direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×C5⋊C16, (C2×C10)⋊2C16, C10⋊2(C2×C16), (C2×C20).9C8, C5⋊2(C22×C16), C20.49(C2×C8), C23.4(C5⋊C8), (C22×C10).5C8, (C22×C4).26F5, C4.54(C22×F5), C20.94(C22×C4), (C22×C20).31C4, C10.16(C22×C8), C5⋊2C8.39C23, C4.16(C2×C5⋊C8), (C2×C4).9(C5⋊C8), C2.1(C22×C5⋊C8), C22.12(C2×C5⋊C8), (C2×C5⋊2C8).36C4, (C2×C10).31(C2×C8), C5⋊2C8.55(C2×C4), (C2×C4).167(C2×F5), (C2×C20).176(C2×C4), (C22×C5⋊2C8).23C2, (C2×C5⋊2C8).350C22, SmallGroup(320,1080)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C22×C5⋊C16 |
Subgroups: 186 in 98 conjugacy classes, 76 normal (13 characteristic)
C1, C2, C2 [×6], C4, C4 [×3], C22 [×7], C5, C8 [×4], C2×C4 [×6], C23, C10, C10 [×6], C16 [×4], C2×C8 [×6], C22×C4, C20, C20 [×3], C2×C10 [×7], C2×C16 [×6], C22×C8, C5⋊2C8, C5⋊2C8 [×3], C2×C20 [×6], C22×C10, C22×C16, C5⋊C16 [×4], C2×C5⋊2C8 [×6], C22×C20, C2×C5⋊C16 [×6], C22×C5⋊2C8, C22×C5⋊C16
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C16 [×4], C2×C8 [×6], C22×C4, F5, C2×C16 [×6], C22×C8, C5⋊C8 [×4], C2×F5 [×3], C22×C16, C5⋊C16 [×4], C2×C5⋊C8 [×6], C22×F5, C2×C5⋊C16 [×6], C22×C5⋊C8, C22×C5⋊C16
Generators and relations
G = < a,b,c,d | a2=b2=c5=d16=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
►Regular action on 320 points
Generators in S320
(1 202)(2 203)(3 204)(4 205)(5 206)(6 207)(7 208)(8 193)(9 194)(10 195)(11 196)(12 197)(13 198)(14 199)(15 200)(16 201)(17 161)(18 162)(19 163)(20 164)(21 165)(22 166)(23 167)(24 168)(25 169)(26 170)(27 171)(28 172)(29 173)(30 174)(31 175)(32 176)(33 318)(34 319)(35 320)(36 305)(37 306)(38 307)(39 308)(40 309)(41 310)(42 311)(43 312)(44 313)(45 314)(46 315)(47 316)(48 317)(49 136)(50 137)(51 138)(52 139)(53 140)(54 141)(55 142)(56 143)(57 144)(58 129)(59 130)(60 131)(61 132)(62 133)(63 134)(64 135)(65 158)(66 159)(67 160)(68 145)(69 146)(70 147)(71 148)(72 149)(73 150)(74 151)(75 152)(76 153)(77 154)(78 155)(79 156)(80 157)(81 247)(82 248)(83 249)(84 250)(85 251)(86 252)(87 253)(88 254)(89 255)(90 256)(91 241)(92 242)(93 243)(94 244)(95 245)(96 246)(97 126)(98 127)(99 128)(100 113)(101 114)(102 115)(103 116)(104 117)(105 118)(106 119)(107 120)(108 121)(109 122)(110 123)(111 124)(112 125)(177 229)(178 230)(179 231)(180 232)(181 233)(182 234)(183 235)(184 236)(185 237)(186 238)(187 239)(188 240)(189 225)(190 226)(191 227)(192 228)(209 296)(210 297)(211 298)(212 299)(213 300)(214 301)(215 302)(216 303)(217 304)(218 289)(219 290)(220 291)(221 292)(222 293)(223 294)(224 295)(257 281)(258 282)(259 283)(260 284)(261 285)(262 286)(263 287)(264 288)(265 273)(266 274)(267 275)(268 276)(269 277)(270 278)(271 279)(272 280)
(1 305)(2 306)(3 307)(4 308)(5 309)(6 310)(7 311)(8 312)(9 313)(10 314)(11 315)(12 316)(13 317)(14 318)(15 319)(16 320)(17 154)(18 155)(19 156)(20 157)(21 158)(22 159)(23 160)(24 145)(25 146)(26 147)(27 148)(28 149)(29 150)(30 151)(31 152)(32 153)(33 199)(34 200)(35 201)(36 202)(37 203)(38 204)(39 205)(40 206)(41 207)(42 208)(43 193)(44 194)(45 195)(46 196)(47 197)(48 198)(49 229)(50 230)(51 231)(52 232)(53 233)(54 234)(55 235)(56 236)(57 237)(58 238)(59 239)(60 240)(61 225)(62 226)(63 227)(64 228)(65 165)(66 166)(67 167)(68 168)(69 169)(70 