direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: Q16×C2×C10, C20.80C24, C40.78C23, C4.20(D4×C10), (C2×C20).434D4, C20.327(C2×D4), C4.3(C23×C10), C8.9(C22×C10), C23.62(C5×D4), (C22×C40).28C2, (C22×C8).10C10, C22.67(D4×C10), Q8.1(C22×C10), (C5×Q8).35C23, (C22×Q8).9C10, (C2×C20).973C23, (C2×C40).429C22, (C22×C10).223D4, C10.201(C22×D4), (Q8×C10).280C22, (C22×C20).603C22, C2.25(D4×C2×C10), (C2×C4).90(C5×D4), (Q8×C2×C10).19C2, (C2×C8).87(C2×C10), (C2×C10).688(C2×D4), (C2×Q8).68(C2×C10), (C2×C4).143(C22×C10), (C22×C4).130(C2×C10), SmallGroup(320,1573)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16×C2×C10
G = < a,b,c,d | a2=b10=c8=1, d2=c4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 338 in 258 conjugacy classes, 178 normal (14 characteristic)
C1, C2, C2, C4, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C23, C10, C10, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C20, C20, C20, C2×C10, C22×C8, C2×Q16, C22×Q8, C40, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×C10, C22×Q16, C2×C40, C5×Q16, C22×C20, C22×C20, Q8×C10, Q8×C10, C22×C40, C10×Q16, Q8×C2×C10, Q16×C2×C10
Quotients: C1, C2, C22, C5, D4, C23, C10, Q16, C2×D4, C24, C2×C10, C2×Q16, C22×D4, C5×D4, C22×C10, C22×Q16, C5×Q16, D4×C10, C23×C10, C10×Q16, D4×C2×C10, Q16×C2×C10
►Regular action on 320 points
Generators in S320
(1 100)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 299)(12 300)(13 291)(14 292)(15 293)(16 294)(17 295)(18 296)(19 297)(20 298)(21 278)(22 279)(23 280)(24 271)(25 272)(26 273)(27 274)(28 275)(29 276)(30 277)(31 303)(32 304)(33 305)(34 306)(35 307)(36 308)(37 309)(38 310)(39 301)(40 302)(41 79)(42 80)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 104)(52 105)(53 106)(54 107)(55 108)(56 109)(57 110)(58 101)(59 102)(60 103)(61 90)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(111 163)(112 164)(113 165)(114 166)(115 167)(116 168)(117 169)(118 170)(119 161)(120 162)(121 159)(122 160)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 184)(132 185)(133 186)(134 187)(135 188)(136 189)(137 190)(138 181)(139 182)(140 183)(141 178)(142 179)(143 180)(144 171)(145 172)(146 173)(147 174)(148 175)(149 176)(150 177)(191 243)(192 244)(193 245)(194 246)(195 247)(196 248)(197 249)(198 250)(199 241)(200 242)(201 239)(202 240)(203 231)(204 232)(205 233)(206 234)(207 235)(208 236)(209 237)(210 238)(211 264)(212 265)(213 266)(214 267)(215 268)(216 269)(217 270)(218 261)(219 262)(220 263)(221 258)(222 259)(223 260)(224 251)(225 252)(226 253)(227 254)(228 255)(229 256)(230 257)(281 319)(282 320)(283 311)(284 312)(285 313)(286 314)(287 315)(288 316)(289 317)(290 318)
(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)
