direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8×Dic5, C10⋊4(C4×Q8), (Q8×C10)⋊21C4, C22.36(Q8×D5), C10.67(C23×C4), (C2×Q8).206D10, (C22×Q8).10D5, C10.52(C22×Q8), C2.8(C23×Dic5), (C2×C10).303C24, C20.154(C22×C4), (C2×C20).550C23, (C22×C4).383D10, C4.18(C22×Dic5), C22.46(C23×D5), C4⋊Dic5.388C22, (Q8×C10).232C22, C23.340(C22×D5), (C22×C20).283C22, (C22×C10).421C23, C22.38(Q8⋊2D5), (C4×Dic5).291C22, (C2×Dic5).297C23, C22.32(C22×Dic5), (C22×Dic5).290C22, C5⋊5(C2×C4×Q8), C2.4(C2×Q8×D5), (Q8×C2×C10).8C2, (C5×Q8)⋊27(C2×C4), (C2×C10).97(C2×Q8), C2.4(C2×Q8⋊2D5), (C2×C4×Dic5).17C2, (C2×C20).306(C2×C4), C10.126(C2×C4○D4), (C2×C4⋊Dic5).46C2, (C2×C4).55(C2×Dic5), (C2×C4).631(C22×D5), (C2×C10).199(C4○D4), (C2×C10).310(C22×C4), SmallGroup(320,1483)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 654 in 298 conjugacy classes, 215 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C4 [×10], C22, C22 [×6], C5, C2×C4 [×18], C2×C4 [×18], Q8 [×16], C23, C10 [×3], C10 [×4], C42 [×12], C4⋊C4 [×12], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×12], Dic5 [×4], Dic5 [×6], C20 [×12], C2×C10, C2×C10 [×6], C2×C42 [×3], C2×C4⋊C4 [×3], C4×Q8 [×8], C22×Q8, C2×Dic5 [×12], C2×Dic5 [×6], C2×C20 [×18], C5×Q8 [×16], C22×C10, C2×C4×Q8, C4×Dic5 [×12], C4⋊Dic5 [×12], C22×Dic5, C22×Dic5 [×3], C22×C20 [×3], Q8×C10 [×12], C2×C4×Dic5 [×3], C2×C4⋊Dic5 [×3], Q8×Dic5 [×8], Q8×C2×C10, C2×Q8×Dic5
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×4], C23 [×15], D5, C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, Dic5 [×8], D10 [×7], C4×Q8 [×4], C23×C4, C22×Q8, C2×C4○D4, C2×Dic5 [×28], C22×D5 [×7], C2×C4×Q8, Q8×D5 [×2], Q8⋊2D5 [×2], C22×Dic5 [×14], C23×D5, Q8×Dic5 [×4], C2×Q8×D5, C2×Q8⋊2D5, C23×Dic5, C2×Q8×Dic5
Generators and relations
G = < a,b,c,d,e | a2=b4=d10=1, c2=b2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
►Regular action on 320 points
Generators in S320
(1 108)(2 109)(3 110)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 233)(12 234)(13 235)(14 236)(15 237)(16 238)(17 239)(18 240)(19 231)(20 232)(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(27 91)(28 92)(29 93)(30 94)(31 127)(32 128)(33 129)(34 130)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 111)(48 112)(49 113)(50 114)(51 147)(52 148)(53 149)(54 150)(55 141)(56 142)(57 143)(58 144)(59 145)(60 146)(61 135)(62 136)(63 137)(64 138)(65 139)(66 140)(67 131)(68 132)(69 133)(70 134)(71 164)(72 165)(73 166)(74 167)(75 168)(76 169)(77 170)(78 161)(79 162)(80 163)(81 155)(82 156)(83 157)(84 158)(85 159)(86 160)(87 151)(88 152)(89 153)(90 154)(171 270)(172 261)(173 262)(174 263)(175 264)(176 265)(177 266)(178 267)(179 268)(180 269)(181 252)(182 253)(183 254)(184 255)(185 256)(186 257)(187 258)(188 259)(189 260)(190 251)(191 290)(192 281)(193 282)(194 283)(195 284)(196 285)(197 286)(198 287)(199 288)(200 289)(201 272)(202 273)(203 274)(204 275)(205 276)(206 277)(207 278)(208 279)(209 280)(210 271)(211 310)(212 301)(213 302)(214 303)(215 304)(216 305)(217 306)(218 307)(219 308)(220 309)(221 292)(222 293)(223 294)(224 295)(225 296)(226 297)(227 298)(228 299)(229 300)(230 291)(241 312)(242 313)(243 314)(244 315)(245 316)(246 317)(247 318)(248 319)(249 320)(250 311)