170)(71 171)(72 172)(73 173)(74 174)(75 175)(76 176)(77 161)(78 162)(79 163)(80 164)(81 122)(82 123)(83 124)(84 125)(85 126)(86 127)(87 128)(88 113)(89 114)(90 115)(91 116)(92 117)(93 118)(94 119)(95 120)(96 121)(97 251)(98 252)(99 253)(100 254)(101 255)(102 256)(103 241)(104 242)(105 243)(106 244)(107 245)(108 246)(109 247)(110 248)(111 249)(112 250)(129 186)(130 187)(131 188)(132 189)(133 190)(134 191)(135 192)(136 177)(137 178)(138 179)(139 180)(140 181)(141 182)(142 183)(143 184)(144 185)(209 263)(210 264)(211 265)(212 266)(213 267)(214 268)(215 269)(216 270)(217 271)(218 272)(219 257)(220 258)(221 259)(222 260)(223 261)(224 262)(273 298)(274 299)(275 300)(276 301)(277 302)(278 303)(279 304)(280 289)(281 290)(282 291)(283 292)(284 293)(285 294)(286 295)(287 296)(288 297)
(1 175 228 243 296)(2 244 176 297 229)(3 298 245 230 161)(4 231 299 162 246)(5 163 232 247 300)(6 248 164 301 233)(7 302 249 234 165)(8 235 303 166 250)(9 167 236 251 304)(10 252 168 289 237)(11 290 253 238 169)(12 239 291 170 254)(13 171 240 255 292)(14 256 172 293 225)(15 294 241 226 173)(16 227 295 174 242)(17 204 211 95 178)(18 96 205 179 212)(19 180 81 213 206)(20 214 181 207 82)(21 208 215 83 182)(22 84 193 183 216)(23 184 85 217 194)(24 218 185 195 86)(25 196 219 87 186)(26 88 197 187 220)(27 188 89 221 198)(28 222 189 199 90)(29 200 223 91 190)(30 92 201 191 224)(31 192 93 209 202)(32 210 177 203 94)(33 115 149 260 132)(34 261 116 133 150)(35 134 262 151 117)(36 152 135 118 263)(37 119 153 264 136)(38 265 120 137 154)(39 138 266 155 121)(40 156 139 122 267)(41 123 157 268 140)(42 269 124 141 158)(43 142 270 159 125)(44 160 143 126 271)(45 127 145 272 144)(46 257 128 129 146)(47 130 258 147 113)(48 148 131 114 259)(49 306 106 76 288)(50 77 307 273 107)(51 274 78 108 308)(52 109 275 309 79)(53 310 110 80 276)(54 65 311 277 111)(55 278 66 112 312)(56 97 279 313 67)(57 314 98 68 280)(58 69 315 281 99)(59 282 70 100 316)(60 101 283 317 71)(61 318 102 72 284)(62 73 319 285 103)(63 286 74 104 320)(64 105 287 305 75)
(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)
G:=sub<Sym(320)| (1,202)(2,203)(3,204)(4,205)(5,206)(6,207)(7,208)(8,193)(9,194)(10,195)(11,196)(12,197)(13,198)(14,199)(15,200)(16,201)(17,161)(18,162)(19,163)(20,164)(21,165)(22,166)(23,167)(24,168)(25,169)(26,170)(27,171)(28,172)(29,173)(30,174)(31,175)(32,176)(33,318)(34,319)(35,320)(36,305)(37,306)(38,307)(39,308)(40,309)(41,310)(42,311)(43,312)(44,313)(45,314)(46,315)(47,316)(48,317)(49,136)(50,137)(51,138)(52,139)(53,140)(54,141)(55,142)(56,143)(57,144)(58,129)(59,130)(60,131)(61,132)(62,133)(63,134)(64,135)(65,158)(66,159)(67,160)(68,145)(69,146)(70,147)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,155)(79,156)(80,157)(81,247)(82,248)(83,249)(84,250)(85,251)(86,252)(87,253)(88,254)(89,255)(90,256)(91,241)(92,242)(93,243)(94,244)(95,245)(96,246)(97,126)(98,127)(99,128)(100,113)(101,114)(102,115)(103,116)(104,117)(105,118)(106,119)(107,120)(108,121)(109,122)(110,123)(111,124)(112,125)(177,229)(178,230)(179,231)(180,232)(181,233)(182,234)(183,235)(184,236)(185,237)(186,238)(187,239)(188,240)(189,225)(190,226)(191,227)(192,228)(209,296)(210,297)(211,298)(