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G = C425Dic5order 320 = 26·5

2nd semidirect product of C42 and Dic5 acting via Dic5/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C425Dic5, (C4×C20)⋊17C4, (C2×C42).6D5, C53(C425C4), (C22×C4).398D10, C2.2(C422D5), C10.4(C422C2), C22.46(C4○D20), C23.270(C22×D5), C10.62(C42⋊C2), (C22×C20).476C22, (C22×C10).312C23, C22.38(C22×Dic5), C10.10C42.13C2, C2.7(C23.21D10), (C22×Dic5).31C22, (C2×C4×C20).3C2, (C2×C20).451(C2×C4), (C2×C4).63(C2×Dic5), (C2×C10).71(C4○D4), (C2×C10).278(C22×C4), SmallGroup(320,564)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C425Dic5
C1C5C10C2×C10C22×C10C22×Dic5C10.10C42 — C425Dic5
C5C2×C10 — C425Dic5
C1C23C2×C42

Generators and relations for C425Dic5
 G = < a,b,c,d | a4=b4=c10=1, d2=c5, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b-1, dcd-1=c-1 >

Subgroups: 414 in 138 conjugacy classes, 71 normal (9 characteristic)
C1, C2 [×7], C4 [×10], C22, C22 [×6], C5, C2×C4 [×6], C2×C4 [×18], C23, C10 [×7], C42 [×4], C22×C4 [×3], C22×C4 [×4], Dic5 [×4], C20 [×6], C2×C10, C2×C10 [×6], C2.C42 [×6], C2×C42, C2×Dic5 [×12], C2×C20 [×6], C2×C20 [×6], C22×C10, C425C4, C4×C20 [×4], C22×Dic5 [×4], C22×C20 [×3], C10.10C42 [×6], C2×C4×C20, C425Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, C22×C4, C4○D4 [×6], Dic5 [×4], D10 [×3], C42⋊C2 [×3], C422C2 [×4], C2×Dic5 [×6], C22×D5, C425C4, C4○D20 [×6], C22×Dic5, C422D5 [×4], C23.21D10 [×3], C425Dic5