(1 43 28 37)(2 44 29 38)(3 45 30 39)(4 46 21 40)(5 47 22 31)(6 48 23 32)(7 49 24 33)(8 50 25 34)(9 41 26 35)(10 42 27 36)(11 298 313 307)(12 299 314 308)(13 300 315 309)(14 291 316 310)(15 292 317 301)(16 293 318 302)(17 294 319 303)(18 295 320 304)(19 296 311 305)(20 297 312 306)(51 76 67 82)(52 77 68 83)(53 78 69 84)(54 79 70 85)(55 80 61 86)(56 71 62 87)(57 72 63 88)(58 73 64 89)(59 74 65 90)(60 75 66 81)(91 122 107 116)(92 123 108 117)(93 124 109 118)(94 125 110 119)(95 126 101 120)(96 127 102 111)(97 128 103 112)(98 129 104 113)(99 130 105 114)(100 121 106 115)(131 156 147 169)(132 157 148 170)(133 158 149 161)(134 159 150 162)(135 160 141 163)(136 151 142 164)(137 152 143 165)(138 153 144 166)(139 154 145 167)(140 155 146 168)(171 205 190 196)(172 206 181 197)(173 207 182 198)(174 208 183 199)(175 209 184 200)(176 210 185 191)(177 201 186 192)(178 202 187 193)(179 203 188 194)(180 204 189 195)(211 236 230 245)(212 237 221 246)(213 238 222 247)(214 239 223 248)(215 240 224 249)(216 231 225 250)(217 232 226 241)(218 233 227 242)(219 234 228 243)(220 235 229 244)(251 285 270 276)(252 286 261 277)(253 287 262 278)(254 288 263 279)(255 289 264 280)(256 290 265 271)(257 281 266 272)(258 282 267 273)(259 283 268 274)(260 284 269 275)
(1 63 28 57)(2 64 29 58)(3 65 30 59)(4 66 21 60)(5 67 22 51)(6 68 23 52)(7 69 24 53)(8 70 25 54)(9 61 26 55)(10 62 27 56)(11 287 313 278)(12 288 314 279)(13 289 315 280)(14 290 316 271)(15 281 317 272)(16 282 318 273)(17 283 319 274)(18 284 320 275)(19 285 311 276)(20 286 312 277)(31 82 47 76)(32 83 48 77)(33 84 49 78)(34 85 50 79)(35 86 41 80)(36 87 42 71)(37 88 43 72)(38 89 44 73)(39 90 45 74)(40 81 46 75)(91 142 107 136)(92 143 108 137)(93 144 109 138)(94 145 110 139)(95 146 101 140)(96 147 102 131)(97 148 103 132)(98 149 104 133)(99 150 105 134)(100 141 106 135)(111 169 127 156)(112 170 128 157)(113 161 129 158)(114 162 130 159)(115 163 121 160)(116 164 122 151)(117 165 123 152)(118 166 124 153)(119 167 125 154)(120 168 126 155)(171 225 190 216)(172 226 181 217)(173 227 182 218)(174 228 183 219)(175 229 184 220)(176 230 185 211)(177 221 186 212)(178 222 187 213)(179 223 188 214)(180 224 189 215)(191 245 210 236)(192 246 201 237)(193 247 202 238)(194 248 203 239)(195 249 204 240)(196 250 205 231)(197 241 206 232)(198 242 207 233)(199 243 208 234)(200 244 209 235)(251 305 270 296)(252 306 261 297)(253 307 262 298)(254 308 263 299)(255 309 264 300)(256 310 265 291)(257 301 266 292)(258 302 267 293)(259 303 268 294)(260 304 269 295)
(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)