212,299)(213,300)(214,301)(215,302)(216,303)(217,304)(218,289)(219,290)(220,291)(221,292)(222,293)(223,294)(224,295)(257,281)(258,282)(259,283)(260,284)(261,285)(262,286)(263,287)(264,288)(265,273)(266,274)(267,275)(268,276)(269,277)(270,278)(271,279)(272,280), (1,305)(2,306)(3,307)(4,308)(5,309)(6,310)(7,311)(8,312)(9,313)(10,314)(11,315)(12,316)(13,317)(14,318)(15,319)(16,320)(17,154)(18,155)(19,156)(20,157)(21,158)(22,159)(23,160)(24,145)(25,146)(26,147)(27,148)(28,149)(29,150)(30,151)(31,152)(32,153)(33,199)(34,200)(35,201)(36,202)(37,203)(38,204)(39,205)(40,206)(41,207)(42,208)(43,193)(44,194)(45,195)(46,196)(47,197)(48,198)(49,229)(50,230)(51,231)(52,232)(53,233)(54,234)(55,235)(56,236)(57,237)(58,238)(59,239)(60,240)(61,225)(62,226)(63,227)(64,228)(65,165)(66,166)(67,167)(68,168)(69,169)(70,170)(71,171)(72,172)(73,173)(74,174)(75,175)(76,176)(77,161)(78,162)(79,163)(80,164)(81,122)(82,123)(83,124)(84,125)(85,126)(86,127)(87,128)(88,113)(89,114)(90,115)(91,116)(92,117)(93,118)(94,119)(95,120)(96,121)(97,251)(98,252)(99,253)(100,254)(101,255)(102,256)(103,241)(104,242)(105,243)(106,244)(107,245)(108,246)(109,247)(110,248)(111,249)(112,250)(129,186)(130,187)(131,188)(132,189)(133,190)(134,191)(135,192)(136,177)(137,178)(138,179)(139,180)(140,181)(141,182)(142,183)(143,184)(144,185)(209,263)(210,264)(211,265)(212,266)(213,267)(214,268)(215,269)(216,270)(217,271)(218,272)(219,257)(220,258)(221,259)(222,260)(223,261)(224,262)(273,298)(274,299)(275,300)(276,301)(277,302)(278,303)(279,304)(280,289)(281,290)(282,291)(283,292)(284,293)(285,294)(286,295)(287,296)(288,297), (1,175,228,243,296)(2,244,176,297,229)(3,298,245,230,161)(4,231,299,162,246)(5,163,232,247,300)(6,248,164,301,233)(7,302,249,234,165)(8,235,303,166,250)(9,167,236,251,304)(10,252,168,289,237)(11,290,253,238,169)(12,239,291,170,254)(13,171,240,255,292)(14,256,172,293,225)(15,294,241,226,173)(16,227,295,174,242)(17,204,211,95,178)(18,96,205,179,212)(19,180,81,213,206)(20,214,181,207,82)(21,208,215,83,182)(22,84,193,183,216)(23,184,85,217,194)(24,218,185,195,86)(25,196,219,87,186)(26,88,197,187,220)(27,188,89,221,198)(28,222,189,199,90)(29,200,223,91,190)(30,92,201,191,224)(31,192,93,209,202)(32,210,177,203,94)(33,115,149,260,132)(34,261,116,133,150)(35,134,262,151,117)(36,152,135,118,263)(37,119,153,264,136)(38,265,120,137,154)(39,138,266,155,121)(40,156,139,122,267)(41,123,157,268,140)(42,269,124,141,158)(43,142,270,159,125)(44,160,143,126,271)(45,127,145,272,144)(46,257,128,129,146)(47,130,258,147,113)(48,148,131,114,259)(49,306,106,76,288)(50,77,307,273,107)(51,274,78,108,308)(52,109,275,309,79)(53,310,110,80,276)(54,65,311,277,111)(55,278,66,112,312)(56,97,279,313,67)(57,314,98,68,280)(58,69,315,281,99)(59,282,70,100,316)(60,101,283,317,71)(61,318,102,72,284)(62,73,319,285,103)(63,286,74,104,320)(64,105,287,305,75), (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)>;
