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G = C5×C429C4order 320 = 26·5

Direct product of C5 and C429C4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×C429C4, C429C20, (C4×C20)⋊29C4, C2011(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(C41D4), C23.57(C22×C10), C22.33(C22×C20), (C22×C20).574C22, (C22×C10).448C23, C41(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×C41D4), (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

C1C22 — C5×C429C4
C1C2C22C23C22×C10C22×C20C10×C4⋊C4 — C5×C429C4
C1C22 — C5×C429C4
C1C22×C10 — C5×C429C4

Generators and relations for C5×C429C4
 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 [×6], C4 [×12], C4 [×4], C22, C22 [×6], C5, C2×C4 [×18], C2×C4 [×12], C23, C10, C10 [×6], C42 [×4], C4⋊C4 [×12], C22×C4 [×7], C20 [×12], C20 [×4], C2×C10, C2×C10 [×6], C2×C42, C2×C4⋊C4 [×6], C2×C20 [×18], C2×C20 [×12], C22×C10, C429C4, C4×C20 [×4], C5×C4⋊C4 [×12], C22×C20 [×7], C2×C4×C20, C10×C4⋊C4 [×6], C5×C429C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×6], Q8 [×6], C23, C10 [×7], C4⋊C4 [×12], C22×C4, C2×D4 [×3], C2×Q8 [×3], C20 [×4], C2×C10 [×7], C2×C4⋊C4 [×3], C41D4, C4⋊Q8 [×3], C2×C20 [×6], C5×D4 [×6], C5×Q8 [×6], C22×C10, C429C4, C5×C4⋊C4 [×12], C22×C20, D4×C10 [×3], Q8×C10 [×3], C10×C4⋊C4 [×3], C5×C41D4, C5×C4⋊Q8 [×3], C5×C429C4