(1 253 6 258)(2 252 7 257)(3 251 8 256)(4 260 9 255)(5 259 10 254)(11 83 16 88)(12 82 17 87)(13 81 18 86)(14 90 19 85)(15 89 20 84)(21 269 26 264)(22 268 27 263)(23 267 28 262)(24 266 29 261)(25 265 30 270)(31 274 36 279)(32 273 37 278)(33 272 38 277)(34 271 39 276)(35 280 40 275)(41 289 46 284)(42 288 47 283)(43 287 48 282)(44 286 49 281)(45 285 50 290)(51 294 56 299)(52 293 57 298)(53 292 58 297)(54 291 59 296)(55 300 60 295)(61 309 66 304)(62 308 67 303)(63 307 68 302)(64 306 69 301)(65 305 70 310)(71 314 76 319)(72 313 77 318)(73 312 78 317)(74 311 79 316)(75 320 80 315)(91 174 96 179)(92 173 97 178)(93 172 98 177)(94 171 99 176)(95 180 100 175)(101 189 106 184)(102 188 107 183)(103 187 108 182)(104 186 109 181)(105 185 110 190)(111 194 116 199)(112 193 117 198)(113 192 118 197)(114 191 119 196)(115 200 120 195)(121 209 126 204)(122 208 127 203)(123 207 128 202)(124 206 129 201)(125 205 130 210)(131 214 136 219)(132 213 137 218)(133 212 138 217)(134 211 139 216)(135 220 140 215)(141 229 146 224)(142 228 147 223)(143 227 148 222)(144 226 149 221)(145 225 150 230)(151 234 156 239)(152 233 157 238)(153 232 158 237)(154 231 159 236)(155 240 160 235)(161 246 166 241)(162 245 167 250)(163 244 168 249)(164 243 169 248)(165 242 170 247)
G:=sub<Sym(320)| (1,108)(2,109)(3,110)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,233)(12,234)(13,235)(14,236)(15,237)(16,238)(17,239)(18,240)(19,231)(20,232)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,91)(28,92)(29,93)(30,94)(31,127)(32,128)(33,129)(34,130)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,111)(48,112)(49,113)(50,114)(51,147)(52,148)(53,149)(54,150)(55,141)(56,142)(57,143)(58,144)(59,145)(60,146)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,131)(68,132)(69,133)(70,134)(71,164)(72,165)(73,166)(74,167)(75,168)(76,169)(77,170)(78,161)(79,162)(80,163)(81,155)(82,156)(83,157)(84,158)(85,159)(86,160)(87,151)(88,152)(89,153)(90,154)(171,270)(172,261)(173,262)(174,263)(175,264)(176,265)(177,266)(178,267)(179,268)(180,269)(181,252)(182,253)(183,254)(184,255)(185,256)(186,257)(187,258)(188,259)(189,260)(190,251)(191,290)(192,281)(193,282)(194,283)(195,284)(196,285)(197,286)(198,287)(199,288)(200,289)(201,272)(202,273)(203,274)(204,275)(205,276)(206,277)(207,278)(208,279)(209,280)(210,271)(211,310)(212,301)(213,302)(214,303)(215,304)(216,305)(217,306)(218,307)(219,308)(220,309)(221,292)(222,293)(223,294)(224,295)(225,296)(226,297)(227,298)(228,299)(229,300)(230,291)(241,312)(242,313)(243,314)(244,315)(245,316)(246,317)(247,318)(248,319)(249,320)(250,311), (1,43,28,37)(2,44,29,38)(3,45,30,39)(4,46,21,40)(5,47,22,31)(6,48,23,32)(7,49,24,33)(8,50,25,34)(9,41,26,35)(10,42,27,36)(11,298,313,307)(12,299,314,308)(13,300,315,309)(14,291,316,310)(15,292,317,301)(16,293,318,302)(17,294,319,303)(18,295,320,304)(19,296,311,305)(20,297,312,306)(51,76,67,82)(52,77,68,83)(53,78,69,84)(54,79,70,85)(55,80,61,86)(56,71,62,87)(57,72,63,88)(58,73,64,89)(59,74,65,90)(60,75,66,81)(91,122,107,116)(92,123,108,117)(93,124,109,118)(94,125,110,119)(95,126,101,120)(96,127,102,111)(97,128,103,112)(98,129,104,113)(99,130,105,114)(100,121,106,115)(131,156,147,169)(132,157,148,170)(133,158,149,161)(134,159,150,162)(135,160