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## G = C5×C42⋊8C4order 320 = 26·5

### Direct product of C5 and C42⋊8C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C42⋊8C4
 Chief series C1 — C2 — C22 — C23 — C22×C10 — C22×C20 — C5×C2.C42 — C5×C42⋊8C4
 Lower central C1 — C22 — C5×C42⋊8C4
 Upper central C1 — C22×C10 — C5×C42⋊8C4

Generators and relations for C5×C428C4
G = < a,b,c,d | a5=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >

Subgroups: 226 in 154 conjugacy classes, 98 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C42, C4⋊C4, C22×C4, C22×C4, C20, C20, C2×C10, C2×C10, C2.C42, C2×C42, C2×C4⋊C4, C2×C20, C2×C20, C22×C10, C428C4, C4×C20, C5×C4⋊C4, C22×C20, C22×C20, C5×C2.C42, C2×C4×C20, C10×C4⋊C4, C5×C428C4
Quotients:

Smallest permutation representation of C5×C428C4
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)
(1 117 66 105)(2 118 67 101)(3 119 68 102)(4 120 69 103)(5 116 70 104)(6 253 23 246)(7 254 24 247)(8 255 25 248)(9 251 21 249)(10 252 22 250)(11 150 36 125)(12 146 37 121)(13 147 38 122)(14 148 39 123)(15 149 40 124)(16 242 33 267)(17 243 34 268)(18 244 35 269)(19 245 31 270)(20 241 32 266)(26 132 45 130)(27 133 41 126)(28 134 42 127)(29 135 43 128)(30 131 44 129)(46 100 53 112)(47 96 54 113)(48 97 55 114)(49 98 51 115)(50 99 52 111)(56 167 73 174)(57 168 74 175)(58 169 75 171)(59 170 71 172)(60 166 72 173)(61 187 78 162)(62 188 79 163)(63 189 80 164)(64 190 76 165)(65 186 77 161)(81 157 106 145)(82 158 107 141)(83 159 108 142)(84 160 109 143)(85 156 110 144)(86 140 93 152)(87 136 94 153)(88 137 95 154)(89 138 91 155)(90 139 92 151)(176 277 193 265)(177 278 194 261)(178 279 195 262)(179 280 191 263)(180 276 192 264)(181 274 198 257)(182 275 199 258)(183 271 200 259)(184 272 196 260)(185 273 197 256)(201 287 226 294)(202 288 227 295)(203 289 228 291)(204 290 229 292)(205 286 230 293)(206 285 213 310)(207 281 214 306)(208 282 215 307)(209 283 211 308)(210 284 212 309)(216 317 233 305)(217 318 234 301)(218 319 235 302)(219 320 231 303)(220 316 232 304)(221 314 238 297)(222 315 239 298)(223 311 240 299)(224 312 236 300)(225 313 237 296)
(1 94 42 65)(2 95 43 61)(3 91 44 62)(4 92 45 63)(5 93 41 64)(6 276 300 286)(7 277 296 287)(8 278 297 288)(9 279 298 289)(10 280 299 290)(11 60 53 85)(12 56 54 81)(13 57 55 82)(14 58 51 83)(15 59 52 84)(16 274 301 307)(17 275 302 308)(18 271 303 309)(19 272 304 310)(20 273 305 306)(21 262 315 291)(22 263 311 292)(23 264 312 293)(24 265 313 294)(25 261 314 295)(26 80 69 90)(27 76 70 86)(28 77 66 87)(29 78 67 88)(30 79 68 89)(31 260 316 285)(32 256 317 281)(33 257 318 282)(34 258 319 283)(35 259 320 284)(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)(176 237 201 247)(177 238 202 248)(178 239 203 249)(179 240 204 250)(180 236 205 246)(181 234 215 267)(182 235 211 268)(183 231 212 269)(184 232 213 270)(185 233 214 266)(191 223 229 252)(192 224 230 253)(193 225 226 254)(194 221 227 255)(195 222 228 251)(196 220 206 245)(197 216 207 241)(198 217 208 242)(199 218 209 243)(200 219 210 244)
(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), (1,117,66,105)(2,118,67,101)(3,119,68,102)(4,120,69,103)(5,116,70,104)(6,253,23,246)(7,254,24,247)(8,255,25,248)(9,251,21,249)(10,252,22,250)(11,150,36,125)(12,146,37,121)(13,147,38,122)(14,148,39,123)(15,149,40,124)(16,242,33,267)(17,243,34,268)(18,244,35,269)(19,245,31,270)(20,241,32,266)(26,132,45,130)(27,133,41,126)(28,134,42,127)(29,135,43,128)(30,131,44,129)(46,100,53,112)(47,96,54,113)(48,97,55,114)(49,98,51,115)(50,99,52,111)(56,167,73,174)(57,168,74,175)(58,169,75,171)(59,170,71,172)(60,166,72,173)(61,187,78,162)(62,188,79,163)(63,189,80,164)(64,190,76,165)(65,186,77,161)(81,157,106,145