(1 274 67 295 55 281 41 303)(2 275 68 296 56 282 42 304)(3 276 69 297 57 283 43 305)(4 277 70 298 58 284 44 306)(5 278 61 299 59 285 45 307)(6 279 62 300 60 286 46 308)(7 280 63 291 51 287 47 309)(8 271 64 292 52 288 48 310)(9 272 65 293 53 289 49 301)(10 273 66 294 54 290 50 302)(11 102 313 73 35 94 21 90)(12 103 314 74 36 95 22 81)(13 104 315 75 37 96 23 82)(14 105 316 76 38 97 24 83)(15 106 317 77 39 98 25 84)(16 107 318 78 40 99 26 85)(17 108 319 79 31 100 27 86)(18 109 320 80 32 91 28 87)(19 110 311 71 33 92 29 88)(20 101 312 72 34 93 30 89)(111 212 150 191 128 230 132 208)(112 213 141 192 129 221 133 209)(113 214 142 193 130 222 134 210)(114 215 143 194 121 223 135 201)(115 216 144 195 122 224 136 202)(116 217 145 196 123 225 137 203)(117 218 146 197 124 226 138 204)(118 219 147 198 125 227 139 205)(119 220 148 199 126 228 140 206)(120 211 149 200 127 229 131 207)(151 252 190 231 168 270 172 248)(152 253 181 232 169 261 173 249)(153 254 182 233 170 262 174 250)(154 255 183 234 161 263 175 241)(155 256 184 235 162 264 176 242)(156 257 185 236 163 265 177 243)(157 258 186 237 164 266 178 244)(158 259 187 238 165 267 179 245)(159 260 188 239 166 268 180 246)(160 251 189 240 167 269 171 247)
(1 175 55 183)(2 176 56 184)(3 177 57 185)(4 178 58 186)(5 179 59 187)(6 180 60 188)(7 171 51 189)(8 172 52 190)(9 173 53 181)(10 174 54 182)(11 210 35 193)(12 201 36 194)(13 202 37 195)(14 203 38 196)(15 204 39 197)(16 205 40 198)(17 206 31 199)(18 207 32 200)(19 208 33 191)(20 209 34 192)(21 214 313 222)(22 215 314 223)(23 216 315 224)(24 217 316 225)(25 218 317 226)(26 219 318 227)(27 220 319 228)(28 211 320 229)(29 212 311 230)(30 213 312 221)(41 154 67 161)(42 155 68 162)(43 156 69 163)(44 157 70 164)(45 158 61 165)(46 159 62 166)(47 160 63 167)(48 151 64 168)(49 152 65 169)(50 153 66 170)(71 128 88 111)(72 129 89 112)(73 130 90 113)(74 121 81 114)(75 122 82 115)(76 123 83 116)(77 124 84 117)(78 125 85 118)(79 126 86 119)(80 127 87 120)(91 149 109 131)(92 150 110 132)(93 141 101 133)(94 142 102 134)(95 143 103 135)(96 144 104 136)(97 145 105 137)(98 146 106 138)(99 147 107 139)(100 148 108 140)(231 310 248 292)(232 301 249 293)(233 302 250 294)(234 303 241 295)(235 304 242 296)(236 305 243 297)(237 306 244 298)(238 307 245 299)(239 308 246 300)(240 309 247 291)(251 280 269 287)(252 271 270 288)(253 272 261 289)(254 273 262 290)(255 274 263 281)(256 275 264 282)(257 276 265 283)(258 277 266 284)(259 278 267 285)(260 279 268 286)
G:=sub<Sym(320)| (1,100)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,299)(12,300)(13,291)(14,292)(15,293)(16,294)(17,295)(18,296)(19,297)(20,298)(21,278)(22,279)(23,280)(24,271)(25,272)(26,273)(27,274)(28,275)(29,276)(30,277)(31,303)(32,304)(33,305)(34,306)(35,307)(36,308)(37,309)(38,310)(39,301)(40,302)(41,79)(42,80)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109)(57,110)(58,101)(59,102)(60,103)(61,90)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(111,163)(112,164)(113,165)(114,166)(115,167)(116,168)(117,169)(118,170)(119,161)(120,162)(121,159)(122,160)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,184)(132,185)(133,186)(134,187)(135,188)(136,189)(137,190)(138,181)(139,182)(140,183)(141,178)(142,179)(143,180)(144,171)(145,172)(146,173)(147,174)(148,175)(149,176)(150,177)(191,243)(192