,141,163)(136,151,142,164)(137,152,143,165)(138,153,144,166)(139,154,145,167)(140,155,146,168)(171,205,190,196)(172,206,181,197)(173,207,182,198)(174,208,183,199)(175,209,184,200)(176,210,185,191)(177,201,186,192)(178,202,187,193)(179,203,188,194)(180,204,189,195)(211,236,230,245)(212,237,221,246)(213,238,222,247)(214,239,223,248)(215,240,224,249)(216,231,225,250)(217,232,226,241)(218,233,227,242)(219,234,228,243)(220,235,229,244)(251,285,270,276)(252,286,261,277)(253,287,262,278)(254,288,263,279)(255,289,264,280)(256,290,265,271)(257,281,266,272)(258,282,267,273)(259,283,268,274)(260,284,269,275), (1,63,28,57)(2,64,29,58)(3,65,30,59)(4,66,21,60)(5,67,22,51)(6,68,23,52)(7,69,24,53)(8,70,25,54)(9,61,26,55)(10,62,27,56)(11,287,313,278)(12,288,314,279)(13,289,315,280)(14,290,316,271)(15,281,317,272)(16,282,318,273)(17,283,319,274)(18,284,320,275)(19,285,311,276)(20,286,312,277)(31,82,47,76)(32,83,48,77)(33,84,49,78)(34,85,50,79)(35,86,41,80)(36,87,42,71)(37,88,43,72)(38,89,44,73)(39,90,45,74)(40,81,46,75)(91,142,107,136)(92,143,108,137)(93,144,109,138)(94,145,110,139)(95,146,101,140)(96,147,102,131)(97,148,103,132)(98,149,104,133)(99,150,105,134)(100,141,106,135)(111,169,127,156)(112,170,128,157)(113,161,129,158)(114,162,130,159)(115,163,121,160)(116,164,122,151)(117,165,123,152)(118,166,124,153)(119,167,125,154)(120,168,126,155)(171,225,190,216)(172,226,181,217)(173,227,182,218)(174,228,183,219)(175,229,184,220)(176,230,185,211)(177,221,186,212)(178,222,187,213)(179,223,188,214)(180,224,189,215)(191,245,210,236)(192,246,201,237)(193,247,202,238)(194,248,203,239)(195,249,204,240)(196,250,205,231)(197,241,206,232)(198,242,207,233)(199,243,208,234)(200,244,209,235)(251,305,270,296)(252,306,261,297)(253,307,262,298)(254,308,263,299)(255,309,264,300)(256,310,265,291)(257,301,266,292)(258,302,267,293)(259,303,268,294)(260,304,269,295), (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), (1,253,6,258)(2,252,7,257)(3,251,8,256)(4,260,9,255)(5,259,10,254)(11,83,16,88)(12,82,17,87)(13,81,18,86)(14,90,19,85)(15,89,20,84)(21,269,26,264)(22,268,27,263)(23,267,28,262)(24,266,29,261)(25,265,30,270)(31,274,36,279)(32,273,37,278)(33,272,38,277)(34,271,39,276)(35,280,40,275)(41,289,46,284)(42,288,47,283)(43,287,48,282)(44,286,49,281)(45,285,50,290)(51,294,56,299)(52,293,57,298)(53,292,58,297)(54,291,59,296)(55,300,60,295)(61,309,66,304)(62,308,67,303)(63,307,68,302)(64,306,69,301)(65,305,70,310)(71,314,76,319)(72,313,77,318)(73,312,78,317)(74,311,79,316)(75,320,80,315)(91,174,96,179)(92,173,97,178)(93,172,98,177)(94,171,99,176)(95,180,100,175)(101,189,106,184)(102,188,107,183)(103,187,108,182)(104,186,109,181)(105,185,110,190)(111,194,116,199)(112,193,117,198)(113,192,118,197)(114,191,119,196)(115,200,120,195)(121,209,126,204)(122,208,127,203)(123,207,128,202)(124,206,129,201)(125,205,130,210)(131,214,136,219)(132,213,137,218)(133,212,138,217)(134,211,139,216)(135,220,140,215)(141,229,146,224)(142,228,147,223)(143,227,148,222)(144,226,149,221)(145,225,150,230)(151,234,156,239)(152,233,157,238)(153,232,158,237)(154,231,159,236)(155,240,160,235)(161,246,166,241)(162,245,167,250)(163,244,168,249)(164,243,169,248)(165,242,170,247)>;
