direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C42⋊5C4, C42⋊5C20, (C4×C20)⋊19C4, (C2×C42).4C10, C2.C42.2C10, C22.32(C22×C20), C23.56(C22×C10), C10.75(C42⋊C2), C10.31(C42⋊2C2), (C22×C10).447C23, (C22×C20).492C22, (C2×C4×C20).6C2, (C2×C4).57(C2×C20), (C2×C20).459(C2×C4), C2.8(C5×C42⋊C2), C2.1(C5×C42⋊2C2), C22.18(C5×C4○D4), (C22×C4).85(C2×C10), (C2×C10).208(C4○D4), (C2×C10).320(C22×C4), (C5×C2.C42).4C2, SmallGroup(320,884)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C42⋊5C4
G = < a,b,c,d | a5=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c-1 >
Subgroups: 210 in 138 conjugacy classes, 82 normal (10 characteristic)
C1, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C42, C22×C4, C20, C2×C10, C2×C10, C2.C42, C2×C42, C2×C20, C2×C20, C22×C10, C42⋊5C4, C4×C20, C22×C20, C5×C2.C42, C2×C4×C20, C5×C42⋊5C4
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C22×C4, C4○D4, C20, C2×C10, C42⋊C2, C42⋊2C2, C2×C20, C22×C10, C42⋊5C4, C22×C20, C5×C4○D4, C5×C42⋊C2, C5×C42⋊2C2, C5×C42⋊5C4
►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)
(1 127 42 97)(2 128 43 98)(3 129 44 99)(4 130 45 100)(5 126 41 96)(6 253 312 236)(7 254 313 237)(8 255 314 238)(9 251 315 239)(10 252 311 240)(11 116 70 133)(12 117 66 134)(13 118 67 135)(14 119 68 131)(15 120 69 132)(16 247 304 217)(17 248 305 218)(18 249 301 219)(19 250 302 220)(20 246 303 216)(21 244 320 231)(22 245 316 232)(23 241 317 233)(24 242 318 234)(25 243 319 235)(26 115 49 123)(27 111 50 124)(28 112 46 125)(29 113 47 121)(30 114 48 122)(31 270 296 223)(32 266 297 224)(33 267 298 225)(34 268 299 221)(35 269 300 222)(36 103 53 150)(37 104 54 146)(38 105 55 147)(39 101 51 148)(40 102 52 149)(56 152 86 165)(57 153 87 161)(58 154 88 162)(59 155 89 163)(60 151 90 164)(61 159 108 171)(62 160 109 172)(63 156 110 173)(64 157 106 174)(65 158 107 175)(71 140 84 170)(72 136 85 166)(73 137 81 167)(74 138 82 168)(75 139 83 169)(76 143 93 190)(77 144 94 186)(78 145 95 187)(79 141 91 188)(80 142 92 189)(176 310 206 263)(177 306 207 264)(178 307 208 265)(179 308 209 261)(180 309 210 262)(181 289 228 259)(182 290 229 260)(183 286 230 256)(184 287 226 257)(185 288 227 258)(191 292 204 280)(192 293 205 276)(193 294 201 277)(194 295 202 278)(195 291 203 279)(196 285 213 272)(197 281 214 273)(198 282 215 274)(199 283 211 275)(200 284 212 271)
(1 94 12 57)(2 95 13 58)(3 91 14 59)(4 92 15 60)(5 93 11 56)(6 276 303 286)(7 277 304 287)(8 278 305 288)(9 279 301 289)(10 280 302 290)(16 257 313 294)(17 258 314 295)(18 259 315 291)(19 260 311 292)(20 256 312 293)(21 271 300 309)(22 272 296 310)(23 273 297 306)(24 274 298 307)(25 275 299 308)(26 61 51 83)(27 62 52 84)(28 63 53 85)(29 64 54 81)(30 65 55 82)(31 263 316 285)(32 264 317 281)(33 265 318 282)(34 261 319 283)(35 262 320 284)(36 72 46 110)(37 73 47 106)(38 74 48 107)(39 75 49 108)(40 71 50 109)(41 76 70 86)(42 77 66 87)(43 78 67 88)(44 79 68 