,244)(193,245)(194,246)(195,247)(196,248)(197,249)(198,250)(199,241)(200,242)(201,239)(202,240)(203,231)(204,232)(205,233)(206,234)(207,235)(208,236)(209,237)(210,238)(211,264)(212,265)(213,266)(214,267)(215,268)(216,269)(217,270)(218,261)(219,262)(220,263)(221,258)(222,259)(223,260)(224,251)(225,252)(226,253)(227,254)(228,255)(229,256)(230,257)(281,319)(282,320)(283,311)(284,312)(285,313)(286,314)(287,315)(288,316)(289,317)(290,318), (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), (1,274,67,295,55,281,41,303)(2,275,68,296,56,282,42,304)(3,276,69,297,57,283,43,305)(4,277,70,298,58,284,44,306)(5,278,61,299,59,285,45,307)(6,279,62,300,60,286,46,308)(7,280,63,291,51,287,47,309)(8,271,64,292,52,288,48,310)(9,272,65,293,53,289,49,301)(10,273,66,294,54,290,50,302)(11,102,313,73,35,94,21,90)(12,103,314,74,36,95,22,81)(13,104,315,75,37,96,23,82)(14,105,316,76,38,97,24,83)(15,106,317,77,39,98,25,84)(16,107,318,78,40,99,26,85)(17,108,319,79,31,100,27,86)(18,109,320,80,32,91,28,87)(19,110,311,71,33,92,29,88)(20,101,312,72,34,93,30,89)(111,212,150,191,128,230,132,208)(112,213,141,192,129,221,133,209)(113,214,142,193,130,222,134,210)(114,215,143,194,121,223,135,201)(115,216,144,195,122,224,136,202)(116,217,145,196,123,225,137,203)(117,218,146,197,124,226,138,204)(118,219,147,198,125,227,139,205)(119,220,148,199,126,228,140,206)(120,211,149,200,127,229,131,207)(151,252,190,231,168,270,172,248)(152,253,181,232,169,261,173,249)(153,254,182,233,170,262,174,250)(154,255,183,234,161,263,175,241)(155,256,184,235,162,264,176,242)(156,257,185,236,163,265,177,243)(157,258,186,237,164,266,178,244)(158,259,187,238,165,267,179,245)(159,260,188,239,166,268,180,246)(160,251,189,240,167,269,171,247), (1,175,55,183)(2,176,56,184)(3,177,57,185)(4,178,58,186)(5,179,59,187)(6,180,60,188)(7,171,51,189)(8,172,52,190)(9,173,53,181)(10,174,54,182)(11,210,35,193)(12,201,36,194)(13,202,37,195)(14,203,38,196)(15,204,39,197)(16,205,40,198)(17,206,31,199)(18,207,32,200)(19,208,33,191)(20,209,34,192)(21,214,313,222)(22,215,314,223)(23,216,315,224)(24,217,316,225)(25,218,317,226)(26,219,318,227)(27,220,319,228)(28,211,320,229)(29,212,311,230)(30,213,312,221)(41,154,67,161)(42,155,68,162)(43,156,69,163)(44,157,70,164)(45,158,61,165)(46,159,62,166)(47,160,63,167)(48,151,64,168)(49,152,65,169)(50,153,66,170)(71,128,88,111)(72,129,89,112)(73,130,90,113)(74,121,81,114)(75,122,82,115)(76,123,83,116)(77,124,84,117)(78,125,85,118)(79,126,86,119)(80,127,87,120)(91,149,109,131)(92,150,110,132)(93,141,101,133)(94,142,102,134)(95,143,103,135)(96,144,104,136)(97,145,105,137)(98,146,106,138)(99,147,107,139)(100,148,108,140)(231,310,248,292)(232,301,249,293)(233,302,250,294)(234,303,241,295)(235,304,242,296)(236,305,243,297)(237,306,244,298)(238,307,245,299)(239,308,246,300)(240,309,247,291)(251,280,269,287)(252,271,270,288)(253,272,261,289)(254,273,262,290)(255,274,263,281)(256,275,264,282)(257,276,265,283)(258,277,266,284)(259,278,267,285)(260,279,268,286)>;
