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G = C429Dic5order 320 = 26·5

6th semidirect product of C42 and Dic5 acting via Dic5/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C429Dic5, (C4×C20)⋊21C4, (C2×C4).91D20, C20.63(C4⋊C4), (C2×C20).54Q8, C53(C428C4), (C2×C20).384D4, (C2×C42).17D5, C4.15(C4⋊Dic5), (C2×C4).46Dic10, C22.37(C2×D20), C2.2(C4.D20), (C22×C4).397D10, C10.4(C42.C2), C2.3(C20.6Q8), C10.11(C4.4D4), C22.45(C4○D20), C22.22(C2×Dic10), C23.269(C22×D5), C10.61(C42⋊C2), (C22×C20).475C22, (C22×C10).311C23, C22.37(C22×Dic5), C10.10C42.12C2, C2.6(C23.21D10), (C22×Dic5).30C22, (C2×C4×C20).12C2, C10.51(C2×C4⋊C4), C2.6(C2×C4⋊Dic5), (C2×C10).29(C2×Q8), (C2×C20).470(C2×C4), (C2×C10).147(C2×D4), (C2×C4⋊Dic5).17C2, (C2×C4).62(C2×Dic5), (C2×C10).70(C4○D4), (C2×C10).277(C22×C4), SmallGroup(320,563)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C429Dic5
C1C5C10C2×C10C22×C10C22×Dic5C10.10C42 — C429Dic5
C5C2×C10 — C429Dic5
C1C23C2×C42

Generators and relations for C429Dic5
 G = < a,b,c,d | a4=b4=c10=1, d2=c5, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 462 in 154 conjugacy classes, 87 normal (17 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C5, C2×C4 [×10], C2×C4 [×16], C23, C10, C10 [×6], C42 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], Dic5 [×4], C20 [×4], C20 [×4], C2×C10 [×3], C2×C10 [×4], C2.C42 [×4], C2×C42, C2×C4⋊C4 [×2], C2×Dic5 [×12], C2×C20 [×10], C2×C20 [×4], C22×C10, C428C4, C4⋊Dic5 [×4], C4×C20 [×4], C22×Dic5 [×4], C22×C20, C22×C20 [×2], C10.10C42 [×4], C2×C4⋊Dic5 [×2], C2×C4×C20, C429Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], Dic5 [×4], D10 [×3], C2×C4⋊C4, C42⋊C2 [×2], C4.4D4 [×2], C42.C2 [×2], Dic10 [×2], D20 [×2], C2×Dic5 [×6], C22×D5, C428C4, C4⋊Dic5 [×4], C2×Dic10, C2×D20, C4○D20 [×4], C22×Dic5, C20.6Q8 [×2], C4.D20 [×2], C2×C4⋊Dic5, C23.21D10 [×2], C429Dic5