)(82,158,107,141)(83,159,108,142)(84,160,109,143)(85,156,110,144)(86,140,93,152)(87,136,94,153)(88,137,95,154)(89,138,91,155)(90,139,92,151)(176,277,193,265)(177,278,194,261)(178,279,195,262)(179,280,191,263)(180,276,192,264)(181,274,198,257)(182,275,199,258)(183,271,200,259)(184,272,196,260)(185,273,197,256)(201,287,226,294)(202,288,227,295)(203,289,228,291)(204,290,229,292)(205,286,230,293)(206,285,213,310)(207,281,214,306)(208,282,215,307)(209,283,211,308)(210,284,212,309)(216,317,233,305)(217,318,234,301)(218,319,235,302)(219,320,231,303)(220,316,232,304)(221,314,238,297)(222,315,239,298)(223,311,240,299)(224,312,236,300)(225,313,237,296), (1,94,42,65)(2,95,43,61)(3,91,44,62)(4,92,45,63)(5,93,41,64)(6,276,300,286)(7,277,296,287)(8,278,297,288)(9,279,298,289)(10,280,299,290)(11,60,53,85)(12,56,54,81)(13,57,55,82)(14,58,51,83)(15,59,52,84)(16,274,301,307)(17,275,302,308)(18,271,303,309)(19,272,304,310)(20,273,305,306)(21,262,315,291)(22,263,311,292)(23,264,312,293)(24,265,313,294)(25,261,314,295)(26,80,69,90)(27,76,70,86)(28,77,66,87)(29,78,67,88)(30,79,68,89)(31,260,316,285)(32,256,317,281)(33,257,318,282)(34,258,319,283)(35,259,320,284)(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)(176,237,201,247)(177,238,202,248)(178,239,203,249)(179,240,204,250)(180,236,205,246)(181,234,215,267)(182,235,211,268)(183,231,212,269)(184,232,213,270)(185,233,214,266)(191,223,229,252)(192,224,230,253)(193,225,226,254)(194,221,227,255)(195,222,228,251)(196,220,206,245)(197,216,207,241)(198,217,208,242)(199,218,209,243)(200,219,210,244), (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), (1,117,66,105)(2,118,67,101)(3,119,68,102)(4,120,69,103)(5,116,70,104)(6,253,23,246)(7,254,24,247)(8,255,25,248)(9,251,21,249)(10,252,22,250)(11,150,36,125)(12,146,37,121)(13,147,38,122)(14,148,39,123)(15,149,40,124)(16,242,33,267)(17,243,34,268)(18,244,35,269)(19,245,31,270)(20,241,32,266)(26,132,45,130)(27,133,41,126)(28,134,42,127)(29,135,43,128)(30,131,44,129)(46,100,53,112)(47,96,54,113)(48,97,55,114)(49,98,51,115)(50,99,52,111)(56,167,73,174)(57,168,74,175)(58,169,75,171)(59,170,71,172)(60,166,72,173)(61,187,78,162)(62,188,79,163)(63,189,80,164)(64,190,76,165)(65,186,77,161)(81,157,106,145)(82,158,107,141)(83,159,108,142)(84,160,109,143)(85,156,110,144)(86,140,93,152)(87,136,94,153)(88,137,95,154)(89,138,91,155)(90,139,92,151)(176,277,193,265)(177,278,194,261)(178,279,195,262)(179,280,191,263)(180,276,192,264)(181,274,198,257)(182,275,199,258)(183,271,200,259)(184,272,196,260)(185,273,197,256)(201,287,226,294)(202,288,227,295)(203,289,228,291)(204,290,229,292)(205,286,230,293)(206,285,213,310)(207,281,214,306)(208,282,215,307)(209,283,211,308)(210,284,212,309)(216,317,233,305)(217,318,234,301)(218,319,235,302)(219,320,231,303)(220,316,232,304)(221,314,238,297)(222,315,239,298)(223,311,240,299)(224,312,236,300)(225,313,237,296), (1,94,42,65)(2,95,43,61)(3,91,44,62)(4,92,45,63)(5,93,41,64)(6,276,300,286)(7,277,296,287)(8,278,297,288)(9,279,298,289)(10,280,299,290)(11,60,53,85)(12,56,54,81)(13,57,55,82)(14,58,51,83)(15,59,52,84)(16,274,301,307)(17,275,302,308)(18,271,303,309)(19,272,304,310)(20,273,305,306)(21,262,315,291)(22,263,311,292)(23,264,312,293)(24,265,313,294)(25,261,314,295)(26,80,69,90)(27,76,70,86)(28,77,66,87)(29,78,67,88)(30,79,68,89)(31,260,316,285)(32,256,317,281)(33,257,318,282)(34,258,319,283)(35,259,320,284)(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)(176,237,201,247)(177,238,202,248)(178,239,203,249)(179,240,204,250)(180,236,205,246)(181,234,215,267)(182,235,211,268)(183,231,212,269)(184,232,213,270)(185,233,214,266)(191,223,229,252)(192,224,230,253)(193,225,226,254)(194,221,227,255)(195,222