G:=Group( (1,108)(2,109)(3,110)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,233)(12,234)(13,235)(14,236)(15,237)(16,238)(17,239)(18,240)(19,231)(20,232)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,91)(28,92)(29,93)(30,94)(31,127)(32,128)(33,129)(34,130)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,111)(48,112)(49,113)(50,114)(51,147)(52,148)(53,149)(54,150)(55,141)(56,142)(57,143)(58,144)(59,145)(60,146)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,131)(68,132)(69,133)(70,134)(71,164)(72,165)(73,166)(74,167)(75,168)(76,169)(77,170)(78,161)(79,162)(80,163)(81,155)(82,156)(83,157)(84,158)(85,159)(86,160)(87,151)(88,152)(89,153)(90,154)(171,270)(172,261)(173,262)(174,263)(175,264)(176,265)(177,266)(178,267)(179,268)(180,269)(181,252)(182,253)(183,254)(184,255)(185,256)(186,257)(187,258)(188,259)(189,260)(190,251)(191,290)(192,281)(193,282)(194,283)(195,284)(196,285)(197,286)(198,287)(199,288)(200,289)(201,272)(202,273)(203,274)(204,275)(205,276)(206,277)(207,278)(208,279)(209,280)(210,271)(211,310)(212,301)(213,302)(214,303)(215,304)(216,305)(217,306)(218,307)(219,308)(220,309)(221,292)(222,293)(223,294)(224,295)(225,296)(226,297)(227,298)(228,299)(229,300)(230,291)(241,312)(242,313)(243,314)(244,315)(245,316)(246,317)(247,318)(248,319)(249,320)(250,311), (1,43,28,37)(2,44,29,38)(3,45,30,39)(4,46,21,40)(5,47,22,31)(6,48,23,32)(7,49,24,33)(8,50,25,34)(9,41,26,35)(10,42,27,36)(11,298,313,307)(12,299,314,308)(13,300,315,309)(14,291,316,310)(15,292,317,301)(16,293,318,302)(17,294,319,303)(18,295,320,304)(19,296,311,305)(20,297,312,306)(51,76,67,82)(52,77,68,83)(53,78,69,84)(54,79,70,85)(55,80,61,86)(56,71,62,87)(57,72,63,88)(58,73,64,89)(59,74,65,90)(60,75,66,81)(91,122,107,116)(92,123,108,117)(93,124,109,118)(94,125,110,119)(95,126,101,120)(96,127,102,111)(97,128,103,112)(98,129,104,113)(99,130,105,114)(100,121,106,115)(131,156,147,169)(132,157,148,170)(133,158,149,161)(134,159,150,162)(135,160,141,163)(136,151,142,164)(137,152,143,165)(138,153,144,166)(139,154,145,167)(140,155,146,168)(171,205,190,196)(172,206,181,197)(173,207,182,198)(174,208,183,199)(175,209,184,200)(176,210,185,191)(177,201,186,192)(178,202,187,193)(179,203,188,194)(180,204,189,195)(211,236,230,245)(212,237,221,246)(213,238,222,247)(214,239,223,248)(215,240,224,249)(216,231,225,250)(217,232,226,241)(218,233,227,242)(219,234,228,243)(220,235,229,244)(251,285,270,276)(252,286,261,277)(253,287,262,278)(254,288,263,279)(255,289,264,280)(256,290,265,271)(257,281,266,272)(258,282,267,273)(259,283,268,274)(260,284,269,275), (1,63,28,57)(2,64,29,58)(3,65,30,59)(4,66,21,60)(5,67,22,51)(6,68,23,52)(7,69,24,53)(8,70,25,54)(9,61,26,55)(10,62,27,56)(11,287,313,278)(12,288,314,279)(13,289,315,280)(14,290,316,271)(15,281,317,272)(16,282,318,273)(17,283,319,274)(18,284,320