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