direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C42⋊9C4, C42⋊9C20, (C4×C20)⋊29C4, C20⋊11(C4⋊C4), (C2×C20).72Q8, (C2×C20).412D4, C10.35(C4⋊Q8), (C2×C42).9C10, C22.33(D4×C10), C22.11(Q8×C10), C10.38(C4⋊1D4), C23.57(C22×C10), C22.33(C22×C20), (C22×C20).574C22, (C22×C10).448C23, C4⋊1(C5×C4⋊C4), C2.1(C5×C4⋊Q8), C2.6(C10×C4⋊C4), (C2×C4×C20).32C2, (C2×C4⋊C4).5C10, C10.84(C2×C4⋊C4), (C2×C4).66(C5×D4), C2.1(C5×C4⋊1D4), (C10×C4⋊C4).34C2, (C2×C4).15(C5×Q8), (C2×C4).70(C2×C20), (C2×C20).504(C2×C4), (C2×C10).600(C2×D4), (C2×C10).103(C2×Q8), (C2×C10).321(C22×C4), (C22×C4).107(C2×C10), SmallGroup(320,885)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C42⋊9C4
G = < a,b,c,d | a5=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 258 in 186 conjugacy classes, 130 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C42, C4⋊C4, C22×C4, C20, C20, C2×C10, C2×C10, C2×C42, C2×C4⋊C4, C2×C20, C2×C20, C22×C10, C42⋊9C4, C4×C20, C5×C4⋊C4, C22×C20, C2×C4×C20, C10×C4⋊C4, C5×C42⋊9C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C23, C10, C4⋊C4, C22×C4, C2×D4, C2×Q8, C20, C2×C10, C2×C4⋊C4, C4⋊1D4, C4⋊Q8, C2×C20, C5×D4, C5×Q8, C22×C10, C42⋊9C4, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, C10×C4⋊C4, C5×C4⋊1D4, C5×C4⋊Q8, C5×C42⋊9C4
►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)
(1 134 42 105)(2 135 43 101)(3 131 44 102)(4 132 45 103)(5 133 41 104)(6 253 300 224)(7 254 296 225)(8 255 297 221)(9 251 298 222)(10 252 299 223)(11 100 53 125)(12 96 54 121)(13 97 55 122)(14 98 51 123)(15 99 52 124)(16 242 301 217)(17 243 302 218)(18 244 303 219)(19 245 304 220)(20 241 305 216)(21 249 315 239)(22 250 311 240)(23 246 312 236)(24 247 313 237)(25 248 314 238)(26 120 69 130)(27 116 70 126)(28 117 66 127)(29 118 67 128)(30 119 68 129)(31 270 316 232)(32 266 317 233)(33 267 318 234)(34 268 319 235)(35 269 320 231)(36 112 46 150)(37 113 47 146)(38 114 48 147)(39 115 49 148)(40 111 50 149)(56 145 81 174)(57 141 82 175)(58 142 83 171)(59 143 84 172)(60 144 85 173)(61 137 95 162)(62 138 91 163)(63 139 92 164)(64 140 93 165)(65 136 94 161)(71 160 109 170)(72 156 110 166)(73 157 106 167)(74 158 107 168)(75 159 108 169)(76 152 86 190)(77 153 87 186)(78 154 88 187)(79 155 89 188)(80 151 90 189)(176 294 201 265)(177 295 202 261)(178 291 203 262)(179 292 204 263)(180 293 205 264)(181 282 215 257)(182 283 211 258)(183 284 212 259)(184 285 213 260)(185 281 214 256)(191 290 229 280)(192 286 230 276)(193 287 226 277)(194 288 227 278)(195 289 228 279)(196 310 206 272)(197 306 207 273)(198 307 208 274)(199 308 209 275)(200 309 210 271)
(1 77 66 65)(2 78 67 61)(3 79 68 62)(4 80 69 63)(5 76 70 64)(6 276 23 264)(7 277 24 265)(8 278 25 261)(9 279 21 262)(10 280 22 263)(11 110 36 85)(12 106 37 81)(13 107 38 82)(14 108 39 83)(15 109 40 84)(16 274 33 257)(17 275 34 258)(18 271 35 259)(19 272 31 260)(20 273 32 256)(26 92 45 90)(27 93 41 86)(28 94 42 87)(29 95 43 88)(30 91 44 89)(46 60 53 72)(47 56 54 73)(48 57 55 74)(49 58 51 75)(50 59 52 71)(96 167 113 174)(97 168 114 175)(98 169 115 171)(99 170 111 172)(100 166 112 173)(101 187 118 162)(102 188 119 163)(103 189 120 164)(104 190 116 165)(105 186 117 161)(121 157 146 145)(122 158 147 141)(123 159 148 142)(124 160 149 143)(125 156 150 144)(126 140 133 152)(127 136 134 153)(128 137 135 154)(129 