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G = C4⋊C4×Dic5order 320 = 26·5

Direct product of C4⋊C4 and Dic5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4×Dic5, C204C42, C41(C4×Dic5), C4⋊Dic520C4, C10.88(C4×D4), C10.31(C4×Q8), C2.2(D4×Dic5), C2.1(Q8×Dic5), (C4×Dic5)⋊10C4, C22.21(Q8×D5), C10.40(C2×C42), (C2×Dic5).40Q8, C22.106(D4×D5), C2.3(D208C4), (C2×Dic5).279D4, (C22×C4).333D10, C2.3(Dic53Q8), C23.289(C22×D5), C10.53(C42⋊C2), C22.54(D42D5), (C22×C10).339C23, (C22×C20).346C22, C22.22(Q82D5), C22.20(C22×Dic5), C10.10C42.34C2, (C22×Dic5).210C22, C56(C4×C4⋊C4), (C5×C4⋊C4)⋊14C4, C2.4(D5×C4⋊C4), C10.42(C2×C4⋊C4), C2.9(C2×C4×Dic5), (C2×C4⋊C4).28D5, (C10×C4⋊C4).11C2, (C2×C4×Dic5).5C2, C22.55(C2×C4×D5), (C2×C4).151(C4×D5), (C2×C10).75(C2×Q8), (C2×C20).255(C2×C4), C2.4(C4⋊C47D5), (C2×C10).328(C2×D4), (C2×C4⋊Dic5).33C2, (C2×C4).32(C2×Dic5), (C2×C10).151(C4○D4), (C2×C10).216(C22×C4), (C2×Dic5).107(C2×C4), SmallGroup(320,602)

Series: Derived Chief Lower central Upper central

C1C10 — C4⋊C4×Dic5
C1C5C10C2×C10C22×C10C22×Dic5C2×C4×Dic5 — C4⋊C4×Dic5
C5C10 — C4⋊C4×Dic5
C1C23C2×C4⋊C4

Generators and relations for C4⋊C4×Dic5
 G = < a,b,c,d | a4=b4=c10=1, d2=c5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 510 in 194 conjugacy classes, 111 normal (41 characteristic)
C1, C2 [×7], C4 [×4], C4 [×14], C22 [×7], C5, C2×C4 [×10], C2×C4 [×22], C23, C10 [×7], C42 [×8], C4⋊C4 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], Dic5 [×4], Dic5 [×6], C20 [×4], C20 [×4], C2×C10 [×7], C2.C42 [×2], C2×C42 [×3], C2×C4⋊C4, C2×C4⋊C4, C2×Dic5 [×12], C2×Dic5 [×6], C2×C20 [×10], C2×C20 [×4], C22×C10, C4×C4⋊C4, C4×Dic5 [×4], C4×Dic5 [×4], C4⋊Dic5 [×4], C5×C4⋊C4 [×4], C22×Dic5 [×2], C22×Dic5 [×2], C22×C20, C22×C20 [×2], C10.10C42 [×2], C2×C4×Dic5, C2×C4×Dic5 [×2], C2×C4⋊Dic5, C10×C4⋊C4, C4⋊C4×Dic5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×2], Q8 [×2], C23, D5, C42 [×4], C4⋊C4 [×4], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], Dic5 [×4], D10 [×3], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C4×D5 [×4], C2×Dic5 [×6], C22×D5, C4×C4⋊C4, C4×Dic5 [×4], C2×C4×D5 [×2], D4×D5, D42D5, Q8×D5, Q82D5, C22×Dic5, Dic53Q8, D5×C4⋊C4, C4⋊C47D5, D208C4, C2×C4×Dic5, D4×Dic5, Q8×Dic5, C4⋊C4×Dic5