G:=Group( (1,100)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,299)(12,300)(13,291)(14,292)(15,293)(16,294)(17,295)(18,296)(19,297)(20,298)(21,278)(22,279)(23,280)(24,271)(25,272)(26,273)(27,274)(28,275)(29,276)(30,277)(31,303)(32,304)(33,305)(34,306)(35,307)(36,308)(37,309)(38,310)(39,301)(40,302)(41,79)(42,80)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109)(57,110)(58,101)(59,102)(60,103)(61,90)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(111,163)(112,164)(113,165)(114,166)(115,167)(116,168)(117,169)(118,170)(119,161)(120,162)(121,159)(122,160)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,184)(132,185)(133,186)(134,187)(135,188)(136,189)(137,190)(138,181)(139,182)(140,183)(141,178)(142,179)(143,180)(144,171)(145,172)(146,173)(147,174)(148,175)(149,176)(150,177)(191,243)(192,244)(193,245)(194,246)(195,247)(196,248)(197,249)(198,250)(199,241)(200,242)(201,239)(202,240)(203,231)(204,232)(205,233)(206,234)(207,235)(208,236)(209,237)(210,238)(211,264)(212,265)(213,266)(214,267)(215,268)(216,269)(217,270)(218,261)(219,262)(220,263)(221,258)(222,259)(223,260)(224,251)(225,252)(226,253)(227,254)(228,255)(229,256)(230,257)(281,319)(282,320)(283,311)(284,312)(285,313)(286,314)(287,315)(288,316)(289,317)(290,318), (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), (1,274,67,295,55,281,41,303)(2,275,68,296,56,282,42,304)(3,276,69,297,57,283,43,305)(4,277,70,298,58,284,44,306)(5,278,61,299,59,285,45,307)(6,279,62,300,60,286,46,308)(7,280,63,291,51,287,47,309)(8,271,64,292,52,288,48,310)(9,272,65,293,53,289,49,301)(10,273,66,294,54,290,50,302)(11,102,313,73,35,94,21,90)(12,103,314,74,36,95,22,81)(13,104,315,75,37,96,23,82)(14,105,316,76,38,97,24,83)(15,106,317,77,39,98,25,84)(16,107,318,78,40,99,26,85)(17,108,319,79,31,100,27,86)(18,109,320,80,32,91,28,87)(19,110,311,71,33,92,29,88)(20,101,312,72,34,93,30,89)(111,212,150,191,128,230,132,208)(112,213,141,192,129,221,133,209)(113,214,142,193,130,222,134,210)(114,215,143,194,121,223,135,201)(115,216,144,195,122,224,136,202)(116,217,145,196,123,225,137,203)(117,218,146,197,124,226,138,204)(118,219,147,198,125,227,139,205)(119,220,148,199,126,228,140,206)(120,211,149,200,127,229,131,207)(151,252,190,231,168,270,172,248)(152,253,181,232,169,261,173,249)(153,254,182,233,170,262,174,250)(154,255,183,234,161,263,175,241)(155,256,184,235,162,264,176,242)(156,257,185,236,163,265,177,243)(157,258,186,237,164,266,178,244)(158,259,187,238,165,267,179,245)(159,260,188,239,166,268,180,246)(160,251,189,240,167,269,171,247), (1,175,55,183)(2,176,56,184)(3,177,57,185)(4,178,58,186)(5,179,59,187)(6,180,60,188)(7,171,51,189)(8,172,52,190)(9,173,53,181)(10,174,54,182)(11,210,35,193)(12,201,36,194)(13,202,37,195)(14,203,38,196)(15,204,39,197)(16,205,40,198)(17,206,31,199)(18,207,32,200)(19,208,33,191)(20,209,34,192)(21,214,313,222)(22,215,314,223)(23,216,315,224)(24,217,316,225)(25,218,317,226)(26,219,318,227)(27,220,319,228)(28,211,320,229)(29,212,311,230)(30,213,312,221)(41,154,67,161)(42,