,275)(19,285,311,276)(20,286,312,277)(31,82,47,76)(32,83,48,77)(33,84,49,78)(34,85,50,79)(35,86,41,80)(36,87,42,71)(37,88,43,72)(38,89,44,73)(39,90,45,74)(40,81,46,75)(91,142,107,136)(92,143,108,137)(93,144,109,138)(94,145,110,139)(95,146,101,140)(96,147,102,131)(97,148,103,132)(98,149,104,133)(99,150,105,134)(100,141,106,135)(111,169,127,156)(112,170,128,157)(113,161,129,158)(114,162,130,159)(115,163,121,160)(116,164,122,151)(117,165,123,152)(118,166,124,153)(119,167,125,154)(120,168,126,155)(171,225,190,216)(172,226,181,217)(173,227,182,218)(174,228,183,219)(175,229,184,220)(176,230,185,211)(177,221,186,212)(178,222,187,213)(179,223,188,214)(180,224,189,215)(191,245,210,236)(192,246,201,237)(193,247,202,238)(194,248,203,239)(195,249,204,240)(196,250,205,231)(197,241,206,232)(198,242,207,233)(199,243,208,234)(200,244,209,235)(251,305,270,296)(252,306,261,297)(253,307,262,298)(254,308,263,299)(255,309,264,300)(256,310,265,291)(257,301,266,292)(258,302,267,293)(259,303,268,294)(260,304,269,295), (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), (1,253,6,258)(2,252,7,257)(3,251,8,256)(4,260,9,255)(5,259,10,254)(11,83,16,88)(12,82,17,87)(13,81,18,86)(14,90,19,85)(15,89,20,84)(21,269,26,264)(22,268,27,263)(23,267,28,262)(24,266,29,261)(25,265,30,270)(31,274,36,279)(32,273,37,278)(33,272,38,277)(34,271,39,276)(35,280,40,275)(41,289,46,284)(42,288,47,283)(43,287,48,282)(44,286,49,281)(45,285,50,290)(51,294,56,299)(52,293,57,298)(53,292,58,297)(54,291,59,296)(55,300,60,295)(61,309,66,304)(62,308,67,303)(63,307,68,302)(64,306,69,301)(65,305,70,310)(71,314,76,319)(72,313,77,318)(73,312,78,317)(74,311,79,316)(75,320,80,315)(91,174,96,179)(92,173,97,178)(93,172,98,177)(94,171,99,176)(95,180,100,175)(101,189,106,184)(102,188,107,183)(103,187,108,182)(104,186,109,181)(105,185,110,190)(111,194,116,199)(112,193,117,198)(113,192,118,197)(114,191,119,196)(115,200,120,195)(121,209,126,204)(122,208,127,203)(123,207,128,202)(124,206,129,201)(125,205,130,210)(131,214,136,219)(132,213,137,218)(133,212,138,217)(134,211,139,216)(135,220,140,215)(141,229,146,224)(142,228,147,223)(143,227,148,222)(144,226,149,221)(145,225,150,230)(151,234,156,239)(152,233,157,238)(153,232,158,237)(154,231,159,236)(155,240,160,235)(161,246,166,241)(162,245,167,250)(163,244,168,249)(164,243,169,248)(165,242,170,247) );
G=PermutationGroup([(1,108),(2,109),(3,110),(4,101),(5,102),(6,103),(7,104),(8,105),(9,106),(10,107),(11,233),(12,234),(13,235),(14,236),(15,237),(16,238),(17,239),(18,240),(19,231),(20,232),(21,95),(22,96),(23,97),(24,98),(25,99),(26,100),(27,91),(28,92),(29,93),(30,94),(31,127),(32,128),(33,129),(34,130),(35,121),(36,122),(37,123),(38,124),(39,125),(40,126),(41,115),(42,116),(43,117),(44,118),(45,119),(46,120),(47,111),(48,112),(49,113),(50,114),(51,147),(52,148),(53,149),(54,150),(55,141),(56,142),(57,143),(58,144),(59,145),(60,146),(61,135),(62,136),(63,137),(64,138),(65,139),(66,140),(67,131),(68,132),(69,133),(70,134),(71,164),(72,165),(73,166),(74,167