89)(45 80 69 90)(96 143 133 165)(97 144 134 161)(98 145 135 162)(99 141 131 163)(100 142 132 164)(101 139 123 171)(102 140 124 172)(103 136 125 173)(104 137 121 174)(105 138 122 175)(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)(176 232 213 270)(177 233 214 266)(178 234 215 267)(179 235 211 268)(180 231 212 269)(181 239 203 249)(182 240 204 250)(183 236 205 246)(184 237 201 247)(185 238 202 248)(191 220 229 252)(192 216 230 253)(193 217 226 254)(194 218 227 255)(195 219 228 251)(196 223 206 245)(197 224 207 241)(198 225 208 242)(199 221 209 243)(200 222 210 244)
(1 177 29 201)(2 178 30 202)(3 179 26 203)(4 180 27 204)(5 176 28 205)(6 165 296 136)(7 161 297 137)(8 162 298 138)(9 163 299 139)(10 164 300 140)(11 213 53 183)(12 214 54 184)(13 215 55 185)(14 211 51 181)(15 212 52 182)(16 186 317 157)(17 187 318 158)(18 188 319 159)(19 189 320 160)(20 190 316 156)(21 172 302 142)(22 173 303 143)(23 174 304 144)(24 175 305 145)(25 171 301 141)(31 166 312 152)(32 167 313 153)(33 168 314 154)(34 169 315 155)(35 170 311 151)(36 230 70 196)(37 226 66 197)(38 227 67 198)(39 228 68 199)(40 229 69 200)(41 206 46 192)(42 207 47 193)(43 208 48 194)(44 209 49 195)(45 210 50 191)(56 245 85 216)(57 241 81 217)(58 242 82 218)(59 243 83 219)(60 244 84 220)(61 251 91 221)(62 252 92 222)(63 253 93 223)(64 254 94 224)(65 255 95 225)(71 250 90 231)(72 246 86 232)(73 247 87 233)(74 248 88 234)(75 249 89 235)(76 270 110 236)(77 266 106 237)(78 267 107 238)(79 268 108 239)(80 269 109 240)(96 285 125 256)(97 281 121 257)(98 282 122 258)(99 283 123 259)(100 284 124 260)(101 291 131 261)(102 292 132 262)(103 293 133 263)(104 294 134 264)(105 295 135 265)(111 290 130 271)(112 286 126 272)(113 287 127 273)(114 288 128 274)(115 289 129 275)(116 310 150 276)(117 306 146 277)(118 307 147 278)(119 308 148 279)(120 309 149 280)
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), (1,127,42,97)(2,128,43,98)(3,129,44,99)(4,130,45,100)(5,126,41,96)(6,253,312,236)(7,254,313,237)(8,255,314,238)(9,251,315,239)(10,252,311,240)(11,116,70,133)(12,117,66,134)(13,118,67,135)(14,119,68,131)(15,120,69,132)(16,247,304,217)(17,248,305,218)(18,249,301,219)(19,250,302,220)(20,246,303,216)(21,244,320,231)(22,245,316,232)(23,241,317,233)(24,242,318,234)(25,243,319,235)(26,115,49,123)(27,111,50,124)(28,112,46,125)(29,113,47,121)(30,114,48,122)(31,270,296,223)(32,266,297,224)(33,267,298,225)(34,268,299,221)(35,269,300,222)(36,103,53,150)(37,104,54,146)(38,105,55,147)(39,101,51,148)(40,102,52,149)(56,152,86,165)(57,153,87,161)(58,154,88,162)(59,155,89,163)(60,151,90,164)(61,159,108,171)(62,160,109,172)(63,156,110,173)(64,157,106,174)(65,158,107,175)(71,140,84,170)(72,136,85,166)(73,137,81,167)(74,138,82,168)(75,139,83,169)(76,143,93,190)(77,144,94,186)(78,145,95,187)(79,141,91,188)(80,142,92,189)(176,310,206,263)(177,306,207,264)(178,307,208,265)(179,308,209,261)(180,309,210,262)