direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C42⋊4C4, C42⋊4C20, C20.66C42, (C4×C20)⋊18C4, C4.7(C4×C20), (C2×C42).2C10, C10.50(C2×C42), C22.13(C22×C20), C23.49(C22×C10), C10.69(C42⋊C2), C2.C42.11C10, (C22×C20).487C22, (C22×C10).440C23, C2.2(C2×C4×C20), (C2×C4×C20).4C2, (C2×C4).54(C2×C20), (C2×C20).456(C2×C4), C2.1(C5×C42⋊C2), C22.12(C5×C4○D4), (C22×C4).81(C2×C10), (C2×C10).202(C4○D4), (C2×C10).313(C22×C4), (C5×C2.C42).29C2, SmallGroup(320,877)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C42⋊4C4
G = < a,b,c,d | a5=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, cd=dc >
Subgroups: 226 in 178 conjugacy classes, 130 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, C2×C4, C23, C10, C10, C42, C22×C4, C22×C4, C20, C20, C2×C10, C2.C42, C2×C42, C2×C20, C2×C20, C22×C10, C42⋊4C4, C4×C20, C22×C20, C22×C20, C5×C2.C42, C2×C4×C20, C5×C42⋊4C4
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C42, C22×C4, C4○D4, C20, C2×C10, C2×C42, C42⋊C2, C2×C20, C22×C10, C42⋊4C4, C4×C20, C22×C20, C5×C4○D4, C2×C4×C20, C5×C42⋊C2, C5×C42⋊4C4
►Regular action on 320 points
Generators in S320
(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)
(1 273 47 265)(2 274 48 261)(3 275 49 262)(4 271 50 263)(5 272 46 264)(6 64 31 72)(7 65 32 73)(8 61 33 74)(9 62 34 75)(10 63 35 71)(11 286 27 285)(12 287 28 281)(13 288 29 282)(14 289 30 283)(15 290 26 284)(16 57 25 78)(17 58 21 79)(18 59 22 80)(19 60 23 76)(20 56 24 77)(36 293 41 310)(37 294 42 306)(38 295 43 307)(39 291 44 308)(40 292 45 309)(51 279 68 258)(52 280 69 259)(53 276 70 260)(54 277 66 256)(55 278 67 257)(81 313 87 305)(82 314 88 301)(83 315 89 302)(84 311 90 303)(85 312 86 304)(91 319 108 298)(92 320 109 299)(93 316 110 300)(94 317 106 296)(95 318 107 297)(96 197 117 176)(97 198 118 177)(98 199 119 178)(99 200 120 179)(100 196 116 180)(101 194 114 181)(102 195 115 182)(103 191 111 183)(104 192 112 184)(105 193 113 185)(121 207 127 201)(122 208 128 202)(123 209 129 203)(124 210 130 204)(125 206 126 205)(131 228 148 211)(132 229 149 212)(133 230 150 213)(134 226 146 214)(135 227 147 215)(136 237 157 216)(137 238 158 217)(138 239 159 218)(139 240 160 219)(140 236 156 220)(141 234 154 221)(142 235 155 222)(143 231 151 223)(144 232 152 224)(145 233 153 225)(161 247 167 241)(162 248 168 242)(163 249 169 243)(164 250 170 244)(165 246 166 245)(171 268 188 251)(172 269 189 252)(173 270 190 253)(174 266 186 254)(175 267 187 255)
(1 94 42 65)(2 95 43 61)(3 91 44 62)(4 92 45 63)(5 93 41 64)(6 264 300 293)(7 265 296 294)(8 261 297 295)(9 262 298 291)(10 263 299 292)(11 60 53 85)(12 56 54 81)(13 57 55 82)(14 58 51 83)(15 59 52 84)(16 257 301 282)(17 258 302 283)(18 259 303 284)(19 260 304 285)(20 256 305 281)(21 279 315 289)(22 280 311 290)(23 276 312 286)(24 277 313 287)(25 278 314 288)(26 80 69 90)(27 76 70 86)(28 77 66 87)(29 78 67 88)(30 79 68 89)(31 272 316 310)(32 273 317 306)(33 274 318 307)(34 275 319 308)(35 271 320 309)(36 72 46 110)(37 73 47 106)(38 