),(75,168),(76,169),(77,170),(78,161),(79,162),(80,163),(81,155),(82,156),(83,157),(84,158),(85,159),(86,160),(87,151),(88,152),(89,153),(90,154),(171,270),(172,261),(173,262),(174,263),(175,264),(176,265),(177,266),(178,267),(179,268),(180,269),(181,252),(182,253),(183,254),(184,255),(185,256),(186,257),(187,258),(188,259),(189,260),(190,251),(191,290),(192,281),(193,282),(194,283),(195,284),(196,285),(197,286),(198,287),(199,288),(200,289),(201,272),(202,273),(203,274),(204,275),(205,276),(206,277),(207,278),(208,279),(209,280),(210,271),(211,310),(212,301),(213,302),(214,303),(215,304),(216,305),(217,306),(218,307),(219,308),(220,309),(221,292),(222,293),(223,294),(224,295),(225,296),(226,297),(227,298),(228,299),(229,300),(230,291),(241,312),(242,313),(243,314),(244,315),(245,316),(246,317),(247,318),(248,319),(249,320),(250,311)], [(1,43,28,37),(2,44,29,38),(3,45,30,39),(4,46,21,40),(5,47,22,31),(6,48,23,32),(7,49,24,33),(8,50,25,34),(9,41,26,35),(10,42,27,36),(11,298,313,307),(12,299,314,308),(13,300,315,309),(14,291,316,310),(15,292,317,301),(16,293,318,302),(17,294,319,303),(18,295,320,304),(19,296,311,305),(20,297,312,306),(51,76,67,82),(52,77,68,83),(53,78,69,84),(54,79,70,85),(55,80,61,86),(56,71,62,87),(57,72,63,88),(58,73,64,89),(59,74,65,90),(60,75,66,81),(91,122,107,116),(92,123,108,117),(93,124,109,118),(94,125,110,119),(95,126,101,120),(96,127,102,111),(97,128,103,112),(98,129,104,113),(99,130,105,114),(100,121,106,115),(131,156,147,169),(132,157,148,170),(133,158,149,161),(134,159,150,162),(135,160,141,163),(136,151,142,164),(137,152,143,165),(138,153,144,166),(139,154,145,167),(140,155,146,168),(171,205,190,196),(172,206,181,197),(173,207,182,198),(174,208,183,199),(175,209,184,200),(176,210,185,191),(177,201,186,192),(178,202,187,193),(179,203,188,194),(180,204,189,195),(211,236,230,245),(212,237,221,246),(213,238,222,247),(214,239,223,248),(215,240,224,249),(216,231,225,250),(217,232,226,241),(218,233,227,242),(219,234,228,243),(220,235,229,244),(251,285,270,276),(252,286,261,277),(253,287,262,278),(254,288,263,279),(255,289,264,280),(256,290,265,271),(257,281,266,272),(258,282,267,273),(259,283,268,274),(260,284,269,275)], [(1,63,28,57),(2,64,29,58),(3,65,30,59),(4,66,21,60),(5,67,22,51),(6,68,23,52),(7,69,24,53),(8,70,25,54),(9,61,26,55),(10,62,27,56),(11,287,313,278),(12,288,314,279),(13,289,315,280),(14,290,316,271),(15,281,317,272),(16,282,318,273),(17,283,319,274),(18,284,320,275),(19,285,311,276),(20,286,312,277),(31,82,47,76),(32,83,48,77),(33,84,49,78),(34,85,50,79),(35,86,41,80),(36,87,42,71),(37,88,43,72),(38,89,44,73),(39,90,45,74),(40,81,46,75),(91,142,107,136),(92,143,108,137),(93,144,109,138),(94,145,110,139),(95,146,101,140),(96,147,102,131),(97,148,103,132),(98,149,104,133),(99,150,105,134),(100,141,106,135),(111,169,127,156),(112,170,128,157),(113,161,129,158),(114,162,130,159),(115,163,121,160),(116,164,122,151),(117,165,123,152),(118,166,124,153),(119,167,125,154),(120,168,126,155),(171,225,190