(181,289,228,259)(182,290,229,260)(183,286,230,256)(184,287,226,257)(185,288,227,258)(191,292,204,280)(192,293,205,276)(193,294,201,277)(194,295,202,278)(195,291,203,279)(196,285,213,272)(197,281,214,273)(198,282,215,274)(199,283,211,275)(200,284,212,271), (1,94,12,57)(2,95,13,58)(3,91,14,59)(4,92,15,60)(5,93,11,56)(6,276,303,286)(7,277,304,287)(8,278,305,288)(9,279,301,289)(10,280,302,290)(16,257,313,294)(17,258,314,295)(18,259,315,291)(19,260,311,292)(20,256,312,293)(21,271,300,309)(22,272,296,310)(23,273,297,306)(24,274,298,307)(25,275,299,308)(26,61,51,83)(27,62,52,84)(28,63,53,85)(29,64,54,81)(30,65,55,82)(31,263,316,285)(32,264,317,281)(33,265,318,282)(34,261,319,283)(35,262,320,284)(36,72,46,110)(37,73,47,106)(38,74,48,107)(39,75,49,108)(40,71,50,109)(41,76,70,86)(42,77,66,87)(43,78,67,88)(44,79,68,89)(45,80,69,90)(96,143,133,165)(97,144,134,161)(98,145,135,162)(99,141,131,163)(100,142,132,164)(101,139,123,171)(102,140,124,172)(103,136,125,173)(104,137,121,174)(105,138,122,175)(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)(176,232,213,270)(177,233,214,266)(178,234,215,267)(179,235,211,268)(180,231,212,269)(181,239,203,249)(182,240,204,250)(183,236,205,246)(184,237,201,247)(185,238,202,248)(191,220,229,252)(192,216,230,253)(193,217,226,254)(194,218,227,255)(195,219,228,251)(196,223,206,245)(197,224,207,241)(198,225,208,242)(199,221,209,243)(200,222,210,244), (1,177,29,201)(2,178,30,202)(3,179,26,203)(4,180,27,204)(5,176,28,205)(6,165,296,136)(7,161,297,137)(8,162,298,138)(9,163,299,139)(10,164,300,140)(11,213,53,183)(12,214,54,184)(13,215,55,185)(14,211,51,181)(15,212,52,182)(16,186,317,157)(17,187,318,158)(18,188,319,159)(19,189,320,160)(20,190,316,156)(21,172,302,142)(22,173,303,143)(23,174,304,144)(24,175,305,145)(25,171,301,141)(31,166,312,152)(32,167,313,153)(33,168,314,154)(34,169,315,155)(35,170,311,151)(36,230,70,196)(37,226,66,197)(38,227,67,198)(39,228,68,199)(40,229,69,200)(41,206,46,192)(42,207,47,193)(43,208,48,194)(44,209,49,195)(45,210,50,191)(56,245,85,216)(57,241,81,217)(58,242,82,218)(59,243,83,219)(60,244,84,220)(61,251,91,221)(62,252,92,222)(63,253,93,223)(64,254,94,224)(65,255,95,225)(71,250,90,231)(72,246,86,232)(73,247,87,233)(74,248,88,234)(75,249,89,235)(76,270,110,236)(77,266,106,237)(78,267,107,238)(79,268,108,239)(80,269,109,240)(96,285,125,256)(97,281,121,257)(98,282,122,258)(99,283,123,259)(100,284,124,260)(101,291,131,261)(102,292,132,262)(103,293,133,263)(104,294,134,264)(105,295,135,265)(111,290,130,271)(112,286,126,272)(113,287,127,273)(114,288,128,274)(115,289,129,275)(116,310,150,276)(117,306,146,277)(118,307,147,278)(119,308,148,279)(120,309,149,280)>;