74 48 107)(39 75 49 108)(40 71 50 109)(96 145 121 174)(97 141 122 175)(98 142 123 171)(99 143 124 172)(100 144 125 173)(101 137 135 162)(102 138 131 163)(103 139 132 164)(104 140 133 165)(105 136 134 161)(111 160 149 170)(112 156 150 166)(113 157 146 167)(114 158 147 168)(115 159 148 169)(116 152 126 190)(117 153 127 186)(118 154 128 187)(119 155 129 188)(120 151 130 189)(176 225 201 254)(177 221 202 255)(178 222 203 251)(179 223 204 252)(180 224 205 253)(181 217 215 242)(182 218 211 243)(183 219 212 244)(184 220 213 245)(185 216 214 241)(191 240 229 250)(192 236 230 246)(193 237 226 247)(194 238 227 248)(195 239 228 249)(196 232 206 270)(197 233 207 266)(198 234 208 267)(199 235 209 268)(200 231 210 269)
(1 105 12 121)(2 101 13 122)(3 102 14 123)(4 103 15 124)(5 104 11 125)(6 220 304 253)(7 216 305 254)(8 217 301 255)(9 218 302 251)(10 219 303 252)(16 221 297 242)(17 222 298 243)(18 223 299 244)(19 224 300 245)(20 225 296 241)(21 235 319 249)(22 231 320 250)(23 232 316 246)(24 233 317 247)(25 234 318 248)(26 130 50 111)(27 126 46 112)(28 127 47 113)(29 128 48 114)(30 129 49 115)(31 236 312 270)(32 237 313 266)(33 238 314 267)(34 239 315 268)(35 240 311 269)(36 150 70 116)(37 146 66 117)(38 147 67 118)(39 148 68 119)(40 149 69 120)(41 133 53 100)(42 134 54 96)(43 135 55 97)(44 131 51 98)(45 132 52 99)(56 174 94 136)(57 175 95 137)(58 171 91 138)(59 172 92 139)(60 173 93 140)(61 162 82 141)(62 163 83 142)(63 164 84 143)(64 165 85 144)(65 161 81 145)(71 170 90 151)(72 166 86 152)(73 167 87 153)(74 168 88 154)(75 169 89 155)(76 190 110 156)(77 186 106 157)(78 187 107 158)(79 188 108 159)(80 189 109 160)(176 265 214 281)(177 261 215 282)(178 262 211 283)(179 263 212 284)(180 264 213 285)(181 257 202 295)(182 258 203 291)(183 259 204 292)(184 260 205 293)(185 256 201 294)(191 280 210 309)(192 276 206 310)(193 277 207 306)(194 278 208 307)(195 279 209 308)(196 272 230 286)(197 273 226 287)(198 274 227 288)(199 275 228 289)(200 271 229 290)
G:=sub<Sym(320)| (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), (1,273,47,265)(2,274,48,261)(3,275,49,262)(4,271,50,263)(5,272,46,264)(6,64,31,72)(7,65,32,73)(8,61,33,74)(9,62,34,75)(10,63,35,71)(11,286,27,285)(12,287,28,281)(13,288,29,282)(14,289,30,283)(15,290,26,284)(16,57,25,78)(17,58,21,79)(18,59,22,80)(19,60,23,76)(20,56,24,77)(36,293,41,310)(37,294,42,306)(38,295,43,307)(39,291,44,308)(40,292,45,309)(51,279,68,258)(52,280,69,259)(53,276,70,260)(54,277,66,256)(55,278,67,257)(81,313,87,305)(82,314,88,301)(83,315,89,302)(84,311,90,303)(85,312,86,304)(91,319,108,298)(92,320,109,299)(93,316,110,300)(94,317,106,296)(95,318,107,297)(96,197,117,176)(97,198,118,177)(98,199,119,178)(99,200,120,179)(100,196,116,180)(101,194,114,181)(102,195,115,182)(103,191,111,183)(104,192,112,184)(105,193,113,185)(121,207,127,201)(122,208,128,202)(123,209,129,203)(124,210,130,204)(125,206,126,205)(131,228,148,211)(132,229,149,212)(133,230,150,213)(134,226,146,214)(135,227,147,215)(