G:=Group( (1,202)(2,203)(3,204)(4,205)(5,206)(6,207)(7,208)(8,193)(9,194)(10,195)(11,196)(12,197)(13,198)(14,199)(15,200)(16,201)(17,161)(18,162)(19,163)(20,164)(21,165)(22,166)(23,167)(24,168)(25,169)(26,170)(27,171)(28,172)(29,173)(30,174)(31,175)(32,176)(33,318)(34,319)(35,320)(36,305)(37,306)(38,307)(39,308)(40,309)(41,310)(42,311)(43,312)(44,313)(45,314)(46,315)(47,316)(48,317)(49,136)(50,137)(51,138)(52,139)(53,140)(54,141)(55,142)(56,143)(57,144)(58,129)(59,130)(60,131)(61,132)(62,133)(63,134)(64,135)(65,158)(66,159)(67,160)(68,145)(69,146)(70,147)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,155)(79,156)(80,157)(81,247)(82,248)(83,249)(84,250)(85,251)(86,252)(87,253)(88,254)(89,255)(90,256)(91,241)(92,242)(93,243)(94,244)(95,245)(96,246)(97,126)(98,127)(99,128)(100,113)(101,114)(102,115)(103,116)(104,117)(105,118)(106,119)(107,120)(108,121)(109,122)(110,123)(111,124)(112,125)(177,229)(178,230)(179,231)(180,232)(181,233)(182,234)(183,235)(184,236)(185,237)(186,238)(187,239)(188,240)(189,225)(190,226)(191,227)(192,228)(209,296)(210,297)(211,298)(212,299)(213,300)(214,301)(215,302)(216,303)(217,304)(218,289)(219,290)(220,291)(221,292)(222,293)(223,294)(224,295)(257,281)(258,282)(259,283)(260,284)(261,285)(262,286)(263,287)(264,288)(265,273)(266,274)(267,275)(268,276)(269,277)(270,278)(271,279)(272,280), (1,305)(2,306)(3,307)(4,308)(5,309)(6,310)(7,311)(8,312)(9,313)(10,314)(11,315)(12,316)(13,317)(14,318)(15,319)(16,320)(17,154)(18,155)(19,156)(20,157)(21,158)(22,159)(23,160)(24,145)(25,146)(26,147)(27,148)(28,149)(29,150)(30,151)(31,152)(32,153)(33,199)(34,200)(35,201)(36,202)(37,203)(38,204)(39,205)(40,206)(41,207)(42,208)(43,193)(44,194)(45,195)(46,196)(47,197)(48,198)(49,229)(50,230)(51,231)(52,232)(53,233)(54,234)(55,235)(56,236)(57,237)(58,238)(59,239)(60,240)(61,225)(62,226)(63,227)(64,228)(65,165)(66,166)(67,167)(68,168)(69,169)(70,170)(71,171)(72,172)(73,173)(74,174)(75,175)(76,176)(77,161)(78,162)(79,163)(80,164)(81,122)(82,123)(83,124)(84,125)(85,126)(86,127)(87,128)(88,113)(89,114)(90,115)(91,116)(92,117)(93,118)(94,119)(95,120)(96,121)(97,251)(98,252)(99,253)(100,254)(101,255)(102,256)(103,241)(104,242)(105,243)(106,244)(107,245)(108,246)(109,247)(110,248)(111,249)(112,250)(129,186)(130,187)(131,188)(132,189)(133,190)(134,191)(135,192)(136,177)(137,178)(138,179)(139,180)(140,181)(141,182)(142,183)(143,184)(144,185)(209,263)(210,264)(211,265)(212,266)(213,267)(214,268)(215,269)(216,270)(217,271)(218,272)(219,257)(220,258)(221,259)(222,260)(223,261)(224,262)(273,298)(274,299)(275,300)(276,301)(277,302)(278,303)(279,304)(280,289)(281,290)(282,291)(283,292)(284,293)(285,294)(286,295)(287,296)(288,297), (1,175,228,243,296)(2,244,176,297,229)(3,298,245,230,161)(4,231,299,162,246)(5,163,232,247,300)(6,248,164,301,233)(7,302,249,234,165)(8,235,303,166,250)(9,167,236,251,304)(10,252,168,289,237)(11,290,253,238,169)(12,239,291,170,254)(13,171,240,255,292)(14,256,172,293,225)(15,294,241,226,173)(16,227,295,174,242)(17,204,211,95,178)(18,96,205,179,212)(19,180,81,213,206)(20,214,181,207,82)(21,208,215,83,182)(22,84,193,183,216)(23,184,85,217,194)(24,218,185,195,86)(25,196,219,87,186)(26,88,197,187,220)(27,188,89,221,198)(28,222,189,199,90)(29,200,223,91,190)(30,92,201,191,224)(31,192,93,209,202)(32,210,177,203,94)(33,115,149,260,132)(34,261,116,133,150)(35,134,262,151,117)(36,152,135,118,263)(37,119,153,264,136)(38,265,120,137,154)(39,138,266,155,121)(40,156,139,