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), (1,127,42,97)(2,128,43,98)(3,129,44,99)(4,130,45,100)(5,126,41,96)(6,253,312,236)(7,254,313,237)(8,255,314,238)(9,251,315,239)(10,252,311,240)(11,116,70,133)(12,117,66,134)(13,118,67,135)(14,119,68,131)(15,120,69,132)(16,247,304,217)(17,248,305,218)(18,249,301,219)(19,250,302,220)(20,246,303,216)(21,244,320,231)(22,245,316,232)(23,241,317,233)(24,242,318,234)(25,243,319,235)(26,115,49,123)(27,111,50,124)(28,112,46,125)(29,113,47,121)(30,114,48,122)(31,270,296,223)(32,266,297,224)(33,267,298,225)(34,268,299,221)(35,269,300,222)(36,103,53,150)(37,104,54,146)(38,105,55,147)(39,101,51,148)(40,102,52,149)(56,152,86,165)(57,153,87,161)(58,154,88,162)(59,155,89,163)(60,151,90,164)(61,159,108,171)(62,160,109,172)(63,156,110,173)(64,157,106,174)(65,158,107,175)(71,140,84,170)(72,136,85,166)(73,137,81,167)(74,138,82,168)(75,139,83,169)(76,143,93,190)(77,144,94,186)(78,145,95,187)(79,141,91,188)(80,142,92,189)(176,310,206,263)(177,306,207,264)(178,307,208,265)(179,308,209,261)(180,309,210,262)(181,289,228,259)(182,290,229,260)(183,286,230,256)(184,287,226,257)(185,288,227,258)(191,292,204,280)(192,293,205,276)(193,294,201,277)(194,295,202,278)(195,291,203,279)(196,285,213,272)(197,281,214,273)(198,282,215,274)(199,283,211,275)(200,284,212,271), (1,94,12,57)(2,95,13,58)(3,91,14,59)(4,92,15,60)(5,93,11,56)(6,276,303,286)(7,277,304,287)(8,278,305,288)(9,279,301,289)(10,280,302,290)(16,257,313,294)(17,258,314,295)(18,259,315,291)(19,260,311,292)(20,256,312,293)(21,271,300,309)(22,272,296,310)(23,273,297,306)(24,274,298,307)(25,275,299,308)(26,61,51,83)(27,62,52,84)(28,63,53,85)(29,64,54,81)(30,65,55,82)(31,263,316,285)(32,264,317,281)(33,265,318,282)(34,261,319,283)(35,262,320,284)(36,72,46,110)(37,73,47,106)(38,74,48,107)(39,75,49,108)(40,71,50,109)(41,76,70,86)(42,77,66,87)(43,78,67,88)(44,79,68,89)(45,80,69,90)(96,143,133,165)(97,144,134,161)(98,145,135,162)(99,141,131,163)(100,142,132,164)(101,139,123,171)(102,140,124,172)(103,136,125,173)(104,137,121,174)(105,138,122,175)(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)(176,232,213,270)(177,233,214,266)(178,234,215,267)(179,235,211,268)(180,231,212,269)(181,239,203,249)(182,240,204,250)(183,236,205,246)(184,237,201,247)(185,238,202,248)(191,220,229,252)(192,216,230,253)(193,217,226,254)(194,218,227,255)(195,219,228,251)(196,223,206,245)(197,224,207,241)(198,225,208,242)(199,221,209,243)(200,222,210,244), (1,177,29,201)(2,178,30,202)(3,179,26,203)(4,180,27,204)(5,176,28,205)(6,165,296,136)(7,161,297,137)(8,162,298,138)(9,163,299,139)(10,164,300,140)(11,213,53,183)(12,214,54,184)(13,215,55,185)(14,211,51,181)(15,212,52,182)(16,186,317,157)(17,187,318,158)(18,188,319,159)(19,189,320,160)(20,190,316,156)(21,172,302,142)(22,173,303,143)(23,174,304,144)(24,175,305,145)(25,171,301,141)(31,166,312,152)(32,167,313,153)(33,168,314,154)(34,169,315,155)(35,170,311,151)(36,230,70,196)(37,226,66,197)(38,227,67,198)(39,228,68,199)(40,229,69,200)(41,206,46,192)(42,207,47,193)(43,208,48,194)(44,209,49,195)(45,210,50,191)(56,245,85,216)(57,241,81,217)(58,242,82,218)(59,243,