136,237,157,216)(137,238,158,217)(138,239,159,218)(139,240,160,219)(140,236,156,220)(141,234,154,221)(142,235,155,222)(143,231,151,223)(144,232,152,224)(145,233,153,225)(161,247,167,241)(162,248,168,242)(163,249,169,243)(164,250,170,244)(165,246,166,245)(171,268,188,251)(172,269,189,252)(173,270,190,253)(174,266,186,254)(175,267,187,255), (1,94,42,65)(2,95,43,61)(3,91,44,62)(4,92,45,63)(5,93,41,64)(6,264,300,293)(7,265,296,294)(8,261,297,295)(9,262,298,291)(10,263,299,292)(11,60,53,85)(12,56,54,81)(13,57,55,82)(14,58,51,83)(15,59,52,84)(16,257,301,282)(17,258,302,283)(18,259,303,284)(19,260,304,285)(20,256,305,281)(21,279,315,289)(22,280,311,290)(23,276,312,286)(24,277,313,287)(25,278,314,288)(26,80,69,90)(27,76,70,86)(28,77,66,87)(29,78,67,88)(30,79,68,89)(31,272,316,310)(32,273,317,306)(33,274,318,307)(34,275,319,308)(35,271,320,309)(36,72,46,110)(37,73,47,106)(38,74,48,107)(39,75,49,108)(40,71,50,109)(96,145,121,174)(97,141,122,175)(98,142,123,171)(99,143,124,172)(100,144,125,173)(101,137,135,162)(102,138,131,163)(103,139,132,164)(104,140,133,165)(105,136,134,161)(111,160,149,170)(112,156,150,166)(113,157,146,167)(114,158,147,168)(115,159,148,169)(116,152,126,190)(117,153,127,186)(118,154,128,187)(119,155,129,188)(120,151,130,189)(176,225,201,254)(177,221,202,255)(178,222,203,251)(179,223,204,252)(180,224,205,253)(181,217,215,242)(182,218,211,243)(183,219,212,244)(184,220,213,245)(185,216,214,241)(191,240,229,250)(192,236,230,246)(193,237,226,247)(194,238,227,248)(195,239,228,249)(196,232,206,270)(197,233,207,266)(198,234,208,267)(199,235,209,268)(200,231,210,269), (1,105,12,121)(2,101,13,122)(3,102,14,123)(4,103,15,124)(5,104,11,125)(6,220,304,253)(7,216,305,254)(8,217,301,255)(9,218,302,251)(10,219,303,252)(16,221,297,242)(17,222,298,243)(18,223,299,244)(19,224,300,245)(20,225,296,241)(21,235,319,249)(22,231,320,250)(23,232,316,246)(24,233,317,247)(25,234,318,248)(26,130,50,111)(27,126,46,112)(28,127,47,113)(29,128,48,114)(30,129,49,115)(31,236,312,270)(32,237,313,266)(33,238,314,267)(34,239,315,268)(35,240,311,269)(36,150,70,116)(37,146,66,117)(38,147,67,118)(39,148,68,119)(40,149,69,120)(41,133,53,100)(42,134,54,96)(43,135,55,97)(44,131,51,98)(45,132,52,99)(56,174,94,136)(57,175,95,137)(58,171,91,138)(59,172,92,139)(60,173,93,140)(61,162,82,141)(62,163,83,142)(63,164,84,143)(64,165,85,144)(65,161,81,145)(71,170,90,151)(72,166,86,152)(73,167,87,153)(74,168,88,154)(75,169,89,155)(76,190,110,156)(77,186,106,157)(78,187,107,158)(79,188,108,159)(80,189,109,160)(176,265,214,281)(177,261,215,282)(178,262,211,283)(179,263,212,284)(180,264,213,285)(181,257,202,295)(182,258,203,291)(183,259,204,292)(184,260,205,293)(185,256,201,294)(191,280,210,309)(192,276,206,310)(193,277,207,306)(194,278,208,307)(195,279,209,308)(196,272,230,286)(197,273,226,287)(198,274,227,288)(199,275,228,289)(200,271,229,290)>;