155,68,162)(43,156,69,163)(44,157,70,164)(45,158,61,165)(46,159,62,166)(47,160,63,167)(48,151,64,168)(49,152,65,169)(50,153,66,170)(71,128,88,111)(72,129,89,112)(73,130,90,113)(74,121,81,114)(75,122,82,115)(76,123,83,116)(77,124,84,117)(78,125,85,118)(79,126,86,119)(80,127,87,120)(91,149,109,131)(92,150,110,132)(93,141,101,133)(94,142,102,134)(95,143,103,135)(96,144,104,136)(97,145,105,137)(98,146,106,138)(99,147,107,139)(100,148,108,140)(231,310,248,292)(232,301,249,293)(233,302,250,294)(234,303,241,295)(235,304,242,296)(236,305,243,297)(237,306,244,298)(238,307,245,299)(239,308,246,300)(240,309,247,291)(251,280,269,287)(252,271,270,288)(253,272,261,289)(254,273,262,290)(255,274,263,281)(256,275,264,282)(257,276,265,283)(258,277,266,284)(259,278,267,285)(260,279,268,286) );
G=PermutationGroup([[(1,100),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,299),(12,300),(13,291),(14,292),(15,293),(16,294),(17,295),(18,296),(19,297),(20,298),(21,278),(22,279),(23,280),(24,271),(25,272),(26,273),(27,274),(28,275),(29,276),(30,277),(31,303),(32,304),(33,305),(34,306),(35,307),(36,308),(37,309),(38,310),(39,301),(40,302),(41,79),(42,80),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(50,78),(51,104),(52,105),(53,106),(54,107),(55,108),(56,109),(57,110),(58,101),(59,102),(60,103),(61,90),(62,81),(63,82),(64,83),(65,84),(66,85),(67,86),(68,87),(69,88),(70,89),(111,163),(112,164),(113,165),(114,166),(115,167),(116,168),(117,169),(118,170),(119,161),(120,162),(121,159),(122,160),(123,151),(124,152),(125,153),(126,154),(127,155),(128,156),(129,157),(130,158),(131,184),(132,185),(133,186),(134,187),(135,188),(136,189),(137,190),(138,181),(139,182),(140,183),(141,178),(142,179),(143,180),(144,171),(145,172),(146,173),(147,174),(148,175),(149,176),(150,177),(191,243),(192,244),(193,245),(194,246),(195,247),(196,248),(197,249),(198,250),(199,241),(200,242),(201,239),(202,240),(203,231),(204,232),(205,233),(206,234),(207,235),(208,236),(209,237),(210,238),(211,264),(212,265),(213,266),(214,267),(215,268),(216,269),(217,270),(218,261),(219,262),(220,263),(221,258),(222,259),(223,260),(224,251),(225,252),(226,253),(227,254),(228,255),(229,256),(230,257),(281,319),(282,320),(283,311),(284,312),(285,313),(286,314),(287,315),(288,316),(289,317),(290,318)], [(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)], [(1,274,67,295,55,281,41,303),(2,275,68,296,56,282,42,304),(3,276,69,297,57,283,43,305),(4,277,70,298,58,284,44,306),(5,278,61,299,59,285,45,307),(6,279,62,300,60,286,46,308),(7,280,63,291,51,287,47,309),(8,271,64,292,52,288,48,310),(9,272,65,293,53,289,49,301),(10,273,66,294,54,290,50,302),(11,102,313,73,35,94,21,90),(12,103,314,74,36,95,22,81),(13,104,315,75,37,96,23,82),(14,105,316,76,38,97,24,83),(15,106,317,77,39,98,25,84),(16,107,318,78,40,99,26,85),(17,108,319,79,31,100,27,86),(18,109,320,80,32,91,28,87),(19,110,311,71,33,92,29,88),(20,101,312,72,34,93,30,89),(111,212,150,191,128,230,132,208),(112,213,141,192,129,221,133,209),