138 131 155)(130 139 132 151)(176 254 193 247)(177 255 194 248)(178 251 195 249)(179 252 191 250)(180 253 192 246)(181 242 198 267)(182 243 199 268)(183 244 200 269)(184 245 196 270)(185 241 197 266)(201 225 226 237)(202 221 227 238)(203 222 228 239)(204 223 229 240)(205 224 230 236)(206 232 213 220)(207 233 214 216)(208 234 215 217)(209 235 211 218)(210 231 212 219)(281 305 306 317)(282 301 307 318)(283 302 308 319)(284 303 309 320)(285 304 310 316)(286 312 293 300)(287 313 294 296)(288 314 295 297)(289 315 291 298)(290 311 292 299)
(1 185 12 201)(2 181 13 202)(3 182 14 203)(4 183 15 204)(5 184 11 205)(6 165 304 144)(7 161 305 145)(8 162 301 141)(9 163 302 142)(10 164 303 143)(16 175 297 137)(17 171 298 138)(18 172 299 139)(19 173 300 140)(20 174 296 136)(21 188 319 159)(22 189 320 160)(23 190 316 156)(24 186 317 157)(25 187 318 158)(26 210 50 191)(27 206 46 192)(28 207 47 193)(29 208 48 194)(30 209 49 195)(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)(41 213 53 180)(42 214 54 176)(43 215 55 177)(44 211 51 178)(45 212 52 179)(56 254 94 216)(57 255 95 217)(58 251 91 218)(59 252 92 219)(60 253 93 220)(61 242 82 221)(62 243 83 222)(63 244 84 223)(64 245 85 224)(65 241 81 225)(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)(96 294 134 256)(97 295 135 257)(98 291 131 258)(99 292 132 259)(100 293 133 260)(101 282 122 261)(102 283 123 262)(103 284 124 263)(104 285 125 264)(105 281 121 265)(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), (1,134,42,105)(2,135,43,101)(3,131,44,102)(4,132,45,103)(5,133,41,104)(6,253,300,224)(7,254,296,225)(8,255,297,221)(9,251,298,222)(10,252,299,223)(11,100,53,125)(12,96,54,121)(13,97,55,122)(14,98,51,123)(15,99,52,124)(16,242,301,217)(17,243,302,218)(18,244,303,219)(19,245,304,220)(20,241,305,216)(21,249,315,239)(22,250,311,240)(23,246,312,236)(24,247,313,237)(25,248,314,238)(26,120,69,130)(27,116,70,126)(28,117,66,127)(29,118,67,128)(30,119,68,129)(31,270,316,232)(32,266,317,233)(33,267,318,234)(34,268,319,235)(35,269,320,231)(36,112,46,150)(37,113,47,146)(38,114,48,147)(39,115,49,148)(40,111,50,149)(56,145,81,174)(57,141,82,175)(58,142,83,171)(59,143,84,172)(60,144,85,173)(61,137,95,162)(62,138,91,163)(63,139,92,164)(64,140,93,165)(65,136,94,161)(71,160,109,170)(72,156,110,166)(73,157,106,167)(74,158,107,168)(75,159,108,169)(76,152,86,190)(77,153,87,186)(78,154,88,187)(79,155,89,188)(80,151,90,189)(176,294,201,265)(177,295,202,261)(178,291,203,262)(179,292,204,263)(180,293,205,264)(181,282,215,257)(182,283,211,258)(183,284,212,259)(184,285,213,260)(185,281,214,256)(191,290,229,280)(192,286,230,276)(193,287,226,277)(194,288,227,278)(195,289,228,279)(196,310,206,272)(197,306,207,273)(198,307,208,274)(199,308,209,275)(200,309,210,271), (1,77,66,65)(2,78,67,61)(3,79,68,62)(4,80,69,63)(5,76,70,64)(6,276,23,264)(7,277,24,265)(8,278,25,261)(9,279,21,262)(10,280,22,263)(11,110,36,85)(12,106,37,81)(13,107,38,82)(14,108,39,83)(15,109,40,84)(16,274,33,257)(17,275,34,258)(18,271,35,259)(19,272,31,260)(20,273,32,256)(26,92,45,90)(27,93,41,86)(28,94,42,87)(29,95,43,88)(30,91,44,89)(46,60,53,72)(47,56,54,73)(48,57,55,74)(49,58,51,75)(50,59,52,71)(96,167,113,174)(97,168,114,175)(98,169,115,171)(99,170,111,172)(100,166,112,173)(101,187,118,162)(102,188,119,163)(103,189,120,164)(104