G:=Group( (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), (1,273,47,265)(2,274,48,261)(3,275,49,262)(4,271,50,263)(5,272,46,264)(6,64,31,72)(7,65,32,73)(8,61,33,74)(9,62,34,75)(10,63,35,71)(11,286,27,285)(12,287,28,281)(13,288,29,282)(14,289,30,283)(15,290,26,284)(16,57,25,78)(17,58,21,79)(18,59,22,80)(19,60,23,76)(20,56,24,77)(36,293,41,310)(37,294,42,306)(38,295,43,307)(39,291,44,308)(40,292,45,309)(51,279,68,258)(52,280,69,259)(53,276,70,260)(54,277,66,256)(55,278,67,257)(81,313,87,305)(82,314,88,301)(83,315,89,302)(84,311,90,303)(85,312,86,304)(91,319,108,298)(92,320,109,299)(93,316,110,300)(94,317,106,296)(95,318,107,297)(96,197,117,176)(97,198,118,177)(98,199,119,178)(99,200,120,179)(100,196,116,180)(101,194,114,181)(102,195,115,182)(103,191,111,183)(104,192,112,184)(105,193,113,185)(121,207,127,201)(122,208,128,202)(123,209,129,203)(124,210,130,204)(125,206,126,205)(131,228,148,211)(132,229,149,212)(133,230,150,213)(134,226,146,214)(135,227,147,215)(136,237,157,216)(137,238,158,217)(138,239,159,218)(139,240,160,219)(140,236,156,220)(141,234,154,221)(142,235,155,222)(143,231,151,223)(144,232,152,224)(145,233,153,225)(161,247,167,241)(162,248,168,242)(163,249,169,243)(164,250,170,244)(165,246,166,245)(171,268,188,251)(172,269,189,252)(173,270,190,253)(174,266,186,254)(175,267,187,255), (1,94,42,65)(2,95,43,61)(3,91,44,62)(4,92,45,63)(5,93,41,64)(6,264,300,293)(7,265,296,294)(8,261,297,295)(9,262,298,291)(10,263,299,292)(11,60,53,85)(12,56,54,81)(13,57,55,82)(14,58,51,83)(15,59,52,84)(16,257,301,282)(17,258,302,283)(18,259,303,284)(19,260,304,285)(20,256,305,281)(21,279,315,289)(22,280,311,290)(23,276,312,286)(24,277,313,287)(25,278,314,288)(26,80,69,90)(27,76,70,86)(28,77,66,87)(29,78,67,88)(30,79,68,89)(31,272,316,310)(32,273,317,306)(33,274,318,307)(34,275,319,308)(35,271,320,309)(36,72,46,110)(37,73,47,106)(38,74,48,107)(39,75,49,108)(40,71,50,109)(96,145,121,174)(97,141,122,175)(98,142,123,171)(99,143,124,172)(100,144,125,173)(101,137,135,162)(102,138,131,163)(103,139,132,164)(104,140,133,165)(105,136,134,161)(111,160,149,170)(112,156,150,166)(113,157,146,167)(114,158,147,168)(115,159,148,169)(116,152,126,190)(117,153,127,186)(118,154,128,187)(119,155,129,188)(120,151,130,189)(176,225,201,254)(177,221,202,255)(178,222,203,251)(179,223,204,252)(180,224,205,253)(181,217,215,242)(182,218,211,243)(183,219,212,244)(184,220,213,245)(185,216,214,241)(191,240,229,250)(192,236,230,246)(193,237,226,247)(194,238,227,248)(195,239,228,249)(196,232,206,270)(197,233,207,266)(198,234,208,267)(199,235,209,268)(200,231,210,269), (1,105,12,121)(2,101,13,122)(3,102,14,123)(4,103,15,124)(5,104,11,125)(6,220,304,253)(7,216,305,254)(8,217,301,255)(9,218,302,251)(10,219,303,252)(16,221,297,242)(17,222,298,243)(18,223,299,244)(19,224,300,245)(20,225,296,241)(21,235,319,249)(22,231,320,250)(23,232,316,246)(24,233,317,247)(25,234,318,248)(26,130,50,111)(27,126,46,112)(28,127,47,113)(29,128,48,114)(30,129,49,115)(31,236,312,270)(32,237,313,266)(33,238,314,267)(34,239,315,268)(35,240,311,269)(36,150,70,116)(37,146,