122,267)(41,123,157,268,140)(42,269,124,141,158)(43,142,270,159,125)(44,160,143,126,271)(45,127,145,272,144)(46,257,128,129,146)(47,130,258,147,113)(48,148,131,114,259)(49,306,106,76,288)(50,77,307,273,107)(51,274,78,108,308)(52,109,275,309,79)(53,310,110,80,276)(54,65,311,277,111)(55,278,66,112,312)(56,97,279,313,67)(57,314,98,68,280)(58,69,315,281,99)(59,282,70,100,316)(60,101,283,317,71)(61,318,102,72,284)(62,73,319,285,103)(63,286,74,104,320)(64,105,287,305,75), (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) );
G=PermutationGroup([(1,202),(2,203),(3,204),(4,205),(5,206),(6,207),(7,208),(8,193),(9,194),(10,195),(11,196),(12,197),(13,198),(14,199),(15,200),(16,201),(17,161),(18,162),(19,163),(20,164),(21,165),(22,166),(23,167),(24,168),(25,169),(26,170),(27,171),(28,172),(29,173),(30,174),(31,175),(32,176),(33,318),(34,319),(35,320),(36,305),(37,306),(38,307),(39,308),(40,309),(41,310),(42,311),(43,312),(44,313),(45,314),(46,315),(47,316),(48,317),(49,136),(50,137),(51,138),(52,139),(53,140),(54,141),(55,142),(56,143),(57,144),(58,129),(59,130),(60,131),(61,132),(62,133),(63,134),(64,135),(65,158),(66,159),(67,160),(68,145),(69,146),(70,147),(71,148),(72,149),(73,150),(74,151),(75,152),(76,153),(77,154),(78,155),(79,156),(80,157),(81,247),(82,248),(83,249),(84,250),(85,251),(86,252),(87,253),(88,254),(89,255),(90,256),(91,241),(92,242),(93,243),(94,244),(95,245),(96,246),(97,126),(98,127),(99,128),(100,113),(101,114),(102,115),(103,116),(104,117),(105,118),(106,119),(107,120),(108,121),(109,122),(110,123),(111,124),(112,125),(177,229),(178,230),(179,231),(180,232),(181,233),(182,234),(183,235),(184,236),(185,237),(186,238),(187,239),(188,240),(189,225),(190,226),(191,227),(192,228),(209,296),(210,297),(211,298),(212,299),(213,300),(214,301),(215,302),(216,303),(217,304),(218,289),(219,290),(220,291),(221,292),(222,293),(223,294),(224,295),(257,281),(258,282),(259,283),(260,284),(261,285),(262,286),(263,287),(264,288),(265,273),(266,274),(267,275),(268,276),(269,277),(270,278),(271,279),(272,280)], [(1,305),(2,306),(3,307),(4,308),(5,309),(6,310),(7,311),(8,312),(9,313),(10,314),(11,315),(12,316),(13,317),(14,318),(15,319),(16,320),(17,154),(18,155),(19,156),(20,157),(21,158),(22,159),(23,160),(24,145),(25,146),(26,147),(27,148),(28,149),(29,150),(30,151),(31,152),(32,153),(33,199),(34,200),(35,201),(36,202),(37,203),(38,204),(39,205),(40,206),(41,207),(42,208),(43,193),(44,194),(45,195),(46,196),(47,197),(48,198),(49,229),(50,230),(51,231),(52,232),(53,233),(54,234),(55,235),(56,236),(57,237),(58,238),(59,239),(60,240),(61,225),(62,226),(63,227),(64,228),(65,165),(66,166),(67,167),(68,168),(69,169),(70,170),(71,171),(72,172),(73,173),(74,174),(75,175),(76,176),(77,161),(78,162),(79,163),(80,164),(81,122),(82,123),(83,124),(84,125),(85,126),(86,127),(87,128),(88,113),(89,114),(90,115),(91,116),(92,117),(93,118),(94,119),(95,120),(96,121),(97,251),(98,252),(99,253),(100,254),(101,255),(102,256),(103,241),(104,242),(105,243),(106,244),(107,245),(108,246),(109,247),(110,248),(111,249),(112,250),(129,186),(130,187),(131,188),(132,189),(133,190),(134,191),(135,192),(136,177),(137,178),(138,179),(139,180