Smallest permutation representation of C4⋊C4×Dic5
Regular action on 320 points
Generators in S320
(1 128 39 110)(2 129 40 101)(3 130 31 102)(4 121 32 103)(5 122 33 104)(6 123 34 105)(7 124 35 106)(8 125 36 107)(9 126 37 108)(10 127 38 109)(11 116 45 91)(12 117 46 92)(13 118 47 93)(14 119 48 94)(15 120 49 95)(16 111 50 96)(17 112 41 97)(18 113 42 98)(19 114 43 99)(20 115 44 100)(21 236 305 216)(22 237 306 217)(23 238 307 218)(24 239 308 219)(25 240 309 220)(26 231 310 211)(27 232 301 212)(28 233 302 213)(29 234 303 214)(30 235 304 215)(51 162 76 147)(52 163 77 148)(53 164 78 149)(54 165 79 150)(55 166 80 141)(56 167 71 142)(57 168 72 143)(58 169 73 144)(59 170 74 145)(60 161 75 146)(61 160 89 135)(62 151 90 136)(63 152 81 137)(64 153 82 138)(65 154 83 139)(66 155 84 140)(67 156 85 131)(68 157 86 132)(69 158 87 133)(70 159 88 134)(171 287 191 270)(172 288 192 261)(173 289 193 262)(174 290 194 263)(175 281 195 264)(176 282 196 265)(177 283 197 266)(178 284 198 267)(179 285 199 268)(180 286 200 269)(181 272 208 252)(182 273 209 253)(183 274 210 254)(184 275 201 255)(185 276 202 256)(186 277 203 257)(187 278 204 258)(188 279 205 259)(189 280 206 260)(190 271 207 251)(221 312 248 292)(222 313 249 293)(223 314 250 294)(224 315 241 295)(225 316 242 296)(226 317 243 297)(227 318 244 298)(228 319 245 299)(229 320 246 300)(230 311 247 291)
(1 70 14 54)(2 61 15 55)(3 62 16 56)(4 63 17 57)(5 64 18 58)(6 65 19 59)(7 66 20 60)(8 67 11 51)(9 68 12 52)(10 69 13 53)(21 265 316 256)(22 266 317 257)(23 267 318 258)(24 268 319 259)(25 269 320 260)(26 270 311 251)(27 261 312 252)(28 262 313 253)(29 263 314 254)(30 264 315 255)(31 90 50 71)(32 81 41 72)(33 82 42 73)(34 83 43 74)(35 84 44 75)(36 85 45 76)(37 86 46 77)(38 87 47 78)(39 88 48 79)(40 89 49 80)(91 162 107 156)(92 163 108 157)(93 164 109 158)(94 165 110 159)(95 166 101 160)(96 167 102 151)(97 168 103 152)(98 169 104 153)(99 170 105 154)(100 161 106 155)(111 142 130 136)(112 143 121 137)(113 144 122 138)(114 145 123 139)(115 146 124 140)(116 147 125 131)(117 148 126 132)(118 149 127 133)(119 150 128 134)(120 141 129 135)(171 230 190 211)(172 221 181 212)(173 222 182 213)(174 223 183 214)(175 224 184 215)(176 225 185 216)(177 226 186 217)(178 227 187 218)(179 228 188 219)(180 229 189 220)(191 247 207 231)(192 248 208 232)(193 249 209 233)(194 250 210 234)(195 241 201 235)(196 242 202 236)(197 243 203 237)(198 244 204 238)(199 245 205 239)(200 246 206 240)(271 310 287 291)(272 301 288 292)(273 302 289 293)(274 303 290 294)(275 304 281 295)(276 305 282 296)(277 306 283 297)(278 307 284 298)(279 308 285 299)(280 309 286 300)
(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)
(1 176 6 171)(2 175 7 180)(3 174 8 179)(4 173 9 178)(5 172 10 177)(11 188 16 183)(12 187 17 182)(13 186 18 181)(14 185 19 190)(15 184 20 189)(21 170 26 165)(22 169 27 164)(23 168 28 163)(24 167 29 162)(25 166 30 161)(31 194 36 199)(32 193 37 198)(33 192 38 197)(34 191 39 196)(35 200 40 195)(41 209 46 204)(42 208 47 203)(43 207 48 202)(44 206 49 201)(45 205 50 210)(51 219 56 214)(52 218 57 213)(53 217 58 212)(54 216 59 211)(55 215 60 220)(61 224 66 229)(62 223 67 228)(63 222 68 227)(64 221 69 226)(65 230 70 225)(71 234 76 239)(72 233 77 238)(73 232 78 237)(74 231 79 236)(75 240 80 235)(81 249 86 244)(82 248 87 243)(83 247 88 242)(84 246 89 241)(85 245 90 250)(91 259 96 254)(92 258 97 253)(93 257 98 252)(94 256 99 251)(95 255 100 260)(101 264 106 269)(102 263 107 268)(103 262 108 267)(104 261 109 266)(105 270 110 265)(111 274 116 279)(112 273 117 278)(113 272 118 277)(114 271 119 276)(115 280 120 275)(121 289 126 284)(122 288 127 283)(123 287 128 282)(124 286 129 281)(125 285 130 290)(131 299 136 294)(132 298 137 293)(133 297 138 292)(134 296 139 291)(135 295 140 300)(141 304 146 309)(142 303 147 308)(143 302 148 307)(144 301 149 306)(145 310 150 305)(151 314 156 319)(152 313 157 318)(153 312 158 317)(154 311 159 316)(155 320 160 315)

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

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

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

80 conjugacy classes

class 1 2A···2G4A···4L4M···4T4U···4AF5A5B10A···10N20A···20X
order12···24···44···44···45510···1020···20
size11···12···25···510···10222···24···4

80 irreducible representations

dim1111111122222224444
type++++++-+-++--+
imageC1C2C2C2C2C4C4C4D4Q8D5C4○D4Dic5D10C4×D5D4×D5D42D5Q8×D5Q82D5
kernelC4⋊C4×Dic5C10.10C42C2×C4×Dic5C2×C4⋊Dic5C10×C4⋊C4C4×Dic5C4⋊Dic5C5×C4⋊C4C2×Dic5C2×Dic5C2×C4⋊C4C2×C10C4⋊C4C22×C4C2×C4C22C22C22C22
# reps12311888222486162222

Matrix representation of C4⋊C4×Dic5 in GL6(𝔽41)

4000000
0400000
001000
000100
0000714
00001434
,
3200000
0320000
001000
000100
0000329
000009
,
4010000
5350000
00344000
008100
000010
000001
,
1230000
7290000
00402000
004100
0000400
0000040

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,14,0,0,0,0,14,34],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,9,9],[40,5,0,0,0,0,1,35,0,0,0,0,0,0,34,8,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,7,0,0,0,0,3,29,0,0,0,0,0,0,40,4,0,0,0,0,20,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;

C4⋊C4×Dic5 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\times {\rm Dic}_5
% in TeX

G:=Group("C4:C4xDic5");
// GroupNames label

G:=SmallGroup(320,602);
// by ID

G=gap.SmallGroup(320,602);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,387,100,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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