66,117)(38,147,67,118)(39,148,68,119)(40,149,69,120)(41,133,53,100)(42,134,54,96)(43,135,55,97)(44,131,51,98)(45,132,52,99)(56,174,94,136)(57,175,95,137)(58,171,91,138)(59,172,92,139)(60,173,93,140)(61,162,82,141)(62,163,83,142)(63,164,84,143)(64,165,85,144)(65,161,81,145)(71,170,90,151)(72,166,86,152)(73,167,87,153)(74,168,88,154)(75,169,89,155)(76,190,110,156)(77,186,106,157)(78,187,107,158)(79,188,108,159)(80,189,109,160)(176,265,214,281)(177,261,215,282)(178,262,211,283)(179,263,212,284)(180,264,213,285)(181,257,202,295)(182,258,203,291)(183,259,204,292)(184,260,205,293)(185,256,201,294)(191,280,210,309)(192,276,206,310)(193,277,207,306)(194,278,208,307)(195,279,209,308)(196,272,230,286)(197,273,226,287)(198,274,227,288)(199,275,228,289)(200,271,229,290) );
G=PermutationGroup([[(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)], [(1,273,47,265),(2,274,48,261),(3,275,49,262),(4,271,50,263),(5,272,46,264),(6,64,31,72),(7,65,32,73),(8,61,33,74),(9,62,34,75),(10,63,35,71),(11,286,27,285),(12,287,28,281),(13,288,29,282),(14,289,30,283),(15,290,26,284),(16,57,25,78),(17,58,21,79),(18,59,22,80),(19,60,23,76),(20,56,24,77),(36,293,41,310),(37,294,42,306),(38,295,43,307),(39,291,44,308),(40,292,45,309),(51,279,68,258),(52,280,69,259),(53,276,70,260),(54,277,66,256),(55,278,67,257),(81,313,87,305),(82,314,88,301),(83,315,89,302),(84,311,90,303),(85,312,86,304),(91,319,108,298),(92,320,109,299),(93,316,110,300),(94,317,106,296),(95,318,107,297),(96,197,117,176),(97,198,118,177),(98,199,119,178),(99,200,120,179),(100,196,116,180),(101,194,114,181),(102,195,115,182),(103,191,111,183),(104,192,112,184),(105,193,113,185),(121,207,127,201),(122,208,128,202),(123,209,129,203),(124,210,130,204),(125,206,126,205),(131,228,148,211),(132,229,149,212),(133,230,150,213),(134,226,146,214),(135,227,147,215),(136,237,157,216),(137,238,158,217),(138,239,159,218),(139,240,160,219),(140,236,156,220),(141,234,154,221),(142,235,155,222),(143,231,151,223),(144,232,152,224),(145,233,153,225),(161,247,167,241),(162,248,168,242),(163,249,169,243),(164,250,170,244),(165,246,166,245),(171,268,188,251),(172,269,189,252),(173,270,190,253),(174,266,186,254),(175,267,187,255)], [(1,94,42,65),(2,95,43,61),(3,91,44,62),(4,92,45,63),(5,93,41,64),(6,264,300,293),(7,265,296,294),(8,261,297,295),(9,262,298,291),(10,263,299,292),(11,60,53,85),(12,56,54,81),(13,57,55,82),(14,58,51,83),(15,59,52,84),(16,257,301,282),(17,258,302,283),(18,259,303,284),(19,260,304,285),(20,256,305,281),(21,279,315,289),(22,280,311,290),(23,276,312,286),(24,277,313,287),(25,278,314,288),(26,80,69,90),(27,76,70,86),(28,77,66,87),(29,78,67,88),(30,79,68,89),(31,272,316,310),(32,273,317,306),(33,274,318,307),(34,275,319,308),(35,271,320,309),(36,72,46,110),(37,73,47,106),(38,74,48,107),(39,75,49,108),(40,71,50,109),(96,145,121,174),(97,141,122,175),(98,142,123,171),(99,143,124,172),(100,144,125,173),(101,137,135,162),(102,138,131,163),(103,139,132,164),(104,140,133