(28,117,71,87)(29,118,72,88)(30,119,73,89)(41,122,315,99)(42,123,316,100)(43,124,317,91)(44,125,318,92)(45,126,319,93)(46,127,320,94)(47,128,311,95)(48,129,312,96)(49,130,313,97)(50,121,314,98)(151,280,200,231)(152,271,191,232)(153,272,192,233)(154,273,193,234)(155,274,194,235)(156,275,195,236)(157,276,196,237)(158,277,197,238)(159,278,198,239)(160,279,199,240)(161,308,228,241)(162,309,229,242)(163,310,230,243)(164,301,221,244)(165,302,222,245)(166,303,223,246)(167,304,224,247)(168,305,225,248)(169,306,226,249)(170,307,227,250)(171,284,204,251)(172,285,205,252)(173,286,206,253)(174,287,207,254)(175,288,208,255)(176,289,209,256)(177,290,210,257)(178,281,201,258)(179,282,202,259)(180,283,203,260)(181,294,214,261)(182,295,215,262)(183,296,216,263)(184,297,217,264)(185,298,218,265)(186,299,219,266)(187,300,220,267)(188,291,211,268)(189,292,212,269)(190,293,213,270)>;
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), (1,133)(2,134)(3,135)(4,136)(5,137)(6,138)(7,139)(8,140)(9,131)(10,132)(11,39)(12,40)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,71)(29,72)(30,73)(41,315)(42,316)(43,317)(44,318)(45,319)(46,320)(47,311)(48,312)(49,313)(50,314)(51,67)(52,68)(53,69)(54,70)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(91,124)(92,125)(93,126)(94,127)(95,128)(96,129)(97,130)(98,121)(99,122)(100,123)(101,147)(102,148)(103,149)(104,150)(105,141)(106,142)(107,143)(108,144)(109,145)(110,146)(151,200)(152,191)(153,192)(154,193)(155,194)(156,195)(157,196)(158,197)(159,198)(160,199)(161,228)(162,229)(163,230)(164,221)(165,222)(166,223)(167,224)(168,225)(169,226)(170,227)(171,204)(172,205)(173,206)(174,207)(175,208)(176,209)(177,210)(178,201)(179,202)(180,203)(181,214)(182,215)(183,216)(184,217)(185,218)(186,219)(187,220)(188,211)(189,212)(190,213)(231,280)(232,271)(233,272)(234,273)(235,274)(236,275)(237,276)(238,277)(239,278)(240,279)(241,308)(242,309)(243,310)(244,301)(245,302)(246,303)(247,304)(248,305)(249,306)(250,307)(251,284)(252,285)(253,286)(254,287)(255,288)(256,289)(257,290)(258,281)(259,282)(260,283)(261,294)(262,295)(263,296)(264,297)(265,298)(266,299)(267,300)(268,291)(269,292)(270,293), (1,168,96,218,120,204,150,196)(2,169,97,219,111,205,141,197)(3,170,98,220,112,206,142,198)(4,161,99,211,113,207,143,199)(5,162,100,212,114,208,144,200)(6,163,91,213,115,209,145,191)(7,164,92,214,116,210,146,192)(8,165,93,215,117,201,147,193)(9,166,94,216,118,202,148,194)(10,167,95,217,119,203,149,195)(11,258,71,262,45,302,66,234)(12,259,72,263,46,303,67,235)(13,260,73,264,47,304,68,236)(14,251,74,265,48,305,69,237)(15,252,75,266,49,306,70,238)(16,253,76,267,50,307,61,239)(17,254,77,268,41,308,62,240)(18,255,78,269,42,309,63,231)(19,256,79,270,43,310,64,232)(20,257,80,261,44,301,65,233)(21,298,312,248,53,276,32,284)(22,299,313,249,54,277,33,285)(23,300,314,250,55,278,34,286)(24,291,315,241,56,279,35,287)(25,292,316,242,57,280,36,288)(26,293,317,243,58,271,37,289)(27,294,318,244,59,272,38,290)(28,295,319,245,60,273,39,281)(29,296,320,246,51,274,40,282)(30,297,311,247,52,275,31,283)(81,172,105,158,134,226,130,186)(82,173,106,159,135,227,121,187)(83,174,107,160,136,228,122,188)(84,175,108,151,137,229,123,189)(85,176,109,152,138,230,124,190)(86,177,110,153,139,221,125,181)(87,178,101,154,140,222,126,182)(88,179,102,155,131,223,127,183)(89,180,103,156,132,224,128,184)(90,171,104,157,133,225,129,185), (1,69,133,53)(2,70,134,54)(3,61,135,55)(4,62,136,56)(5,63,137,