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G = C424Dic5order 320 = 26·5

1st semidirect product of C42 and Dic5 acting via Dic5/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C424Dic5, C20.54C42, (C4×C20)⋊25C4, (C2×C42).4D5, C53(C424C4), (C4×Dic5)⋊18C4, C4.19(C4×Dic5), C10.37(C2×C42), (C22×C4).458D10, C2.3(C42⋊D5), C22.42(C4○D20), C23.265(C22×D5), C10.31(C42⋊C2), (C22×C10).307C23, (C22×C20).472C22, C22.17(C22×Dic5), C10.10C42.42C2, C2.2(C23.21D10), (C22×Dic5).202C22, (C2×C4×C20).29C2, C2.6(C2×C4×Dic5), C22.51(C2×C4×D5), (C2×C4).180(C4×D5), (C2×C4×Dic5).33C2, (C2×C20).489(C2×C4), (C2×C4).60(C2×Dic5), (C2×C10).67(C4○D4), (C2×C10).275(C22×C4), (C2×Dic5).146(C2×C4), SmallGroup(320,559)

Series: Derived Chief Lower central Upper central

C1C10 — C424Dic5
C1C5C10C2×C10C22×C10C22×Dic5C2×C4×Dic5 — C424Dic5
C5C10 — C424Dic5
C1C22×C4C2×C42

Generators and relations for C424Dic5
 G = < a,b,c,d | a4=b4=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 462 in 178 conjugacy classes, 103 normal (13 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×12], C22, C22 [×6], C5, C2×C4 [×10], C2×C4 [×20], C23, C10, C10 [×6], C42 [×4], C42 [×8], C22×C4, C22×C4 [×2], C22×C4 [×4], Dic5 [×8], C20 [×4], C20 [×4], C2×C10, C2×C10 [×6], C2.C42 [×4], C2×C42, C2×C42 [×2], C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×10], C2×C20 [×4], C22×C10, C424C4, C4×Dic5 [×8], C4×C20 [×4], C22×Dic5 [×4], C22×C20, C22×C20 [×2], C10.10C42 [×4], C2×C4×Dic5 [×2], C2×C4×C20, C424Dic5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, D5, C42 [×4], C22×C4 [×3], C4○D4 [×4], Dic5 [×4], D10 [×3], C2×C42, C42⋊C2 [×6], C4×D5 [×4], C2×Dic5 [×6], C22×D5, C424C4, C4×Dic5 [×4], C2×C4×D5 [×2], C4○D20 [×4], C22×Dic5, C42⋊D5 [×4], C2×C4×Dic5, C23.21D10 [×2], C424Dic5