,165),(105,136,134,161),(111,160,149,170),(112,156,150,166),(113,157,146,167),(114,158,147,168),(115,159,148,169),(116,152,126,190),(117,153,127,186),(118,154,128,187),(119,155,129,188),(120,151,130,189),(176,225,201,254),(177,221,202,255),(178,222,203,251),(179,223,204,252),(180,224,205,253),(181,217,215,242),(182,218,211,243),(183,219,212,244),(184,220,213,245),(185,216,214,241),(191,240,229,250),(192,236,230,246),(193,237,226,247),(194,238,227,248),(195,239,228,249),(196,232,206,270),(197,233,207,266),(198,234,208,267),(199,235,209,268),(200,231,210,269)], [(1,105,12,121),(2,101,13,122),(3,102,14,123),(4,103,15,124),(5,104,11,125),(6,220,304,253),(7,216,305,254),(8,217,301,255),(9,218,302,251),(10,219,303,252),(16,221,297,242),(17,222,298,243),(18,223,299,244),(19,224,300,245),(20,225,296,241),(21,235,319,249),(22,231,320,250),(23,232,316,246),(24,233,317,247),(25,234,318,248),(26,130,50,111),(27,126,46,112),(28,127,47,113),(29,128,48,114),(30,129,49,115),(31,236,312,270),(32,237,313,266),(33,238,314,267),(34,239,315,268),(35,240,311,269),(36,150,70,116),(37,146,66,117),(38,147,67,118),(39,148,68,119),(40,149,69,120),(41,133,53,100),(42,134,54,96),(43,135,55,97),(44,131,51,98),(45,132,52,99),(56,174,94,136),(57,175,95,137),(58,171,91,138),(59,172,92,139),(60,173,93,140),(61,162,82,141),(62,163,83,142),(63,164,84,143),(64,165,85,144),(65,161,81,145),(71,170,90,151),(72,166,86,152),(73,167,87,153),(74,168,88,154),(75,169,89,155),(76,190,110,156),(77,186,106,157),(78,187,107,158),(79,188,108,159),(80,189,109,160),(176,265,214,281),(177,261,215,282),(178,262,211,283),(179,263,212,284),(180,264,213,285),(181,257,202,295),(182,258,203,291),(183,259,204,292),(184,260,205,293),(185,256,201,294),(191,280,210,309),(192,276,206,310),(193,277,207,306),(194,278,208,307),(195,279,209,308),(196,272,230,286),(197,273,226,287),(198,274,227,288),(199,275,228,289),(200,271,229,290)]])
200 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4AF | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 20A | ··· | 20AF | 20AG | ··· | 20DX |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | C4○D4 | C5×C4○D4 |
| kernel | C5×C42⋊4C4 | C5×C2.C42 | C2×C4×C20 | C4×C20 | C42⋊4C4 | C2.C42 | C2×C42 | C42 | C2×C10 | C22 |
| # reps | 1 | 4 | 3 | 24 | 4 | 16 | 12 | 96 | 8 | 32 |
Matrix representation of C5×C42⋊4C4 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 9 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 32 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[9,0,0,0,0,32,0,0,0,0,0,40,0,0,1,0],[40,0,0,0,0,40,0,0,0,0,32,0,0,0,0,32],[32,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;
C5×C42⋊4C4 in GAP, Magma, Sage, TeX
C_5\times C_4^2\rtimes_4C_4
% in TeX
G:=Group("C5xC4^2:4C4"); // GroupNames label
G:=SmallGroup(320,877);
// by ID
G=gap.SmallGroup(320,877);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1128,226]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^2,c*d=d*c>;
// generators/relations