83,219)(60,244,84,220)(61,251,91,221)(62,252,92,222)(63,253,93,223)(64,254,94,224)(65,255,95,225)(71,250,90,231)(72,246,86,232)(73,247,87,233)(74,248,88,234)(75,249,89,235)(76,270,110,236)(77,266,106,237)(78,267,107,238)(79,268,108,239)(80,269,109,240)(96,285,125,256)(97,281,121,257)(98,282,122,258)(99,283,123,259)(100,284,124,260)(101,291,131,261)(102,292,132,262)(103,293,133,263)(104,294,134,264)(105,295,135,265)(111,290,130,271)(112,286,126,272)(113,287,127,273)(114,288,128,274)(115,289,129,275)(116,310,150,276)(117,306,146,277)(118,307,147,278)(119,308,148,279)(120,309,149,280) );
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)], [(1,127,42,97),(2,128,43,98),(3,129,44,99),(4,130,45,100),(5,126,41,96),(6,253,312,236),(7,254,313,237),(8,255,314,238),(9,251,315,239),(10,252,311,240),(11,116,70,133),(12,117,66,134),(13,118,67,135),(14,119,68,131),(15,120,69,132),(16,247,304,217),(17,248,305,218),(18,249,301,219),(19,250,302,220),(20,246,303,216),(21,244,320,231),(22,245,316,232),(23,241,317,233),(24,242,318,234),(25,243,319,235),(26,115,49,123),(27,111,50,124),(28,112,46,125),(29,113,47,121),(30,114,48,122),(31,270,296,223),(32,266,297,224),(33,267,298,225),(34,268,299,221),(35,269,300,222),(36,103,53,150),(37,104,54,146),(38,105,55,147),(39,101,51,148),(40,102,52,149),(56,152,86,165),(57,153,87,161),(58,154,88,162),(59,155,89,163),(60,151,90,164),(61,159,108,171),(62,160,109,172),(63,156,110,173),(64,157,106,174),(65,158,107,175),(71,140,84,170),(72,136,85,166),(73,137,81,167),(74,138,82,168),(75,139,83,169),(76,143,93,190),(77,144,94,186),(78,145,95,187),(79,141,91,188),(80,142,92,189),(176,310,206,263),(177,306,207,264),(178,307,208,265),(179,308,209,261),(180,309,210,262),(181,289,228,259),(182,290,229,260),(183,286,230,256),(184,287,226,257),(185,288,227,258),(191,292,204,280),(192,293,205,276),(193,294,201,277),(194,295,202,278),(195,291,203,279),(196,285,213,272),(197,281,214,273),(198,282,215,274),(199,283,211,275),(200,284,212,271)], [(1,94,12,57),(2,95,13,58),(3,91,14,59),(4,92,15,60),(5,93,11,56),(6,276,303,286),(7,277,304,287),(8,278,305,288),(9,279,301,289),(10,280,302,290),(16,257,313,294),(17,258,314,295),(18,259,315,291),(19,260,311,292),(20,256,312,293),(21,271,300,309),(22,272,296,310),(23,273,297,306),(24,274,298,307),(25,275,299,308),(26,61,51,83),(27,62,52,84),(28,63,53,85),(29,64,54,81),(30,65,55,82),(31,263,316,285),(32,264,317,281),(33,265,318,282),(34,261,319,283),(35,262,320,284),(36,72,46,110),(37,73,47,106),(38,74,48,107),(39,75,49,108),(40,71,50,109),(41,76,70,86),(42,77,66,87),(43,78,67,88),(44,79,68,89),(45,80,69,90),(96,143,133,165),(97,144,134,161),(98,145,135,162),(99,141,131,163),(100,142,132,164),(101,139,123,171),(102,140,124,172),(103,136,125,173),(104,137,121,174),(105,138,122,175),(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),(176,232,213,270),(177,233,214,266),(178,234,215