Smallest permutation representation of C424Dic5
Regular action on 320 points
Generators in S320
(1 119 39 94)(2 120 40 95)(3 111 31 96)(4 112 32 97)(5 113 33 98)(6 114 34 99)(7 115 35 100)(8 116 36 91)(9 117 37 92)(10 118 38 93)(11 125 45 107)(12 126 46 108)(13 127 47 109)(14 128 48 110)(15 129 49 101)(16 130 50 102)(17 121 41 103)(18 122 42 104)(19 123 43 105)(20 124 44 106)(21 242 305 225)(22 243 306 226)(23 244 307 227)(24 245 308 228)(25 246 309 229)(26 247 310 230)(27 248 301 221)(28 249 302 222)(29 250 303 223)(30 241 304 224)(51 156 76 131)(52 157 77 132)(53 158 78 133)(54 159 79 134)(55 160 80 135)(56 151 71 136)(57 152 72 137)(58 153 73 138)(59 154 74 139)(60 155 75 140)(61 166 89 141)(62 167 90 142)(63 168 81 143)(64 169 82 144)(65 170 83 145)(66 161 84 146)(67 162 85 147)(68 163 86 148)(69 164 87 149)(70 165 88 150)(171 271 191 251)(172 272 192 252)(173 273 193 253)(174 274 194 254)(175 275 195 255)(176 276 196 256)(177 277 197 257)(178 278 198 258)(179 279 199 259)(180 280 200 260)(181 288 208 261)(182 289 209 262)(183 290 210 263)(184 281 201 264)(185 282 202 265)(186 283 203 266)(187 284 204 267)(188 285 205 268)(189 286 206 269)(190 287 207 270)(211 311 231 291)(212 312 232 292)(213 313 233 293)(214 314 234 294)(215 315 235 295)(216 316 236 296)(217 317 237 297)(218 318 238 298)(219 319 239 299)(220 320 240 300)
(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)(91 147 107 131)(92 148 108 132)(93 149 109 133)(94 150 110 134)(95 141 101 135)(96 142 102 136)(97 143 103 137)(98 144 104 138)(99 145 105 139)(100 146 106 140)(111 167 130 151)(112 168 121 152)(113 169 122 153)(114 170 123 154)(115 161 124 155)(116 162 125 156)(117 163 126 157)(118 164 127 158)(119 165 128 159)(120 166 129 160)(171 247 190 231)(172 248 181 232)(173 249 182 233)(174 250 183 234)(175 241 184 235)(176 242 185 236)(177 243 186 237)(178 244 187 238)(179 245 188 239)(180 246 189 240)(191 230 207 211)(192 221 208 212)(193 222 209 213)(194 223 210 214)(195 224 201 215)(196 225 202 216)(197 226 203 217)(198 227 204 218)(199 228 205 219)(200 229 206 220)(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)| (1,119,39,94)(2,120,40,95)(3,111,31,96)(4,112,32,97)(5,113,33,98)(6,114,34,99)(7,115,35,100)(8,116,36,91)(9,117,37,92)(10,118,38,93)(11,125,45,107)(12,126,46,108)(13,127,47,109)(14,128,48,110)(15,129,49,101)(16,130,50,102)(17,121,41,103)(18,122,42,104)(19,123,43,105)(20,124,44,106)(21,242,305,225)(22,243,306,226)(23,244,307,227)(24,245,308,228)(25,246,309,229)(26,247,310,230)(27,248,301,221)(28,249,302,222)(29,250,303,223)(30,241,304,224)(51,156,76,131)(52,157,77,132)(53,158,78,133)(54,159,79,134)(55,160,80,135)(56,151,71,136)(57,152,72,137)(58,153,73,138)(59,154,74,139)(60,155,75,140)(61,166,89,141)(62,167,90,142)(63,168,81,143)(64,169,82,144)(65,170,83,145)(66,161,84,146)(67,162,85,147)(68,163,86,148)(69,164,87,149)(70,165,88,150)(171,271,191,251)(172,272,192,252)(173,273,193,253)(174,274,194,254)(175,275,195,255)(176,276,196,256)(177,277,197,257)(178,278,198,258)(179,279,199,259)(180,280,200,260)(181,288,208,261)(182,289,209,262)(183,290,210,263)(184,281,201,264)(185,282,202,265)(186,283,203,266)(187,284,204,267)(188,285,205,268)(189,286,206,269)(190,287,207,270)(211,311,231,291)(212,312,232,292)(213,313,233,293)(214,314,234,294)(215,315,235,295)(216,316,236,296)(217,317,237,297)(218,318,238,298)(219,319,239,299)(220,320,240,300), (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)(91,147,107,131)(92,148,108,132)(93,149,109,133)(94,150,110,134)(95,141,101,135)(96,142,102,136)(97,143,103,137)(98,144,104,138)(99,145,105,139)(100,146,106,140)(111,167,130,151)(112,168,121,152)(113,169,122,153)(114,170,123,154)(115,161,124,155)(116,162,125,156)(117,163,126,157)(118,164,127,158)(119,165,128,159)(120,166,129,160)(171,247,190,231)(172,248,181,232)(173,249,182,233)(174,250,183,234)(175,241,184,235)(176,242,185,236)(177,243,186,237)(178,244,187,238)(179,245,188,239)(180,246,189,240)(191,230,207,211)(192,221,208,212)(193,222,209,213)(194,223,210,214)(195,224,201,215)(196,225,202,216)(197,226,203,217)(198,227,204,218)(199,228,205,219)(200,229,206,220)(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( (1,119,39,94)(2,120,40,95)(3,111,31,96)(4,112,32,97)(5,113,33,98)(6,114,34,99)(7,115,35,100)(8,116,36,91)(9,117,37,92)(10,118,38,93)(11,125,45,107)(12,126,46,108)(13,127,47,109)(14,128,48,110)(15,129,49,101)(16,130,50,102)(17,121,41,103)(18,122,42,104)(19,123,43,105)(20,124,44,106)(21,242,305,225)(22,243,306,226)(23,244,307,227)(24,245,308,228)(25,246,309,229)(26,247,310,230)(27,248,301,221)(28,249,302,222)(29,250,303,223)(30,241,304,224)(51,156,76,131)(52,157,77,132)(53,158,78,133)(54,159,79,134)(55,160,80,135)(56,151,71,136)(57,152,72,137)(58,153,73,138)(59,154,74,139)(60,155,75,140)(61,