(113,214,142,193,130,222,134,210),(114,215,143,194,121,223,135,201),(115,216,144,195,122,224,136,202),(116,217,145,196,123,225,137,203),(117,218,146,197,124,226,138,204),(118,219,147,198,125,227,139,205),(119,220,148,199,126,228,140,206),(120,211,149,200,127,229,131,207),(151,252,190,231,168,270,172,248),(152,253,181,232,169,261,173,249),(153,254,182,233,170,262,174,250),(154,255,183,234,161,263,175,241),(155,256,184,235,162,264,176,242),(156,257,185,236,163,265,177,243),(157,258,186,237,164,266,178,244),(158,259,187,238,165,267,179,245),(159,260,188,239,166,268,180,246),(160,251,189,240,167,269,171,247)], [(1,175,55,183),(2,176,56,184),(3,177,57,185),(4,178,58,186),(5,179,59,187),(6,180,60,188),(7,171,51,189),(8,172,52,190),(9,173,53,181),(10,174,54,182),(11,210,35,193),(12,201,36,194),(13,202,37,195),(14,203,38,196),(15,204,39,197),(16,205,40,198),(17,206,31,199),(18,207,32,200),(19,208,33,191),(20,209,34,192),(21,214,313,222),(22,215,314,223),(23,216,315,224),(24,217,316,225),(25,218,317,226),(26,219,318,227),(27,220,319,228),(28,211,320,229),(29,212,311,230),(30,213,312,221),(41,154,67,161),(42,155,68,162),(43,156,69,163),(44,157,70,164),(45,158,61,165),(46,159,62,166),(47,160,63,167),(48,151,64,168),(49,152,65,169),(50,153,66,170),(71,128,88,111),(72,129,89,112),(73,130,90,113),(74,121,81,114),(75,122,82,115),(76,123,83,116),(77,124,84,117),(78,125,85,118),(79,126,86,119),(80,127,87,120),(91,149,109,131),(92,150,110,132),(93,141,101,133),(94,142,102,134),(95,143,103,135),(96,144,104,136),(97,145,105,137),(98,146,106,138),(99,147,107,139),(100,148,108,140),(231,310,248,292),(232,301,249,293),(233,302,250,294),(234,303,241,295),(235,304,242,296),(236,305,243,297),(237,306,244,298),(238,307,245,299),(239,308,246,300),(240,309,247,291),(251,280,269,287),(252,271,270,288),(253,272,261,289),(254,273,262,290),(255,274,263,281),(256,275,264,282),(257,276,265,283),(258,277,266,284),(259,278,267,285),(260,279,268,286)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | D4 | Q16 | C5×D4 | C5×D4 | C5×Q16 |
| kernel | Q16×C2×C10 | C22×C40 | C10×Q16 | Q8×C2×C10 | C22×Q16 | C22×C8 | C2×Q16 | C22×Q8 | C2×C20 | C22×C10 | C2×C10 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 12 | 2 | 4 | 4 | 48 | 8 | 3 | 1 | 8 | 12 | 4 | 32 |
Matrix representation of Q16×C2×C10 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 23 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 15 | 14 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 1 | 40 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,23,0,0,0,0,23],[40,0,0,0,0,1,0,0,0,0,3,15,0,0,0,14],[1,0,0,0,0,40,0,0,0,0,1,1,0,0,39,40] >;
Q16×C2×C10 in GAP, Magma, Sage, TeX
Q_{16}\times C_2\times C_{10} % in TeX
G:=Group("Q16xC2xC10"); // GroupNames label
G:=SmallGroup(320,1573);
// by ID
G=gap.SmallGroup(320,1573);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,1128,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^8=1,d^2=c^4,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