Smallest permutation representation of C5×C429C4
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 12 97)(2 135 13 98)(3 131 14 99)(4 132 15 100)(5 133 11 96)(6 253 303 216)(7 254 304 217)(8 255 305 218)(9 251 301 219)(10 252 302 220)(16 247 313 237)(17 248 314 238)(18 249 315 239)(19 250 311 240)(20 246 312 236)(21 244 300 222)(22 245 296 223)(23 241 297 224)(24 242 298 225)(25 243 299 221)(26 101 51 123)(27 102 52 124)(28 103 53 125)(29 104 54 121)(30 105 55 122)(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)(41 116 70 126)(42 117 66 127)(43 118 67 128)(44 119 68 129)(45 120 69 130)(56 143 93 165)(57 144 94 161)(58 145 95 162)(59 141 91 163)(60 142 92 164)(61 139 83 171)(62 140 84 172)(63 136 85 173)(64 137 81 174)(65 138 82 175)(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)(176 285 213 263)(177 281 214 264)(178 282 215 265)(179 283 211 261)(180 284 212 262)(181 291 203 259)(182 292 204 260)(183 293 205 256)(184 294 201 257)(185 295 202 258)(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 57)(2 78 67 58)(3 79 68 59)(4 80 69 60)(5 76 70 56)(6 276 20 256)(7 277 16 257)(8 278 17 258)(9 279 18 259)(10 280 19 260)(11 86 41 93)(12 87 42 94)(13 88 43 95)(14 89 44 91)(15 90 45 92)(21 271 35 262)(22 272 31 263)(23 273 32 264)(24 274 33 265)(25 275 34 261)(26 108 39 83)(27 109 40 84)(28 110 36 85)(29 106 37 81)(30 107 38 82)(46 63 53 72)(47 64 54 73)(48 65 55 74)(49 61 51 75)(50 62 52 71)(96 190 116 165)(97 186 117 161)(98 187 118 162)(99 188 119 163)(100 189 120 164)(101 169 115 171)(102 170 111 172)(103 166 112 173)(104 167 113 174)(105 168 114 175)(121 157 146 137)(122 158 147 138)(123 159 148 139)(124 160 149 140)(125 156 150 136)(126 143 133 152)(127 144 134 153)(128 145 135 154)(129 141 131 155)(130 142 132 151)(176 245 196 270)(177 241 197 266)(178 242 198 267)(179 243 199 268)(180 244 200 269)(181 251 195 249)(182 252 191 250)(183 253 192 246)(184 254 193 247)(185 255 194 248)(201 217 226 237)(202 218 227 238)(203 219 228 239)(204 220 229 240)(205 216 230 236)(206 232 213 223)(207 233 214 224)(208 234 215 225)(209 235 211 221)(210 231 212 222)(281 297 306 317)(282 298 307 318)(283 299 308 319)(284 300 309 320)(285 296 310 316)(286 312 293 303)(287 313 294 304)(288 314 295 305)(289 315 291 301)(290 311 292 302)
(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), (1,134,12,97)(2,135,13,98)(3,131,14,99)(4,132,15,100)(5,133,11,96)(6,253,303,216)(7,254,304,217)(8,255,305,218)(9,251,301,219)(10,252,302,220)(16,247,313,237)(17,248,314,238)(18,249,315,239)(19,250,311,240)(20,246,312,236)(21,244,300,222)(22,245,296,223)(23,241,297,224)(24,242,298,225)(25,243,299,221)(26,101,51,123)(27,102,52,124)(28,103,53,125)(29,104,54,121)(30,105,55,122)(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)(41,116,70,126)(42,117,66,127)(43,118,67,128)(44,119,68,129)(45,120,69,130)(56,143,93,165)(57,144,94,161)(58,145,95,162)(59,141,91,163)(60,142,92,164)(61,139,83,171)(62,140,84,172)(63,136,85,173)(64,137,81,174)(65,138,82,175)(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)(176,285,213,263)(177,281,214,264)(178,282,215,265)(179,283,211,261)(180,284,212,262)(181,291,203,259)(182,292,204,260)(183,293,205,256)(184,294,201,257)(185,295,202,258)(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), (1,77,66,57)(2,78,67,58)(3,79,68,59)(4,80,69,60)(5,76,70,56)(6,276,20,256)(7,277,16,257)(8,278,17,258)(9,279,18,259)(10,280,19,260)(11,86,41,93)(12,87,42,94)(13,88,43,95)(14,89,44,91)(15,90,45,92)(21,271,35,262)(22,272,31,263)(23,273,32,264)(24,274,33,265)(25,275,34,261)(26,108,39,83)(27,109,40,84)(28,110,36,85)(29,106,37,81)(30,107,38,82)(46,63,53,72)(47,64,54,73)(48,65,55,74)(49,61,51,75)(50,62,52,71)(96,190,116,165)(97,186,117,161)(98,187,118,162)(99,188,119,163)(100,189,120,164)(101,169,115,171)(102,170,111,172)(103,166,112,173)(104,167,113,174)(105,168,114,175)(121,157,146,137)(122,158,147,138)(123,159,148,139)(124,160,149,140)(125,156,150,136)(126,143,133,152)(127,144,134,153)(128,145,135,154)(129,141,131,155)(130,142,132,151)(176,245,196,270)(177,241,197,266)(178,242,198,267)(179,243,199,268)(180,244,200,269)(181,251,195,249)(182,252,191,250)(183,253,192,246)(184,254,193,247)(185,255,194,248)(201,217,226,237)(202,218,227,238)(203,219,228,239)(204,220,229,240)(205,216,230,236)(206,232,213,223)(207,233,214,224)(208,234,215,225)(209,235,211,221)(210,231,212,222)(281,297,306,317)(282,298,307,318)(283,299,308,319)(284,300,309,320)(285,296,310,316)(286,312,293,303)(287,313,294,304)(288,314,295,305)(289,315,291,301)(290,311,292,302), (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), (1,134,12,97)(2,135,13,98)(3,131,14,99)(4,132,15,100)(5,133,11,96)(6,253,303,216)(7,254,304,217)(8,255,305,218)(9,251,301,219)(10,252,302,220)(16,247,313,237)(17,248,314,238)(18,249,315,239)(19,250,311,240)(20,246,312,236)(21,244,300,222)(22,245,296,223)(23,241,297,224)(24,242,298,225)(25,243,299,221)(26,101,51,123)(27,102,52,124)(28,103,53,125)(29,104,54,121)(30,105,55,122)(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