166,89,141)(62,167,90,142)(63,168,81,143)(64,169,82,144)(65,170,83,145)(66,161,84,146)(67,162,85,147)(68,163,86,148)(69,164,87,149)(70,165,88,150)(171,271,191,251)(172,272,192,252)(173,273,193,253)(174,274,194,254)(175,275,195,255)(176,276,196,256)(177,277,197,257)(178,278,198,258)(179,279,199,259)(180,280,200,260)(181,288,208,261)(182,289,209,262)(183,290,210,263)(184,281,201,264)(185,282,202,265)(186,283,203,266)(187,284,204,267)(188,285,205,268)(189,286,206,269)(190,287,207,270)(211,311,231,291)(212,312,232,292)(213,313,233,293)(214,314,234,294)(215,315,235,295)(216,316,236,296)(217,317,237,297)(218,318,238,298)(219,319,239,299)(220,320,240,300), (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)(91,147,107,131)(92,148,108,132)(93,149,109,133)(94,150,110,134)(95,141,101,135)(96,142,102,136)(97,143,103,137)(98,144,104,138)(99,145,105,139)(100,146,106,140)(111,167,130,151)(112,168,121,152)(113,169,122,153)(114,170,123,154)(115,161,124,155)(116,162,125,156)(117,163,126,157)(118,164,127,158)(119,165,128,159)(120,166,129,160)(171,247,190,231)(172,248,181,232)(173,249,182,233)(174,250,183,234)(175,241,184,235)(176,242,185,236)(177,243,186,237)(178,244,187,238)(179,245,188,239)(180,246,189,240)(191,230,207,211)(192,221,208,212)(193,222,209,213)(194,223,210,214)(195,224,201,215)(196,225,202,216)(197,226,203,217)(198,227,204,218)(199,228,205,219)(200,229,206,220)(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([(1,119,39,94),(2,120,40,95),(3,111,31,96),(4,112,32,97),(5,113,33,98),(6,114,34,99),(7,115,35,100),(8,116,36,91),(9,117,37,92),(10,118,38,93),(11,125,45,107),(12,126,46,108),(13,127,47,109),(14,128,48,110),(15,129,49,101),(16,130,50,102),(17,121,41,103),(18,122,42,104),(19,123,43,105),(20,124,44,106),(21,242,305,225),(22,243,306,226),(23,244,307,227),(24,245,308,228),(25,246,309,229),(26,247,310,230),(27,248,301,221),(28,249,302,222),(29,250,303,223),(30,241,304,224),(51,156,76,131),(52,157,77,132),(53,158,78,133),(54,159,79,134),(55,160,80,135),(56,151,71,136),(57,152,72,137),(58,153,73,138),(59,154,74,139),(60,155,75,140),(61,166,89,141),(62,167,90,142),(63,168,81,143),(64,169,82,144),(65,170,83,145),(66,161,84,146),(67,162,85,147),(68,163,86,148),(69,164,87,149),(70,165,88,150),(171,271,191,251),(172,272,192,252),(173,273,193,253),(174,274,194,254),(175,275,195,255),(176,276,196,256),(177,277,197,257),(178,278,198,258),(179,279,199,259),(180,280,200,260),(181,288,208,261),(182,289,209,262),(183,290,210,263),(184,281,201,264),(185,282,202,265),(186,283,203,266),(187,284,204,267),(188,285,205,268),(189,286,206,269),(190,287,207,270),(211,311,231,291),(212,312,232,292),(213,313,233,293),(214,314,234,294),(215,315,235,295),(216,316,236,296),(217,317,237,297),(218,318,238,298),(219,319,239,299),(220,320,240,300)], [(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),(91,147,107,131),(92,148,108,132),(93,149,109,133),(94,150,110,134),(95,141,101,135),(96,142,102,136),(97,143,103,137),(98,144,104,138),(99,145,105,139),(100,146,106,140),(111,167,130,151),(112,168,121,152),(113,169,122,153),(114,170,123,154),(115,161,124,155),(116,162,125,156),(117,163,126,157),(118,164,127,158),(119,165,128,159),(120,166,129,160),(171,247,190,231),(172,248,181,232),(173,249,182,233),(174,250,183,234),(175,241,184,235),(176,242,185,236),(177,243,186,237),(178,244,187,238),(179,245,188,239),(180,246,189,240),(191,230,207,211),(192,221,208,212),(193,222,209,213),(194,223,210,214),(195,224,201,215),(196,225,202,216),(197,226,203,217),(198,227,204,218),(199,228,205,219),(200,229,206,220),(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)])

104 conjugacy classes

class 1 2A···2G4A···4H4I···4P4Q···4AF5A5B10A···10N20A···20AV
order12···24···44···44···45510···1020···20
size11···11···12···210···10222···22···2

104 irreducible representations

dim111111222222
type+++++-+
imageC1C2C2C2C4C4D5C4○D4Dic5D10C4×D5C4○D20
kernelC424Dic5C10.10C42C2×C4×Dic5C2×C4×C20C4×Dic5C4×C20C2×C42C2×C10C42C22×C4C2×C4C22
# reps142116828861632

Matrix representation of C424Dic5 in GL5(𝔽41)

400000
032000
003200
00090
00009
,
10000
00100
01000
0003013
0001911
,
400000
040000
004000
000401
000337
,
320000
00100
040000
0001128
0002530

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,30,19,0,0,0,13,11],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,33,0,0,0,1,7],[32,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,11,25,0,0,0,28,30] >;

C424Dic5 in GAP, Magma, Sage, TeX

C_4^2\rtimes_4{\rm Dic}_5
% in TeX

G:=Group("C4^2:4Dic5");
// GroupNames label

G:=SmallGroup(320,559);
// by ID

G=gap.SmallGroup(320,559);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,758,100,12550]);
// Polycyclic

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

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