,190,116,165)(105,186,117,161)(121,157,146,145)(122,158,147,141)(123,159,148,142)(124,160,149,143)(125,156,150,144)(126,140,133,152)(127,136,134,153)(128,137,135,154)(129,138,131,155)(130,139,132,151)(176,254,193,247)(177,255,194,248)(178,251,195,249)(179,252,191,250)(180,253,192,246)(181,242,198,267)(182,243,199,268)(183,244,200,269)(184,245,196,270)(185,241,197,266)(201,225,226,237)(202,221,227,238)(203,222,228,239)(204,223,229,240)(205,224,230,236)(206,232,213,220)(207,233,214,216)(208,234,215,217)(209,235,211,218)(210,231,212,219)(281,305,306,317)(282,301,307,318)(283,302,308,319)(284,303,309,320)(285,304,310,316)(286,312,293,300)(287,313,294,296)(288,314,295,297)(289,315,291,298)(290,311,292,299), (1,185,12,201)(2,181,13,202)(3,182,14,203)(4,183,15,204)(5,184,11,205)(6,165,304,144)(7,161,305,145)(8,162,301,141)(9,163,302,142)(10,164,303,143)(16,175,297,137)(17,171,298,138)(18,172,299,139)(19,173,300,140)(20,174,296,136)(21,188,319,159)(22,189,320,160)(23,190,316,156)(24,186,317,157)(25,187,318,158)(26,210,50,191)(27,206,46,192)(28,207,47,193)(29,208,48,194)(30,209,49,195)(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)(41,213,53,180)(42,214,54,176)(43,215,55,177)(44,211,51,178)(45,212,52,179)(56,254,94,216)(57,255,95,217)(58,251,91,218)(59,252,92,219)(60,253,93,220)(61,242,82,221)(62,243,83,222)(63,244,84,223)(64,245,85,224)(65,241,81,225)(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)(96,294,134,256)(97,295,135,257)(98,291,131,258)(99,292,132,259)(100,293,133,260)(101,282,122,261)(102,283,123,262)(103,284,124,263)(104,285,125,264)(105,281,121,265)(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), (1,134,42,105)(2,135,43,101)(3,131,44,102)(4,132,45,103)(5,133,41,104)(6,253,300,224)(7,254,296,225)(8,255,297,221)(9,251,298,222)(10,252,299,223)(11,100,53,125)(12,96,54,121)(13,97,55,122)(14,98,51,123)(15,99,52,124)(16,242,301,217)(17,243,302,218)(18,244,303,219)(19,245,304,220)(20,241,305,216)(21,249,315,239)(22,250,311,240)(23,246,312,236)(24,247,313,237)(25,248,314,238)(26,120,69,130)(27,116,70,126)(28,117,66,127)(29,118,67,128)(30,119,68,129)(31,270,316,232)(32,266,317,233)(33,267,318,234)(34,268,319,235)(35,269,320,231)(36,112,46,150)(37,113,47,146)(38,114,48,147)(39,115,49,148)(40,111,50,149)(56,145,81,174)(57,141,82,175)(58,142,83,171)(59,143,84,172)(60,144,85,173)(61,137,95,162)(62,138,91,163)(63,139,92,164)(64,140,93,165)(65,136,94,161)(71,160,109,170)(72,156,110,166)(73,157,106,167)(74,158,107,168)(75,159,108,169)(76,152,86,190)(77,153,87,186)(78,154,88,187)(79,155,89,188)(80,151,90,189)(176,294,201,265)(177,295,202,261)(178,291,203,262)(179,292,204,263)(180,293,205,264)(181,282,215,257)(182,283,211,258)(183,284,212,259)(184,285,213,260)(185,281,214,256)(191,290,229,280)(192,286,230,276)(193,287,226,277)(194,288,227,278)(195,289,228,279)(196,310,206,272)(197,306,207,273)(198,307,208,274)(199,308,209,275)(200,309,210,271), (1,77,66,65)(2,78,67,61)(3,79,68,62)(4,80,69,63)(5,76,70,64)(6,276,23,264)(7,277,24,265)(8,278,25,261)(9,279,21,262)(10,280,22,263)(11,110,36,85)(12,106,37,81)(13,107,38,82)(14,108,39,83)(15,109,40,84)(16,274,33,257)(17,275,34,258)(18,271,35,259)(19,272,31,260)(20,273,32,256)(26,92,45,90)(27,93,41,86)(28,94,42,87)(29,95,43,88)(30,91,44,89)(46,60,53,72)(47,56,54,73)(