57)(6,64,138,58)(7,65,139,59)(8,66,140,60)(9,67,131,51)(10,68,132,52)(11,101,39,147)(12,102,40,148)(13,103,31,149)(14,104,32,150)(15,105,33,141)(16,106,34,142)(17,107,35,143)(18,108,36,144)(19,109,37,145)(20,110,38,146)(21,120,74,90)(22,111,75,81)(23,112,76,82)(24,113,77,83)(25,114,78,84)(26,115,79,85)(27,116,80,86)(28,117,71,87)(29,118,72,88)(30,119,73,89)(41,122,315,99)(42,123,316,100)(43,124,317,91)(44,125,318,92)(45,126,319,93)(46,127,320,94)(47,128,311,95)(48,129,312,96)(49,130,313,97)(50,121,314,98)(151,280,200,231)(152,271,191,232)(153,272,192,233)(154,273,193,234)(155,274,194,235)(156,275,195,236)(157,276,196,237)(158,277,197,238)(159,278,198,239)(160,279,199,240)(161,308,228,241)(162,309,229,242)(163,310,230,243)(164,301,221,244)(165,302,222,245)(166,303,223,246)(167,304,224,247)(168,305,225,248)(169,306,226,249)(170,307,227,250)(171,284,204,251)(172,285,205,252)(173,286,206,253)(174,287,207,254)(175,288,208,255)(176,289,209,256)(177,290,210,257)(178,281,201,258)(179,282,202,259)(180,283,203,260)(181,294,214,261)(182,295,215,262)(183,296,216,263)(184,297,217,264)(185,298,218,265)(186,299,219,266)(187,300,220,267)(188,291,211,268)(189,292,212,269)(190,293,213,270) );
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)], [(1,133),(2,134),(3,135),(4,136),(5,137),(6,138),(7,139),(8,140),(9,131),(10,132),(11,39),(12,40),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(21,74),(22,75),(23,76),(24,77),(25,78),(26,79),(27,80),(28,71),(29,72),(30,73),(41,315),(42,316),(43,317),(44,318),(45,319),(46,320),(47,311),(48,312),(49,313),(50,314),(51,67),(52,68),(53,69),(54,70),(55,61),(56,62),(57,63),(58,64),(59,65),(60,66),(81,111),(82,112),(83,113),(84,114),(85,115),(86,116),(87,117),(88,118),(89,119),(90,120),(91,124),(92,125),(93,126),(94,127),(95,128),(96,129),(97,130),(98,121),(99,122),(100,123),(101,147),(102,148),(103,149),(104,150),(105,141),(106,142),(107,143),(108,144),(109,145),(110,146),(151,200),(152,191),(153,192),(154,193),(155,194),(156,195),(157,196),(158,197),(159,198),(160,199),(161,228),(162,229),(163,230),(164,221),(165,222),(166,223),(167,224),(168,225),(169,226),(170,227),(171,204),(172,205),(173,206),(174,207),(175,208),(176,209),(177,210),(178,201),(179,202),(180,203),(181,214),(182,215),(183,216),(184,217),(185,218),(186,219),(187,220),(188,211),(189,212),(190,213),(231,280),(232,271),(233,272),(234,273),(235,274),(236,275),(237,276),(238,277),(239,278),(240,279),(241,308),(242,309),(243,310),(244,301),(245,302),(246,303),(247,304),(248,305),(249,306),(250,307),(251,284),(252,285),(253,286),(254,287),(255,288),(256,289),(257,290),(258,281),(259,282),(260,283),(261,294),(262,295),(263,296),(264,297),(265,298),(266,299),(267,300),(268,291),(269,292),(270,293)], [(1,168,96,218,120,204,150,196),(2,169,97,219,111,205,141,197),(3,170,98,220,112,206,142,198),(4,161,99,211,113,207,143,199),(5,162,100,212,114,208,144,200),(6,163,91,213,115,209,145,191),(7,164,92,214,116,210,146,192),(8,165,93,215,117,201,147,193),(9,166,94,216,118,202,148,194),(10,167,95,217,119,203,149,195),(11,258,71,262,45,302,66,234),(12,259,72,263,46,303,67,235),(13,260,73,264,47,304,68,236),(14,251,74,265,48,305,69,237),(15,252,75,266,49,306,70,238),(16,253,76,267,50,307,61,239),(17,254,77,268,41,308,62,240),(18,255,78,269,42,309,63,231),(19,256,79,270,43,310,64,232),(20,257,80,261,44,301,65,233),(21,298,312,248,53,276,32,284),(22,299,313,249,54,277,33,285),(23,300