Smallest permutation representation of C429Dic5
Regular action on 320 points
Generators in S320
(1 108 16 94)(2 109 17 95)(3 110 18 96)(4 101 19 97)(5 102 20 98)(6 103 11 99)(7 104 12 100)(8 105 13 91)(9 106 14 92)(10 107 15 93)(21 219 316 222)(22 220 317 223)(23 211 318 224)(24 212 319 225)(25 213 320 226)(26 214 311 227)(27 215 312 228)(28 216 313 229)(29 217 314 230)(30 218 315 221)(31 130 50 111)(32 121 41 112)(33 122 42 113)(34 123 43 114)(35 124 44 115)(36 125 45 116)(37 126 46 117)(38 127 47 118)(39 128 48 119)(40 129 49 120)(51 145 65 131)(52 146 66 132)(53 147 67 133)(54 148 68 134)(55 149 69 135)(56 150 70 136)(57 141 61 137)(58 142 62 138)(59 143 63 139)(60 144 64 140)(71 167 90 151)(72 168 81 152)(73 169 82 153)(74 170 83 154)(75 161 84 155)(76 162 85 156)(77 163 86 157)(78 164 87 158)(79 165 88 159)(80 166 89 160)(171 278 184 284)(172 279 185 285)(173 280 186 286)(174 271 187 287)(175 272 188 288)(176 273 189 289)(177 274 190 290)(178 275 181 281)(179 276 182 282)(180 277 183 283)(191 254 207 267)(192 255 208 268)(193 256 209 269)(194 257 210 270)(195 258 201 261)(196 259 202 262)(197 260 203 263)(198 251 204 264)(199 252 205 265)(200 253 206 266)(231 294 247 307)(232 295 248 308)(233 296 249 309)(234 297 250 310)(235 298 241 301)(236 299 242 302)(237 300 243 303)(238 291 244 304)(239 292 245 305)(240 293 246 306)
(1 79 39 54)(2 80 40 55)(3 71 31 56)(4 72 32 57)(5 73 33 58)(6 74 34 59)(7 75 35 60)(8 76 36 51)(9 77 37 52)(10 78 38 53)(11 83 43 63)(12 84 44 64)(13 85 45 65)(14 86 46 66)(15 87 47 67)(16 88 48 68)(17 89 49 69)(18 90 50 70)(19 81 41 61)(20 82 42 62)(21 276 299 262)(22 277 300 263)(23 278 291 264)(24 279 292 265)(25 280 293 266)(26 271 294 267)(27 272 295 268)(28 273 296 269)(29 274 297 270)(30 275 298 261)(91 156 116 131)(92 157 117 132)(93 158 118 133)(94 159 119 134)(95 160 120 135)(96 151 111 136)(97 152 112 137)(98 153 113 138)(99 154 114 139)(100 155 115 140)(101 168 121 141)(102 169 122 142)(103 170 123 143)(104 161 124 144)(105 162 125 145)(106 163 126 146)(107 164 127 147)(108 165 128 148)(109 166 129 149)(110 167 130 150)(171 238 204 224)(172 239 205 225)(173 240 206 226)(174 231 207 227)(175 232 208 228)(176 233 209 229)(177 234 210 230)(178 235 201 221)(179 236 202 222)(180 237 203 223)(181 241 195 218)(182 242 196 219)(183 243 197 220)(184 244 198 211)(185 245 199 212)(186 246 200 213)(187 247 191 214)(188 248 192 215)(189 249 193 216)(190 250 194 217)(251 318 284 304)(252 319 285 305)(253 320 286 306)(254 311 287 307)(255 312 288 308)(256 313 289 309)(257 314 290 310)(258 315 281 301)(259 316 282 302)(260 317 283 303)
(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)
(1 182 6 187)(2 181 7 186)(3 190 8 185)(4 189 9 184)(5 188 10 183)(11 174 16 179)(12 173 17 178)(13 172 18 177)(14 171 19 176)(15 180 20 175)(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)(51 225 56 230)(52 224 57 229)(53 223 58 228)(54 222 59 227)(55 221 60 226)(61 216 66 211)(62 215 67 220)(63 214 68 219)(64 213 69 218)(65 212 70 217)(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)(91 265 96 270)(92 264 97 269)(93 263 98 268)(94 262 99 267)(95 261 100 266)(101 256 106 251)(102 255 107 260)(103 254 108 259)(104 253 109 258)(105 252 110 257)(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)(131 305 136 310)(132 304 137 309)(133 303 138 308)(134 302 139 307)(135 301 140 306)(141 296 146 291)(142 295 147 300)(143 294 148 299)(144 293 149 298)(145 292 150 297)(151 314 156 319)(152 313 157 318)(153 312 158 317)(154 311 159 316)(155 320 160 315)

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

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

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

92 conjugacy classes

class 1 2A···2G4A···4L4M···4T5A5B10A···10N20A···20AV
order12···24···44···45510···1020···20
size11···12···220···20222···22···2

92 irreducible representations

dim11111222222222
type+++++-+-+-+
imageC1C2C2C2C4D4Q8D5C4○D4Dic5D10Dic10D20C4○D20
kernelC429Dic5C10.10C42C2×C4⋊Dic5C2×C4×C20C4×C20C2×C20C2×C20C2×C42C2×C10C42C22×C4C2×C4C2×C4C22
# reps142182228868832

Matrix representation of C429Dic5 in GL5(𝔽41)

400000
024100
0401700
000320
000032
,
10000
032000
003200
0001740
000124
,
400000
004000
01700
000040
00017
,
320000
092200
003200
0001740
000324

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,24,40,0,0,0,1,17,0,0,0,0,0,32,0,0,0,0,0,32],[1,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,17,1,0,0,0,40,24],[40,0,0,0,0,0,0,1,0,0,0,40,7,0,0,0,0,0,0,1,0,0,0,40,7],[32,0,0,0,0,0,9,0,0,0,0,22,32,0,0,0,0,0,17,3,0,0,0,40,24] >;

C429Dic5 in GAP, Magma, Sage, TeX

C_4^2\rtimes_9{\rm Dic}_5
% in TeX

G:=Group("C4^2:9Dic5");
// GroupNames label

G:=SmallGroup(320,563);
// by ID

G=gap.SmallGroup(320,563);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,120,422,184,12550]);
// 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,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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