48,57,55,74)(49,58,51,75)(50,59,52,71)(96,167,113,174)(97,168,114,175)(98,169,115,171)(99,170,111,172)(100,166,112,173)(101,187,118,162)(102,188,119,163)(103,189,120,164)(104,190,116,165)(105,186,117,161)(121,157,146,145)(122,158,147,141)(123,159,148,142)(124,160,149,143)(125,156,150,144)(126,140,133,152)(127,136,134,153)(128,137,135,154)(129,138,131,155)(130,139,132,151)(176,254,193,247)(177,255,194,248)(178,251,195,249)(179,252,191,250)(180,253,192,246)(181,242,198,267)(182,243,199,268)(183,244,200,269)(184,245,196,270)(185,241,197,266)(201,225,226,237)(202,221,227,238)(203,222,228,239)(204,223,229,240)(205,224,230,236)(206,232,213,220)(207,233,214,216)(208,234,215,217)(209,235,211,218)(210,231,212,219)(281,305,306,317)(282,301,307,318)(283,302,308,319)(284,303,309,320)(285,304,310,316)(286,312,293,300)(287,313,294,296)(288,314,295,297)(289,315,291,298)(290,311,292,299), (1,185,12,201)(2,181,13,202)(3,182,14,203)(4,183,15,204)(5,184,11,205)(6,165,304,144)(7,161,305,145)(8,162,301,141)(9,163,302,142)(10,164,303,143)(16,175,297,137)(17,171,298,138)(18,172,299,139)(19,173,300,140)(20,174,296,136)(21,188,319,159)(22,189,320,160)(23,190,316,156)(24,186,317,157)(25,187,318,158)(26,210,50,191)(27,206,46,192)(28,207,47,193)(29,208,48,194)(30,209,49,195)(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)(41,213,53,180)(42,214,54,176)(43,215,55,177)(44,211,51,178)(45,212,52,179)(56,254,94,216)(57,255,95,217)(58,251,91,218)(59,252,92,219)(60,253,93,220)(61,242,82,221)(62,243,83,222)(63,244,84,223)(64,245,85,224)(65,241,81,225)(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)(96,294,134,256)(97,295,135,257)(98,291,131,258)(99,292,132,259)(100,293,133,260)(101,282,122,261)(102,283,123,262)(103,284,124,263)(104,285,125,264)(105,281,121,265)(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)], [(1,134,42,105),(2,135,43,101),(3,131,44,102),(4,132,45,103),(5,133,41,104),(6,253,300,224),(7,254,296,225),(8,255,297,221),(9,251,298,222),(10,252,299,223),(11,100,53,125),(12,96,54,121),(13,97,55,122),(14,98,51,123),(15,99,52,124),(16,242,301,217),(17,243,302,218),(18,244,303,219),(19,245,304,220),(20,241,305,216),(21,249,315,239),(22,250,311,240),(23,246,312,236),(24,247,313,237),(25,248,314,238),(26,120,69,130),(27,116,70,126),(28,117,66,127),(29,118,67,128),(30,119,68,129),(31,270,316,232),(32,266,317,233),(33,267,318,234),(34,268,319,235),(35,269,320,231),(36,112,46,150),(37,113,47,146),(38,114,48,147),(39,115,49,148),(40,111,50,149),(56,145,81,174),(57,141,82,175),(58,142,83,171),(59,143,84,172),(60,144,85,173),(61,137,95,162),(62,138,91,163),(63,139,92,164),(64,140,93,165),(65,136,94,161),(71,160,109,170),(72,156,110,166),(73,157,106,167),(74,158,107,168),(75,159,108,169),(76,152,86,190),(77,153,87,186),(78,154,88,187),(79,155,89,188),(80,151,90,189),(176,294,201,265),(177,295,202,261),(178,291,203,262),(179,292,204,263),(180,293,205,264),(181,282,215,257),(182,283,211,258),(183,284,212,259),(184,285,213,260),(185,281,214,256),(191,290,229,280),(192,286,230,276),(193,287,226,277),(194,288,227,278),(195,289,228,279),(196,310,206,272),(197,306,207,273),(198,307,208,274),(199,308,209,275),(200,309,210,271)], [(1,77,66,65),(2,78,67,61),(3,79,68,62),(4,80,69,63),(5,76,70,64),(6,276,23,264),(7,277,24,265),(8,278,25,261),(9