)(41,116,70,126)(42,117,66,127)(43,118,67,128)(44,119,68,129)(45,120,69,130)(56,143,93,165)(57,144,94,161)(58,145,95,162)(59,141,91,163)(60,142,92,164)(61,139,83,171)(62,140,84,172)(63,136,85,173)(64,137,81,174)(65,138,82,175)(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)(176,285,213,263)(177,281,214,264)(178,282,215,265)(179,283,211,261)(180,284,212,262)(181,291,203,259)(182,292,204,260)(183,293,205,256)(184,294,201,257)(185,295,202,258)(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), (1,77,66,57)(2,78,67,58)(3,79,68,59)(4,80,69,60)(5,76,70,56)(6,276,20,256)(7,277,16,257)(8,278,17,258)(9,279,18,259)(10,280,19,260)(11,86,41,93)(12,87,42,94)(13,88,43,95)(14,89,44,91)(15,90,45,92)(21,271,35,262)(22,272,31,263)(23,273,32,264)(24,274,33,265)(25,275,34,261)(26,108,39,83)(27,109,40,84)(28,110,36,85)(29,106,37,81)(30,107,38,82)(46,63,53,72)(47,64,54,73)(48,65,55,74)(49,61,51,75)(50,62,52,71)(96,190,116,165)(97,186,117,161)(98,187,118,162)(99,188,119,163)(100,189,120,164)(101,169,115,171)(102,170,111,172)(103,166,112,173)(104,167,113,174)(105,168,114,175)(121,157,146,137)(122,158,147,138)(123,159,148,139)(124,160,149,140)(125,156,150,136)(126,143,133,152)(127,144,134,153)(128,145,135,154)(129,141,131,155)(130,142,132,151)(176,245,196,270)(177,241,197,266)(178,242,198,267)(179,243,199,268)(180,244,200,269)(181,251,195,249)(182,252,191,250)(183,253,192,246)(184,254,193,247)(185,255,194,248)(201,217,226,237)(202,218,227,238)(203,219,228,239)(204,220,229,240)(205,216,230,236)(206,232,213,223)(207,233,214,224)(208,234,215,225)(209,235,211,221)(210,231,212,222)(281,297,306,317)(282,298,307,318)(283,299,308,319)(284,300,309,320)(285,296,310,316)(286,312,293,303)(287,313,294,304)(288,314,295,305)(289,315,291,301)(290,311,292,302), (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)], [(1,134,12,97),(2,135,13,98),(3,131,14,99),(4,132,15,100),(5,133,11,96),(6,253,303,216),(7,254,304,217),(8,255,305,218),(9,251,301,219),(10,252,302,220),(16,247,313,237),(17,248,314,238),(18,249,315,239),(19,250,311,240),(20,246,312,236),(21,244,300,222),(22,245,296,223),(23,241,297,224),(24,242,298,225),(25,243,299,221),(26,101,51,123),(27,102,52,124),(28,103,53,125),(29,104,54,121),(30,105,55,122),(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),(41,116,70,126),(42,117,66,127),(43,118,67,128),(44,119,68,129),(45,120,69,130),(56,143,93,165),(57,144,94,161),(58,145,95,162),(59,141,91,163),(60,142,92,164),(61,139,83,171),(62,140,84,172),(63,136,85,173),(64,137,81,174),(65,138,82,175),(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),(176,285,213,263),(177,281,214,264),(178,282,215,265),(179,283,211,261),(180,284,212,262),(181,291,203,259),(182,292,204,260),(183,293,205,256),(184,294,201,257),(185,295,202,258),(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)], [(1,77,66,57),(2,78,67,58),(3,79,68,59),(4,80,69,60),(5,76,70,56),(6,276,20,256),(7,277,16,257),(8,278,17,258),(9,279,18,259),(10,280,19,260),(11,86,41,93),(12,87,42,94),(13,88,43,95),(14,89,44,91),(15,90,45,92),(21,271,35,262),(22,272,31,263),(23,273,32,264),(24,274,33,265),(25,275,34,261),(26,108,39,83),(27,109,40,84),(28,110,36,85),(29,106,37,81),(30,107,38,82),(46,63,53,72),(47,64,54,73),(48,65,55,74),(49,61,51,75),(50,62,52,71),(96,190,116,165),(97,186,117,161),(98,187,118,162),(99,188,119,163),(100,189,120,164),(101,169,115,171),(102,170,111,172),(103,166,112,173),(104,167,113,174),(105,168,114,175),(121,157,146,137),(122,158,147,138),(123,159,148,139),(124,160,149,140),(125,156,150,136),(126,143,133,152),(127,144,134,153),(128,145,135,154),(129,141,131,155),(130,142,132,151),(176,245,196,270),(177,241,197,266),(178,242,198,267),(179,243,199,268),(180,244,200,269),(181,251,195,249),(182,252,191,250),(183,253,192,246),(184,254,193,247),(185,255,194,248),(201,217,226,237),(202,218,227,238),(203,219,228,239),(204,220,229,240),(205,216,230,236),(206,232,213,223),(207,233,214,224),(208,234,215,225),(209,235,211,221),(210,231,212,222),(281,297,306,317),(282,298,307,318),(283,299,308,319),(284,300,309,320),(285,296,310,316),(286,312,293,303),(287,313,294,304),(288,314,295,305),(289,315,291,301),(290,311,292,302)], [(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···2G4A···4L4M···4T5A5B5C5D10A···10AB20A···20AV20AW···20CB
order12···24···44···4555510···1020···2020···20
size11···12···24···411111···12···24···4

140 irreducible representations

dim111111112222
type++++-
imageC1C2C2C4C5C10C10C20D4Q8C5×D4C5×Q8
kernelC5×C429C4C2×C4×C20C10×C4⋊C4C4×C20C429C4C2×C42C2×C4⋊C4C42C2×C20C2×C20C2×C4C2×C4
# reps1168442432662424

Matrix representation of C5×C429C4 in GL6(𝔽41)

100000
010000
0037000
0003700
0000180
0000018
,
0400000
100000
000100
0040000
0000400
0000040
,
4000000
0400000
0040000
0004000
0000402
0000401
,
1110000
1300000
00231300
00131800
0000334
0000358

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×C429C4 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

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