,314,250,55,278,34,286),(24,291,315,241,56,279,35,287),(25,292,316,242,57,280,36,288),(26,293,317,243,58,271,37,289),(27,294,318,244,59,272,38,290),(28,295,319,245,60,273,39,281),(29,296,320,246,51,274,40,282),(30,297,311,247,52,275,31,283),(81,172,105,158,134,226,130,186),(82,173,106,159,135,227,121,187),(83,174,107,160,136,228,122,188),(84,175,108,151,137,229,123,189),(85,176,109,152,138,230,124,190),(86,177,110,153,139,221,125,181),(87,178,101,154,140,222,126,182),(88,179,102,155,131,223,127,183),(89,180,103,156,132,224,128,184),(90,171,104,157,133,225,129,185)], [(1,69,133,53),(2,70,134,54),(3,61,135,55),(4,62,136,56),(5,63,137,57),(6,64,138,58),(7,65,139,59),(8,66,140,60),(9,67,131,51),(10,68,132,52),(11,101,39,147),(12,102,40,148),(13,103,31,149),(14,104,32,150),(15,105,33,141),(16,106,34,142),(17,107,35,143),(18,108,36,144),(19,109,37,145),(20,110,38,146),(21,120,74,90),(22,111,75,81),(23,112,76,82),(24,113,77,83),(25,114,78,84),(26,115,79,85),(27,116,80,86),(28,117,71,87),(29,118,72,88),(30,119,73,89),(41,122,315,99),(42,123,316,100),(43,124,317,91),(44,125,318,92),(45,126,319,93),(46,127,320,94),(47,128,311,95),(48,129,312,96),(49,130,313,97),(50,121,314,98),(151,280,200,231),(152,271,191,232),(153,272,192,233),(154,273,193,234),(155,274,194,235),(156,275,195,236),(157,276,196,237),(158,277,197,238),(159,278,198,239),(160,279,199,240),(161,308,228,241),(162,309,229,242),(163,310,230,243),(164,301,221,244),(165,302,222,245),(166,303,223,246),(167,304,224,247),(168,305,225,248),(169,306,226,249),(170,307,227,250),(171,284,204,251),(172,285,205,252),(173,286,206,253),(174,287,207,254),(175,288,208,255),(176,289,209,256),(177,290,210,257),(178,281,201,258),(179,282,202,259),(180,283,203,260),(181,294,214,261),(182,295,215,262),(183,296,216,263),(184,297,217,264),(185,298,218,265),(186,299,219,266),(187,300,220,267),(188,291,211,268),(189,292,212,269),(190,293,213,270)]])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10AB | 20A | ··· | 20P | 20Q | ··· | 20AV | 40A | ··· | 40AF |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | D4 | Q8 | D4 | D8 | Q16 | C5×D4 | C5×Q8 | C5×D4 | C5×D8 | C5×Q16 |
| kernel | C10×C2.D8 | C5×C2.D8 | C10×C4⋊C4 | C22×C40 | C2×C40 | C2×C2.D8 | C2.D8 | C2×C4⋊C4 | C22×C8 | C2×C8 | C2×C20 | C2×C20 | C22×C10 | C2×C10 | C2×C10 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 4 | 16 | 8 | 4 | 32 | 1 | 2 | 1 | 4 | 4 | 4 | 8 | 4 | 16 | 16 |
Matrix representation of C10×C2.D8 ►in GL6(𝔽41)
| 31 | 0 | 0 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 0 | 0 | 25 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 12 |
| 0 | 0 | 0 | 0 | 29 | 29 |
| 6 | 2 | 0 | 0 | 0 | 0 |
| 2 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 1 | 0 | 0 |
| 0 | 0 | 3 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 22 |
| 0 | 0 | 0 | 0 | 22 | 3 |
G:=sub<GL(6,GF(41))| [31,0,0,0,0,0,0,31,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,25,0,0,0,0,0,0,25],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,40,0,0,0,0,2,40,0,0,0,0,0,0,29,29,0,0,0,0,12,29],[6,2,0,0,0,0,2,35,0,0,0,0,0,0,18,3,0,0,0,0,1,23,0,0,0,0,0,0,38,22,0,0,0,0,22,3] >;
C10×C2.D8 in GAP, Magma, Sage, TeX
C_{10}\times C_2.D_8 % in TeX
G:=Group("C10xC2.D8"); // GroupNames label
G:=SmallGroup(320,927);
// by ID
G=gap.SmallGroup(320,927);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1408,7004,172]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations