direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4⋊C4×C2×C10, (C22×C4)⋊9C20, C4⋊2(C22×C20), (C22×C20)⋊31C4, C20⋊13(C22×C4), C2.2(C23×C20), (C23×C4).6C10, C23.59(C5×D4), C23.11(C5×Q8), C24.36(C2×C10), C10.75(C23×C4), (C23×C20).10C2, C23.39(C2×C20), C22.58(D4×C10), C10.55(C22×Q8), C22.16(Q8×C10), (C22×C10).30Q8, (C2×C20).958C23, (C2×C10).333C24, C10.178(C22×D4), (C22×C10).220D4, C22.6(C23×C10), C22.24(C22×C20), C23.66(C22×C10), (C23×C10).119C22, (C22×C20).507C22, (C22×C10).466C23, C2.2(D4×C2×C10), C2.1(Q8×C2×C10), (C2×C4)⋊10(C2×C20), (C2×C20)⋊52(C2×C4), (C2×C10).680(C2×D4), (C2×C10).114(C2×Q8), (C2×C4).53(C22×C10), (C22×C10).216(C2×C4), (C22×C4).123(C2×C10), (C2×C10).346(C22×C4), SmallGroup(320,1515)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 498 in 418 conjugacy classes, 338 normal (16 characteristic)
C1, C2 [×3], C2 [×12], C4 [×8], C4 [×8], C22, C22 [×34], C5, C2×C4 [×36], C2×C4 [×24], C23 [×15], C10 [×3], C10 [×12], C4⋊C4 [×16], C22×C4 [×26], C22×C4 [×8], C24, C20 [×8], C20 [×8], C2×C10, C2×C10 [×34], C2×C4⋊C4 [×12], C23×C4, C23×C4 [×2], C2×C20 [×36], C2×C20 [×24], C22×C10 [×15], C22×C4⋊C4, C5×C4⋊C4 [×16], C22×C20 [×26], C22×C20 [×8], C23×C10, C10×C4⋊C4 [×12], C23×C20, C23×C20 [×2], C4⋊C4×C2×C10
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C5, C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C10 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C20 [×8], C2×C10 [×35], C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, C2×C20 [×28], C5×D4 [×4], C5×Q8 [×4], C22×C10 [×15], C22×C4⋊C4, C5×C4⋊C4 [×16], C22×C20 [×14], D4×C10 [×6], Q8×C10 [×6], C23×C10, C10×C4⋊C4 [×12], C23×C20, D4×C2×C10, Q8×C2×C10, C4⋊C4×C2×C10
Generators and relations
G = < a,b,c,d | a2=b10=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
►Regular action on 320 points
Generators in S320
(1 86)(2 87)(3 88)(4 89)(5 90)(6 81)(7 82)(8 83)(9 84)(10 85)(11 298)(12 299)(13 300)(14 291)(15 292)(16 293)(17 294)(18 295)(19 296)(20 297)(21 278)(22 279)(23 280)(24 271)(25 272)(26 273)(27 274)(28 275)(29 276)(30 277)(31 310)(32 301)(33 302)(34 303)(35 304)(36 305)(37 306)(38 307)(39 308)(40 309)(41 79)(42 80)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(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)(131 181)(132 182)(133 183)(134 184)(135 185)(136 186)(137 187)(138 188)(139 189)(140 190)(141 171)(142 172)(143 173)(144 174)(145 175)(146 176)(147 177)(148 178)(149 179)(150 180)(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)(211 261)(212 262)(213 263)(214 264)(215 265)(216 266)(217 267)(218 268)(219 269)(220 270)(221 251)(222 252)(223 253)(224 254)(225 255)(226 256)(227 257)(228 258)(229 259)(230 260)(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)
(1 161 58 183)(2 162 59 184)(3 163 60 185)(4 164 51 186)(5 165 52 187)(6 166 53 188)(7 167 54 189)(8 168 55 190)(9 169 56 181)(10 170 57 182)(11 213 27 199)(12 214 28 200)(13 215 29 191)(14 216 30 192)(15 217 21 193)(16 218 22 194)(17 219 23 195)(18 220 24 196)(19 211 25 197)(20 212 26 198)(31 225 316 203)(32 226 317 204)(33 227 318 205)(34 228 319 206)(35 229 320 207)(36 230 311 208)(37 221 312 209)(38 222 313 210)(39 223 314 201)(40 224 315 202)(41 154 63 178)(42 155 64 179)(43 156 65 180)(44 157 66 171)(45 158 67 172)(46 159 68 173)(47 160 69 174)(48 151 70 175)(49 152 61 176)(50 153 62 177)(71 128 95 150)(72 129 96 141)(73 130 97 142)(74 121 98 143)(75 122 99 144)(76 123 100 145)(77 124 91 146)(78 125 92 147)(79 126 93 148)(80 127 94 149)(81 114 103 138)(82 115 104 139)(83 116 105 140)(84 117 106 131)(85 118 107 132)(86 119 108 133)(87 120 109 134)(88 111 110 135)(89 112 101 136)(90 113 102 137)(231 310 255 288)(232 301 256 289)(233 302 257 290)(234 303 258 281)(235 304 259 282)(236 305 260 283)(237 306 251 284)(238 307 252 285)(239 308 253 286)(240 309 254 287)(241 298 263 274)(242 299 264 275)(243 300 265 276)(244 291 266 277)(245 292 267 278)(246 293 268 279)(247 294 269 280)(248 295 270 271)(249 296 261 272)(250 297 262 273)
(1 263 63 234)(2 264 64 235)(3 265 65 236)(4 266 66 237)(5 267 67 238)(6 268 68 239)(7 269 69 240)(8 270 70 231)(9 261 61 232)(10 262 62 233)(11 148 319 119)(12 149 320 120)(13 150 311 111)(14 141 312 112)(15 142 313 113)(16 143 314 114)(17 144 315 115)(18 145 316 116)(19 146 317 117)(20 147 318 118)(21 130 38 137)(22 121 39 138)(23 122 40 139)(24 123 31 140)(25 124 32 131)(26 125 33 132)(27 126 34 133)(28 127 35 134)(29 128 36 135)(30 129 37 136)(41 258 58 241)(42 259 59 242)(43 260 60 243)(44 251 51 244)(45 252 52 245)(46 253 53 246)(47 254 54 247)(48 255 55 248)(49 256 56 249)(50 257 57 250)(71 230 110 191)(72 221 101 192)(73 222 102 193)(74 223 103 194)(75 224 104 195)(76 225 105 196)(77 226 106 197)(78 227 107 198)(79 228 108 199)(80 229 109 200)(81 218 98 201)(82 219 99 202)(83 220 100 203)(84 211 91 204)(85 212 92 205)(86 213 93 206)(87 214 94 207)(88 215 95 208)(89 216 96 209)(90 217 97 210)(151 310 190 271)(152 301 181 272)(153 302 182 273)(154 303 183 274)(155 304 184 275)(156 305 185 276)(157 306 186 277)(158 307 187 278)(159 308 188 279)(160 309 189 280)(161 298 178 281)(162 299 179 282)(163 300 180 283)(164 291 171 284)(165 292 172 285)(166 293 173 286)(167 294 174 287)(168 295 175 288)(169 296 176 289)(170 297 177 290)
G:=sub<Sym(320)| (1,86)(2,87)(3,88)(4,89)(5,90)(6,81)(7,82)(8,83)(9,84)(10,85)(11,298)(12,299)(13,300)(14,291)(15,292)(16,293)(17,294)(18,295)(19,296)(20,297)(21,278)(22,279)(23,280)(24,271)(25,272)(26,273)(27,274)(28,275)(29,276)(30,277)(31,310)(32,301)(33,302)(34,303)(35,304)(36,305)(37,306)(38,307)(39,308)(40,309)(41,79)(42,80)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(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)(131,181)(132,182)(133,183)(134,184)(135,185)(136,186)(137,187)(138,188)(139,189)(140,190)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(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)(211,261)(212,262)(213,263)(214,264)(215,265)(216,266)(217,267)(218,268)(219,269)(220,270)(221,251)(222,252)(223,253)(224,254)(225,255)(226,256)(227,257)(228,258)(229,259)(230,260)(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), (1,161,58,183)(2,162,59,184)(3,163,60,185)(4,164,51,186)(5,165,52,187)(6,166,53,188)(7,167,54,189)(8,168,55,190)(9,169,56,181)(10,170,57,182)(11,213,27,199)(12,214,28,200)(13,215,29,191)(14,216,30,192)(15,217,21,193)(16,218,22,194)(17,219,23,195)(18,220,24,196)(19,211,25,197)(20,212,26,198)(31,225,316,203)(32,226,317,204)(33,227,318,205)(34,228,319,206)(35,229,320,207)(36,230,311,208)(37,221,312,209)(38,222,313,210)(39,223,314,201)(40,224,315,202)(41,154,63,178)(42,155,64,179)(43,156,65,180)(44,157,66,171)(45,158,67,172)(46,159,68,173)(47,160,69,174)(48,151,70,175)(49,152,61,176)(50,153,62,177)(71,128,95,150)(72,129,96,141)(73,130,97,142)(74,121,98,143)(75,122,99,144)(76,123,100,145)(77,124,91,146)(78,125,92,147)(79,126,93,148)(80,127,94,149)(81,114,103,138)(82,115,104,139)(83,116,105,140)(84,117,106,131)(85,118,107,132)(86,119,108,133)(87,120,109,134)(88,111,110,135)(89,112,101,136)(90,113,102,137)(231,310,255,288)(232,301,256,289)(233,302,257,290)(234,303,258,281)(235,304,259,282)(236,305,260,283)(237,306,251,284)(238,307,252,285)(239,308,253,286)(240,309,254,287)(241,298,263,274)(242,299,264,275)(243,300,265,276)(244,291,266,277)(245,292,267,278)(246,293,268,279)(247,294,269,280)(248,295,270,271)(249,296,261,272)(250,297,262,273), (1,263,63,234)(2,264,64,235)(3,265,65,236)(4,266,66,237)(5,267,67,238)(6,268,68,239)(7,269,69,240)(8,270,70,231)(9,261,61,232)(10,262,62,233)(11,148,319,119)(12,149,320,120)(13,150,311,111)(14,141,312,112)(15,142,313,113)(16,143,314,114)(17,144,315,115)(18,145,316,116)(19,146,317,117)(20,147,318,118)(21,130,38,137)(22,121,39,138)(23,122,40,139)(24,123,31,140)(25,124,32,131)(26,125,33,132)(27,126,34,133)(28,127,35,134)(29,128,36,135)(30,129,37,136)(41,258,58,241)(42,259,59,242)(43,260,60,243)(44,251,51,244)(45,252,52,245)(46,253,53,246)(47,254,54,247)(48,255,55,248)(49,256,56,249)(50,257,57,250)(71,230,110,191)(72,221,101,192)(73,222,102,193)(74,223,103,194)(75,224,104,195)(76,225,105,196)(77,226,106,197)(78,227,107,198)(79,228,108,199)(80,229,109,200)(81,218,98,201)(82,219,99,202)(83,220,100,203)(84,211,91,204)(85,212,92,205)(86,213,93,206)(87,214,94,207)(88,215,95,208)(89,216,96,209)(90,217,97,210)(151,310,190,271)(152,301,181,272)(153,302,182,273)(154,303,183,274)(155,304,184,275)(156,305,185,276)(157,306,186,277)(158,307,187,278)(159,308,188,279)(160,309,189,280)(161,298,178,281)(162,299,179,282)(163,300,180,283)(164,291,171,284)(165,292,172,285)(166,293,173,286)(167,294,174,287)(168,295,175,288)(169,296,176,289)(170,297,177,290)>;
G:=Group( (1,86)(2,87)(3,88)(4,89)(5,90)(6,81)(7,82)(8,83)(9,84)(10,85)(11,298)(12,299)(13,300)(14,291)(15,292)(16,293)(17,294)(18,295)(19,296)(20,297)(21,278)(22,279)(23,280)(24,271)(25,272)(26,273)(27,274)(28,275)(29,276)(30,277)(31,310)(32,301)(33,302)(34,303)(35,304)(36,305)(37,306)(38,307)(39,308)(40,309)(41,79)(42,80)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(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)(131,181)(132,182)(133,183)(134,184)(135,185)(136,186)(137,187)(138,188)(139,189)(140,190)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(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)(211,261)(212,262)(213,263)(214,264)(215,265)(216,266)(217,267)(218,268)(219,269)(220,270)(221,251)(222,252)(223,253)(224,254)(225,255)(226,256)(227,257)(228,258)(229,259)(230,260)(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), (1,161,58,183)(2,162,59,184)(3,163,60,185)(4,164,51,186)(5,165,52,187)(6,166,53,188)(7,167,54,189)(8,168,55,190)(9,169,56,181)(10,170,57,182)(11,213,27,199)(12,214,28,200)(13,215,29,191)(14,216,30,192)(15,217,21,193)(16,218,22,194)(17,219,23,195)(18,220,24,196)(19,211,25,197)(20,212,26,198)(31,225,316,203)(32,226,317,204)(33,227,318,205)(34,228,319,206)(35,229,320,207)(36,230,311,208)(37,221,312,209)(38,222,313,210)(39,223,314,201)(40,224,315,202)(41,154,63,178)(42,155,64,179)(43,156,65,180)(44,157,66,171)(45,158,67,172)(46,159,68,173)(47,160,69,174)(48,151,70,175)(49,152,61,176)(50,153,62,177)(71,128,95,150)(72,129,96,141)(73,130,97,142)(74,121,98,143)(75,122,99,144)(76,123,100,145)(77,124,91,146)(78,125,92,147)(79,126,93,148)(80,127,94,149)(81,114,103,138)(82,115,104,139)(83,116,105,140)(84,117,106,131)(85,118,107,132)(86,119,108,133)(87,120,109,134)(88,111,110,135)(89,112,101,136)(90,113,102,137)(231,310,255,288)(232,301,256,289)(233,302,257,290)(234,303,258,281)(235,304,259,282)(236,305,260,283)(237,306,251,284)(238,307,252,285)(239,308,253,286)(240,309,254,287)(241,298,263,274)(242,299,264,275)(243,300,265,276)(244,291,266,277)(245,292,267,278)(246,293,268,279)(247,294,269,280)(248,295,270,271)(249,296,261,272)(250,297,262,273), (1,263,63,234)(2,264,64,235)(3,265,65,236)(4,266,66,237)(5,267,67,238)(6,268,68,239)(7,269,69,240)(8,270,70,231)(9,261,61,232)(10,262,62,233)(11,148,319,119)(12,149,320,120)(13,150,311,111)(14,141,312,112)(15,142,313,113)(16,143,314,114)(17,144,315,115)(18,145,316,116)(19,146,317,117)(20,147,318,118)(21,130,38,137)(22,121,39,138)(23,122,40,139)(24,123,31,140)(25,124,32,131)(26,125,33,132)(27,126,34,133)(28,127,35,134)(29,128,36,135)(30,129,37,136)(41,258,58,241)(42,259,59,242)(43,260,60,243)(44,251,51,244)(45,252,52,245)(46,253,53,246)(47,254,54,247)(48,255,55,248)(49,256,56,249)(50,257,57,250)(71,230,110,191)(72,221,101,192)(73,222,102,193)(74,223,103,194)(75,224,104,195)(76,225,105,196)(77,226,106,197)(78,227,107,198)(79,228,108,199)(80,229,109,200)(81,218,98,201)(82,219,99,202)(83,220,100,203)(84,211,91,204)(85,212,92,205)(86,213,93,206)(87,214,94,207)(88,215,95,208)(89,216,96,209)(90,217,97,210)(151,310,190,271)(152,301,181,272)(153,302,182,273)(154,303,183,274)(155,304,184,275)(156,305,185,276)(157,306,186,277)(158,307,187,278)(159,308,188,279)(160,309,189,280)(161,298,178,281)(162,299,179,282)(163,300,180,283)(164,291,171,284)(165,292,172,285)(166,293,173,286)(167,294,174,287)(168,295,175,288)(169,296,176,289)(170,297,177,290) );
G=PermutationGroup([(1,86),(2,87),(3,88),(4,89),(5,90),(6,81),(7,82),(8,83),(9,84),(10,85),(11,298),(12,299),(13,300),(14,291),(15,292),(16,293),(17,294),(18,295),(19,296),(20,297),(21,278),(22,279),(23,280),(24,271),(25,272),(26,273),(27,274),(28,275),(29,276),(30,277),(31,310),(32,301),(33,302),(34,303),(35,304),(36,305),(37,306),(38,307),(39,308),(40,309),(41,79),(42,80),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(50,78),(51,101),(52,102),(53,103),(54,104),(55,105),(56,106),(57,107),(58,108),(59,109),(60,110),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(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),(131,181),(132,182),(133,183),(134,184),(135,185),(136,186),(137,187),(138,188),(139,189),(140,190),(141,171),(142,172),(143,173),(144,174),(145,175),(146,176),(147,177),(148,178),(149,179),(150,180),(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),(211,261),(212,262),(213,263),(214,264),(215,265),(216,266),(217,267),(218,268),(219,269),(220,270),(221,251),(222,252),(223,253),(224,254),(225,255),(226,256),(227,257),(228,258),(229,259),(230,260),(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)], [(1,161,58,183),(2,162,59,184),(3,163,60,185),(4,164,51,186),(5,165,52,187),(6,166,53,188),(7,167,54,189),(8,168,55,190),(9,169,56,181),(10,170,57,182),(11,213,27,199),(12,214,28,200),(13,215,29,191),(14,216,30,192),(15,217,21,193),(16,218,22,194),(17,219,23,195),(18,220,24,196),(19,211,25,197),(20,212,26,198),(31,225,316,203),(32,226,317,204),(33,227,318,205),(34,228,319,206),(35,229,320,207),(36,230,311,208),(37,221,312,209),(38,222,313,210),(39,223,314,201),(40,224,315,202),(41,154,63,178),(42,155,64,179),(43,156,65,180),(44,157,66,171),(45,158,67,172),(46,159,68,173),(47,160,69,174),(48,151,70,175),(49,152,61,176),(50,153,62,177),(71,128,95,150),(72,129,96,141),(73,130,97,142),(74,121,98,143),(75,122,99,144),(76,123,100,145),(77,124,91,146),(78,125,92,147),(79,126,93,148),(80,127,94,149),(81,114,103,138),(82,115,104,139),(83,116,105,140),(84,117,106,131),(85,118,107,132),(86,119,108,133),(87,120,109,134),(88,111,110,135),(89,112,101,136),(90,113,102,137),(231,310,255,288),(232,301,256,289),(233,302,257,290),(234,303,258,281),(235,304,259,282),(236,305,260,283),(237,306,251,284),(238,307,252,285),(239,308,253,286),(240,309,254,287),(241,298,263,274),(242,299,264,275),(243,300,265,276),(244,291,266,277),(245,292,267,278),(246,293,268,279),(247,294,269,280),(248,295,270,271),(249,296,261,272),(250,297,262,273)], [(1,263,63,234),(2,264,64,235),(3,265,65,236),(4,266,66,237),(5,267,67,238),(6,268,68,239),(7,269,69,240),(8,270,70,231),(9,261,61,232),(10,262,62,233),(11,148,319,119),(12,149,320,120),(13,150,311,111),(14,141,312,112),(15,142,313,113),(16,143,314,114),(17,144,315,115),(18,145,316,116),(19,146,317,117),(20,147,318,118),(21,130,38,137),(22,121,39,138),(23,122,40,139),(24,123,31,140),(25,124,32,131),(26,125,33,132),(27,126,34,133),(28,127,35,134),(29,128,36,135),(30,129,37,136),(41,258,58,241),(42,259,59,242),(43,260,60,243),(44,251,51,244),(45,252,52,245),(46,253,53,246),(47,254,54,247),(48,255,55,248),(49,256,56,249),(50,257,57,250),(71,230,110,191),(72,221,101,192),(73,222,102,193),(74,223,103,194),(75,224,104,195),(76,225,105,196),(77,226,106,197),(78,227,107,198),(79,228,108,199),(80,229,109,200),(81,218,98,201),(82,219,99,202),(83,220,100,203),(84,211,91,204),(85,212,92,205),(86,213,93,206),(87,214,94,207),(88,215,95,208),(89,216,96,209),(90,217,97,210),(151,310,190,271),(152,301,181,272),(153,302,182,273),(154,303,183,274),(155,304,184,275),(156,305,185,276),(157,306,186,277),(158,307,187,278),(159,308,188,279),(160,309,189,280),(161,298,178,281),(162,299,179,282),(163,300,180,283),(164,291,171,284),(165,292,172,285),(166,293,173,286),(167,294,174,287),(168,295,175,288),(169,296,176,289),(170,297,177,290)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 17 | 9 |
| 0 | 0 | 0 | 9 | 24 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,40,0],[9,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,17,9,0,0,0,9,24] >;
200 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4X | 5A | 5B | 5C | 5D | 10A | ··· | 10BH | 20A | ··· | 20CR |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | D4 | Q8 | C5×D4 | C5×Q8 |
| kernel | C4⋊C4×C2×C10 | C10×C4⋊C4 | C23×C20 | C22×C20 | C22×C4⋊C4 | C2×C4⋊C4 | C23×C4 | C22×C4 | C22×C10 | C22×C10 | C23 | C23 |
| # reps | 1 | 12 | 3 | 16 | 4 | 48 | 12 | 64 | 4 | 4 | 16 | 16 |
In GAP, Magma, Sage, TeX
C_4\rtimes C_4\times C_2\times C_{10} % in TeX
G:=Group("C4:C4xC2xC10"); // GroupNames label
G:=SmallGroup(320,1515);
// by ID
G=gap.SmallGroup(320,1515);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,568]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^4=d^4=1,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
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≀