,267),(179,235,211,268),(180,231,212,269),(181,239,203,249),(182,240,204,250),(183,236,205,246),(184,237,201,247),(185,238,202,248),(191,220,229,252),(192,216,230,253),(193,217,226,254),(194,218,227,255),(195,219,228,251),(196,223,206,245),(197,224,207,241),(198,225,208,242),(199,221,209,243),(200,222,210,244)], [(1,177,29,201),(2,178,30,202),(3,179,26,203),(4,180,27,204),(5,176,28,205),(6,165,296,136),(7,161,297,137),(8,162,298,138),(9,163,299,139),(10,164,300,140),(11,213,53,183),(12,214,54,184),(13,215,55,185),(14,211,51,181),(15,212,52,182),(16,186,317,157),(17,187,318,158),(18,188,319,159),(19,189,320,160),(20,190,316,156),(21,172,302,142),(22,173,303,143),(23,174,304,144),(24,175,305,145),(25,171,301,141),(31,166,312,152),(32,167,313,153),(33,168,314,154),(34,169,315,155),(35,170,311,151),(36,230,70,196),(37,226,66,197),(38,227,67,198),(39,228,68,199),(40,229,69,200),(41,206,46,192),(42,207,47,193),(43,208,48,194),(44,209,49,195),(45,210,50,191),(56,245,85,216),(57,241,81,217),(58,242,82,218),(59,243,83,219),(60,244,84,220),(61,251,91,221),(62,252,92,222),(63,253,93,223),(64,254,94,224),(65,255,95,225),(71,250,90,231),(72,246,86,232),(73,247,87,233),(74,248,88,234),(75,249,89,235),(76,270,110,236),(77,266,106,237),(78,267,107,238),(79,268,108,239),(80,269,109,240),(96,285,125,256),(97,281,121,257),(98,282,122,258),(99,283,123,259),(100,284,124,260),(101,291,131,261),(102,292,132,262),(103,293,133,263),(104,294,134,264),(105,295,135,265),(111,290,130,271),(112,286,126,272),(113,287,127,273),(114,288,128,274),(115,289,129,275),(116,310,150,276),(117,306,146,277),(118,307,147,278),(119,308,148,279),(120,309,149,280)]])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 20A | ··· | 20AV | 20AW | ··· | 20CB |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
140 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⋊5C4 | C5×C2.C42 | C2×C4×C20 | C4×C20 | C42⋊5C4 | C2.C42 | C2×C42 | C42 | C2×C10 | C22 |
| # reps | 1 | 6 | 1 | 8 | 4 | 24 | 4 | 32 | 12 | 48 |
Matrix representation of C5×C42⋊5C4 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 37 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 32 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 1 | 30 | 0 | 0 |
| 0 | 30 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,37,0,0,0,0,0,37,0,0,0,0,0,37,0,0,0,0,0,37],[40,0,0,0,0,0,0,9,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,32],[40,0,0,0,0,0,0,1,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,1,0],[9,0,0,0,0,0,1,30,0,0,0,30,40,0,0,0,0,0,1,0,0,0,0,0,40] >;
C5×C42⋊5C4 in GAP, Magma, Sage, TeX
C_5\times C_4^2\rtimes_5C_4
% in TeX
G:=Group("C5xC4^2:5C4"); // GroupNames label
G:=SmallGroup(320,884);
// by ID
G=gap.SmallGroup(320,884);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,848,1766,226]);
// 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,d*c*d^-1=b^2*c^-1>;
// generators/relations