,228,251)(196,220,206,245)(197,216,207,241)(198,217,208,242)(199,218,209,243)(200,219,210,244), (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)], [(1,117,66,105),(2,118,67,101),(3,119,68,102),(4,120,69,103),(5,116,70,104),(6,253,23,246),(7,254,24,247),(8,255,25,248),(9,251,21,249),(10,252,22,250),(11,150,36,125),(12,146,37,121),(13,147,38,122),(14,148,39,123),(15,149,40,124),(16,242,33,267),(17,243,34,268),(18,244,35,269),(19,245,31,270),(20,241,32,266),(26,132,45,130),(27,133,41,126),(28,134,42,127),(29,135,43,128),(30,131,44,129),(46,100,53,112),(47,96,54,113),(48,97,55,114),(49,98,51,115),(50,99,52,111),(56,167,73,174),(57,168,74,175),(58,169,75,171),(59,170,71,172),(60,166,72,173),(61,187,78,162),(62,188,79,163),(63,189,80,164),(64,190,76,165),(65,186,77,161),(81,157,106,145),(82,158,107,141),(83,159,108,142),(84,160,109,143),(85,156,110,144),(86,140,93,152),(87,136,94,153),(88,137,95,154),(89,138,91,155),(90,139,92,151),(176,277,193,265),(177,278,194,261),(178,279,195,262),(179,280,191,263),(180,276,192,264),(181,274,198,257),(182,275,199,258),(183,271,200,259),(184,272,196,260),(185,273,197,256),(201,287,226,294),(202,288,227,295),(203,289,228,291),(204,290,229,292),(205,286,230,293),(206,285,213,310),(207,281,214,306),(208,282,215,307),(209,283,211,308),(210,284,212,309),(216,317,233,305),(217,318,234,301),(218,319,235,302),(219,320,231,303),(220,316,232,304),(221,314,238,297),(222,315,239,298),(223,311,240,299),(224,312,236,300),(225,313,237,296)], [(1,94,42,65),(2,95,43,61),(3,91,44,62),(4,92,45,63),(5,93,41,64),(6,276,300,286),(7,277,296,287),(8,278,297,288),(9,279,298,289),(10,280,299,290),(11,60,53,85),(12,56,54,81),(13,57,55,82),(14,58,51,83),(15,59,52,84),(16,274,301,307),(17,275,302,308),(18,271,303,309),(19,272,304,310),(20,273,305,306),(21,262,315,291),(22,263,311,292),(23,264,312,293),(24,265,313,294),(25,261,314,295),(26,80,69,90),(27,76,70,86),(28,77,66,87),(29,78,67,88),(30,79,68,89),(31,260,316,285),(32,256,317,281),(33,257,318,282),(34,258,319,283),(35,259,320,284),(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),(176,237,201,247),(177,238,202,248),(178,239,203,249),(179,240,204,250),(180,236,205,246),(181,234,215,267),(182,235,211,268),(183,231,212,269),(184,232,213,270),(185,233,214,266),(191,223,229,252),(192,224,230,253),(193,225,226,254),(194,221,227,255),(195,222,228,251),(196,220,206,245),(197,216,207,241),(198,217,208,242),(199,218,209,243),(200,219,210,244)], [(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 1 1 2 2 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 D4 Q8 C4○D4 C5×D4 C5×Q8 C5×C4○D4 kernel C5×C42⋊8C4 C5×C2.C42 C2×C4×C20 C10×C4⋊C4 C4×C20 C42⋊8C4 C2.C42 C2×C42 C2×C4⋊C4 C42 C2×C20 C2×C20 C2×C10 C2×C4 C2×C4 C22 # reps 1 4 1 2 8 4 16 4 8 32 2 2 8 8 8 32

Matrix representation of C5×C428C4 in GL5(𝔽41)

 1 0 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 1 0 0 0 0 0 1
,
 40 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 40 0
,
 1 0 0 0 0 0 0 40 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 1 0
,
 9 0 0 0 0 0 26 29 0 0 0 12 15 0 0 0 0 0 1 11 0 0 0 11 40

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,1,0],[1,0,0,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,40,0],[9,0,0,0,0,0,26,12,0,0,0,29,15,0,0,0,0,0,1,11,0,0,0,11,40] >;

C5×C428C4 in GAP, Magma, Sage, TeX

C_5\times C_4^2\rtimes_8C_4
% in TeX

G:=Group("C5xC4^2:8C4");
// GroupNames label

G:=SmallGroup(320,883);
// by ID

G=gap.SmallGroup(320,883);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,568,1766,226]);
// 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>;
// generators/relations

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