,279,21,262),(10,280,22,263),(11,110,36,85),(12,106,37,81),(13,107,38,82),(14,108,39,83),(15,109,40,84),(16,274,33,257),(17,275,34,258),(18,271,35,259),(19,272,31,260),(20,273,32,256),(26,92,45,90),(27,93,41,86),(28,94,42,87),(29,95,43,88),(30,91,44,89),(46,60,53,72),(47,56,54,73),(48,57,55,74),(49,58,51,75),(50,59,52,71),(96,167,113,174),(97,168,114,175),(98,169,115,171),(99,170,111,172),(100,166,112,173),(101,187,118,162),(102,188,119,163),(103,189,120,164),(104,190,116,165),(105,186,117,161),(121,157,146,145),(122,158,147,141),(123,159,148,142),(124,160,149,143),(125,156,150,144),(126,140,133,152),(127,136,134,153),(128,137,135,154),(129,138,131,155),(130,139,132,151),(176,254,193,247),(177,255,194,248),(178,251,195,249),(179,252,191,250),(180,253,192,246),(181,242,198,267),(182,243,199,268),(183,244,200,269),(184,245,196,270),(185,241,197,266),(201,225,226,237),(202,221,227,238),(203,222,228,239),(204,223,229,240),(205,224,230,236),(206,232,213,220),(207,233,214,216),(208,234,215,217),(209,235,211,218),(210,231,212,219),(281,305,306,317),(282,301,307,318),(283,302,308,319),(284,303,309,320),(285,304,310,316),(286,312,293,300),(287,313,294,296),(288,314,295,297),(289,315,291,298),(290,311,292,299)], [(1,185,12,201),(2,181,13,202),(3,182,14,203),(4,183,15,204),(5,184,11,205),(6,165,304,144),(7,161,305,145),(8,162,301,141),(9,163,302,142),(10,164,303,143),(16,175,297,137),(17,171,298,138),(18,172,299,139),(19,173,300,140),(20,174,296,136),(21,188,319,159),(22,189,320,160),(23,190,316,156),(24,186,317,157),(25,187,318,158),(26,210,50,191),(27,206,46,192),(28,207,47,193),(29,208,48,194),(30,209,49,195),(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),(41,213,53,180),(42,214,54,176),(43,215,55,177),(44,211,51,178),(45,212,52,179),(56,254,94,216),(57,255,95,217),(58,251,91,218),(59,252,92,219),(60,253,93,220),(61,242,82,221),(62,243,83,222),(63,244,84,223),(64,245,85,224),(65,241,81,225),(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),(96,294,134,256),(97,295,135,257),(98,291,131,258),(99,292,132,259),(100,293,133,260),(101,282,122,261),(102,283,123,262),(103,284,124,263),(104,285,125,264),(105,281,121,265),(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 | 2 | 2 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | D4 | Q8 | C5×D4 | C5×Q8 |
| kernel | C5×C42⋊9C4 | C2×C4×C20 | C10×C4⋊C4 | C4×C20 | C42⋊9C4 | C2×C42 | C2×C4⋊C4 | C42 | C2×C20 | C2×C20 | C2×C4 | C2×C4 |
| # reps | 1 | 1 | 6 | 8 | 4 | 4 | 24 | 32 | 6 | 6 | 24 | 24 |
Matrix representation of C5×C42⋊9C4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 11 | 1 | 0 | 0 | 0 | 0 |
| 1 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 13 | 0 | 0 |
| 0 | 0 | 13 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 4 |
| 0 | 0 | 0 | 0 | 35 | 8 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,40,0,0,0,0,2,1],[11,1,0,0,0,0,1,30,0,0,0,0,0,0,23,13,0,0,0,0,13,18,0,0,0,0,0,0,33,35,0,0,0,0,4,8] >;
C5×C42⋊9C4 in GAP, Magma, Sage, TeX
C_5\times C_4^2\rtimes_9C_4
% in TeX
G:=Group("C5xC4^2:9C4"); // GroupNames label
G:=SmallGroup(320,885);
// by ID
G=gap.SmallGroup(320,885);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,288,1766,436]);
// 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^-1,d*c*d^-1=c^-1>;
// generators/relations