),(140,181),(141,182),(142,183),(143,184),(144,185),(209,263),(210,264),(211,265),(212,266),(213,267),(214,268),(215,269),(216,270),(217,271),(218,272),(219,257),(220,258),(221,259),(222,260),(223,261),(224,262),(273,298),(274,299),(275,300),(276,301),(277,302),(278,303),(279,304),(280,289),(281,290),(282,291),(283,292),(284,293),(285,294),(286,295),(287,296),(288,297)], [(1,175,228,243,296),(2,244,176,297,229),(3,298,245,230,161),(4,231,299,162,246),(5,163,232,247,300),(6,248,164,301,233),(7,302,249,234,165),(8,235,303,166,250),(9,167,236,251,304),(10,252,168,289,237),(11,290,253,238,169),(12,239,291,170,254),(13,171,240,255,292),(14,256,172,293,225),(15,294,241,226,173),(16,227,295,174,242),(17,204,211,95,178),(18,96,205,179,212),(19,180,81,213,206),(20,214,181,207,82),(21,208,215,83,182),(22,84,193,183,216),(23,184,85,217,194),(24,218,185,195,86),(25,196,219,87,186),(26,88,197,187,220),(27,188,89,221,198),(28,222,189,199,90),(29,200,223,91,190),(30,92,201,191,224),(31,192,93,209,202),(32,210,177,203,94),(33,115,149,260,132),(34,261,116,133,150),(35,134,262,151,117),(36,152,135,118,263),(37,119,153,264,136),(38,265,120,137,154),(39,138,266,155,121),(40,156,139,122,267),(41,123,157,268,140),(42,269,124,141,158),(43,142,270,159,125),(44,160,143,126,271),(45,127,145,272,144),(46,257,128,129,146),(47,130,258,147,113),(48,148,131,114,259),(49,306,106,76,288),(50,77,307,273,107),(51,274,78,108,308),(52,109,275,309,79),(53,310,110,80,276),(54,65,311,277,111),(55,278,66,112,312),(56,97,279,313,67),(57,314,98,68,280),(58,69,315,281,99),(59,282,70,100,316),(60,101,283,317,71),(61,318,102,72,284),(62,73,319,285,103),(63,286,74,104,320),(64,105,287,305,75)], [(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)])
Matrix representation ►G ⊆ GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 44 | 0 | 0 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 104 | 160 | 85 |
| 0 | 0 | 189 | 189 | 23 | 114 |
| 0 | 0 | 52 | 218 | 127 | 33 |
| 0 | 0 | 156 | 137 | 212 | 137 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,240,240,240,240],[44,0,0,0,0,0,0,177,0,0,0,0,0,0,29,189,52,156,0,0,104,189,218,137,0,0,160,23,127,212,0,0,85,114,33,137] >;
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 5 | 8A | ··· | 8P | 10A | ··· | 10G | 16A | ··· | 16AF | 20A | ··· | 20H |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | 5 | ··· | 5 | 4 | ··· | 4 | 5 | ··· | 5 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | F5 | C5⋊C8 | C2×F5 | C5⋊C8 | C5⋊C16 |
| kernel | C22×C5⋊C16 | C2×C5⋊C16 | C22×C5⋊2C8 | C2×C5⋊2C8 | C22×C20 | C2×C20 | C22×C10 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 6 | 1 | 6 | 2 | 12 | 4 | 32 | 1 | 3 | 3 | 1 | 8 |
In GAP, Magma, Sage, TeX
C_2^2\times C_5\rtimes C_{16} % in TeX
G:=Group("C2^2xC5:C16"); // GroupNames label
G:=SmallGroup(320,1080);
// by ID
G=gap.SmallGroup(320,1080);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,80,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^5=d^16=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^3>;
// generators/relations
×
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ℤ
𝔽
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