,216),(172,226,181,217),(173,227,182,218),(174,228,183,219),(175,229,184,220),(176,230,185,211),(177,221,186,212),(178,222,187,213),(179,223,188,214),(180,224,189,215),(191,245,210,236),(192,246,201,237),(193,247,202,238),(194,248,203,239),(195,249,204,240),(196,250,205,231),(197,241,206,232),(198,242,207,233),(199,243,208,234),(200,244,209,235),(251,305,270,296),(252,306,261,297),(253,307,262,298),(254,308,263,299),(255,309,264,300),(256,310,265,291),(257,301,266,292),(258,302,267,293),(259,303,268,294),(260,304,269,295)], [(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)], [(1,253,6,258),(2,252,7,257),(3,251,8,256),(4,260,9,255),(5,259,10,254),(11,83,16,88),(12,82,17,87),(13,81,18,86),(14,90,19,85),(15,89,20,84),(21,269,26,264),(22,268,27,263),(23,267,28,262),(24,266,29,261),(25,265,30,270),(31,274,36,279),(32,273,37,278),(33,272,38,277),(34,271,39,276),(35,280,40,275),(41,289,46,284),(42,288,47,283),(43,287,48,282),(44,286,49,281),(45,285,50,290),(51,294,56,299),(52,293,57,298),(53,292,58,297),(54,291,59,296),(55,300,60,295),(61,309,66,304),(62,308,67,303),(63,307,68,302),(64,306,69,301),(65,305,70,310),(71,314,76,319),(72,313,77,318),(73,312,78,317),(74,311,79,316),(75,320,80,315),(91,174,96,179),(92,173,97,178),(93,172,98,177),(94,171,99,176),(95,180,100,175),(101,189,106,184),(102,188,107,183),(103,187,108,182),(104,186,109,181),(105,185,110,190),(111,194,116,199),(112,193,117,198),(113,192,118,197),(114,191,119,196),(115,200,120,195),(121,209,126,204),(122,208,127,203),(123,207,128,202),(124,206,129,201),(125,205,130,210),(131,214,136,219),(132,213,137,218),(133,212,138,217),(134,211,139,216),(135,220,140,215),(141,229,146,224),(142,228,147,223),(143,227,148,222),(144,226,149,221),(145,225,150,230),(151,234,156,239),(152,233,157,238),(153,232,158,237),(154,231,159,236),(155,240,160,235),(161,246,166,241),(162,245,167,250),(163,244,168,249),(164,243,169,248),(165,242,170,247)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 7 | 40 |
| 0 | 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 23 | 16 |
| 0 | 0 | 0 | 13 | 18 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,9,0,0,0,0,0,32,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,7,1,0,0,0,40,0],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,23,13,0,0,0,16,18] >;
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 4U | ··· | 4AF | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | Q8 | D5 | C4○D4 | D10 | Dic5 | D10 | Q8×D5 | Q8⋊2D5 |
| kernel | C2×Q8×Dic5 | C2×C4×Dic5 | C2×C4⋊Dic5 | Q8×Dic5 | Q8×C2×C10 | Q8×C10 | C2×Dic5 | C22×Q8 | C2×C10 | C22×C4 | C2×Q8 | C2×Q8 | C22 | C22 |
| # reps | 1 | 3 | 3 | 8 | 1 | 16 | 4 | 2 | 4 | 6 | 16 | 8 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_2\times Q_8\times Dic_5
% in TeX
G:=Group("C2xQ8xDic5"); // GroupNames label
G:=SmallGroup(320,1483);
// by ID
G=gap.SmallGroup(320,1483);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^10=1,c^2=b^2,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