Smallest permutation representation of C425Dic5
Regular action on 320 points
Generators in S320
(1 102 18 96)(2 103 19 97)(3 104 20 98)(4 105 11 99)(5 106 12 100)(6 107 13 91)(7 108 14 92)(8 109 15 93)(9 110 16 94)(10 101 17 95)(21 223 316 214)(22 224 317 215)(23 225 318 216)(24 226 319 217)(25 227 320 218)(26 228 311 219)(27 229 312 220)(28 230 313 211)(29 221 314 212)(30 222 315 213)(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)(51 147 67 131)(52 148 68 132)(53 149 69 133)(54 150 70 134)(55 141 61 135)(56 142 62 136)(57 143 63 137)(58 144 64 138)(59 145 65 139)(60 146 66 140)(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)(171 289 190 273)(172 290 181 274)(173 281 182 275)(174 282 183 276)(175 283 184 277)(176 284 185 278)(177 285 186 279)(178 286 187 280)(179 287 188 271)(180 288 189 272)(191 268 207 259)(192 269 208 260)(193 270 209 251)(194 261 210 252)(195 262 201 253)(196 263 202 254)(197 264 203 255)(198 265 204 256)(199 266 205 257)(200 267 206 258)(231 308 247 299)(232 309 248 300)(233 310 249 291)(234 301 250 292)(235 302 241 293)(236 303 242 294)(237 304 243 295)(238 305 244 296)(239 306 245 297)(240 307 246 298)
(1 79 39 56)(2 80 40 57)(3 71 31 58)(4 72 32 59)(5 73 33 60)(6 74 34 51)(7 75 35 52)(8 76 36 53)(9 77 37 54)(10 78 38 55)(11 81 41 65)(12 82 42 66)(13 83 43 67)(14 84 44 68)(15 85 45 69)(16 86 46 70)(17 87 47 61)(18 88 48 62)(19 89 49 63)(20 90 50 64)(21 254 303 276)(22 255 304 277)(23 256 305 278)(24 257 306 279)(25 258 307 280)(26 259 308 271)(27 260 309 272)(28 251 310 273)(29 252 301 274)(30 253 302 275)(91 154 114 131)(92 155 115 132)(93 156 116 133)(94 157 117 134)(95 158 118 135)(96 159 119 136)(97 160 120 137)(98 151 111 138)(99 152 112 139)(100 153 113 140)(101 164 127 141)(102 165 128 142)(103 166 129 143)(104 167 130 144)(105 168 121 145)(106 169 122 146)(107 170 123 147)(108 161 124 148)(109 162 125 149)(110 163 126 150)(171 230 193 249)(172 221 194 250)(173 222 195 241)(174 223 196 242)(175 224 197 243)(176 225 198 244)(177 226 199 245)(178 227 200 246)(179 228 191 247)(180 229 192 248)(181 212 210 234)(182 213 201 235)(183 214 202 236)(184 215 203 237)(185 216 204 238)(186 217 205 239)(187 218 206 240)(188 219 207 231)(189 220 208 232)(190 211 209 233)(261 292 290 314)(262 293 281 315)(263 294 282 316)(264 295 283 317)(265 296 284 318)(266 297 285 319)(267 298 286 320)(268 299 287 311)(269 300 288 312)(270 291 289 313)
(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)
(1 174 6 179)(2 173 7 178)(3 172 8 177)(4 171 9 176)(5 180 10 175)(11 190 16 185)(12 189 17 184)(13 188 18 183)(14 187 19 182)(15 186 20 181)(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)| (1,102,18,96)(2,103,19,97)(3,104,20,98)(4,105,11,99)(5,106,12,100)(6,107,13,91)(7,108,14,92)(8,109,15,93)(9,110,16,94)(10,101,17,95)(21,223,316,214)(22,224,317,215)(23,225,318,216)(24,226,319,217)(25,227,320,218)(26,228,311,219)(27,229,312,220)(28,230,313,211)(29,221,314,212)(30,222,315,213)(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)(51,147,67,131)(52,148,68,132)(53,149,69,133)(54,150,70,134)(55,141,61,135)(56,142,62,136)(57,143,63,137)(58,144,64,138)(59,145,65,139)(60,146,66,140)(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)(171,289,190,273)(172,290,181,274)(173,281,182,275)(174,282,183,276)(175,283,184,277)(176,284,185,278)(177,285,186,279)(178,286,187,280)(179,287,188,271)(180,288,189,272)(191,268,207,259)(192,269,208,260)(193,270,209,251)(194,261,210,252)(195,262,201,253)(196,263,202,254)(197,264,203,255)(198,265,204,256)(199,266,205,257)(200,267,206,258)(231,308,247,299)(232,309,248,300)(233,310,249,291)(234,301,250,292)(235,302,241,293)(236,303,242,294)(237,304,243,295)(238,305,244,296)(239,306,245,297)(240,307,246,298), (1,79,39,56)(2,80,40,57)(3,71,31,58)(4,72,32,59)(5,73,33,60)(6,74,34,51)(7,75,35,52)(8,76,36,53)(9,77,37,54)(10,78,38,55)(11,81,41,65)(12,82,42,66)(13,83,43,67)(14,84,44,68)(15,85,45,69)(16,86,46,70)(17,87,47,61)(18,88,48,62)(19,89,49,63)(20,90,50,64)(21,254,303,276)(22,255,304,277)(23,256,305,278)(24,257,306,279)(25,258,307,280)(26,259,308,271)(27,260,309,272)(28,251,310,273)(29,252,301,274)(30,253,302,275)(91,154,114,131)(92,155,115,132)(93,156,116,133)(94,157,117,134)(95,158,118,135)(96,159,119,136)(97,160,120,137)(98,151,111,138)(99,152,112,139)(100,153,113,140)(101,164,127,141)(102,165,128,142)(103,166,129,143)(104,167,130,144)(105,168,121,145)(106,169,122,146)(107,170,123,147)(108,161,124,148)(109,162,125,149)(110,163,126,150)(171,230,193,249)(172,221,194,250)(173,222,195,241)(174,223,196,242)(175,224,197,243)(176,225,198,244)(177,226,199,245)(178,227,200,246)(179,228,191,247)(180,229,192,248)(181,212,210,234)(182,213,201,235)(183,214,202,236)(184,215,203,237)(185,216,204,238)(186,217,205,239)(187,218,206,240)(188,219,207,231)(189,220,208,232)(190,211,209,233)(261,292,290,314)(262,293,281,315)(263,294,282,316)(264,295,283,317)(265,296,284,318)(266,297,285,319)(267,298,286,320)(268,299,287,311)(269,300,288,312)(270,291,289,313), (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), (1,174,6,179)(2,173,7,178)(3,172,8,177)(4,171,9,176)(5,180,10,175)(11,190,16,185)(12,189,17,184)(13,188,18,183)(14,187,19,182)(15,186,20,181)(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( (1,102,18,96)(2,103,19,97)(3,104,20,98)(4,105,11,99)(5,106,12,100)(6,107,13,91)(7,108,14,92)(8,109,15,93)(9,110,16,94)(10,101,17,95)(21,223,316,214)(22,224,317,215)(23,225,318,216)(24,226,319,217)(25,227,320,218)(26,228,311,219)(27,229,312,220)(28,230,313,211)(29,221,314,212)(30,222,315,213)(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)(51,147,67,131)(52,148,68,132)(53,149,69,133)(54,150,70,134)(55,141,61,135)(56,142,62,136)(57,143,63,137)(58,144,64,138)(59,145,65,139)(60,146,66,140)(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)(171,289,190,273)(172,290,181,274)(173,281,182,275)(174,282,183,276)(175,283,184,277)(176,284,185,278)(177,285,186,279)(178,286,187,280)(179,287,188,271)(180,288,189,272)(191,268,207,259)(192,269,208,260)(193,270,209,251)(194,261,210,252)(195,262,201,253)(196,263,202,254)(197,264,203,255)(198,265,204,256)(199,266,205,257)(200,267,206,258)(231,308,247,299)(232,309,248,300)(233,310,249,291)(234,301,250,292)(235,302,241,293)(236,303,242,294)(237,304,243,295)(238,305,244,296)(239,306,245,297)(240,307,246,298), (1,79,39,56)(2,80,40,57)(3,71,31,58)(4,72,32,59)(5,73,33,60)(6,74,34,51)(7,75,35,52)(8,76,36,53)(9,77,37,54)(10,78,38,55)(11,81,41,65)(12,82,42,66)(13,83,43,67)(14,84,44,68)(15,85,45,69)(16,86,46,70)(17,87,47,61)(18,88,48,62)(19,89,49,63)(20,90,50,64)(21,254,303,276)(22,255,304,277)(23,256,305,278)(24,257,306,279)(25,258,307,280)(26,259,308,271)(27,260,309,272)(28,251,310,273)(29,252,301,274)(30,253,302,275)(91,154,114,131)(92,155,115,132)(93,156,116,133)(94,157,117,134)(95,158,118,135)(96,159,119,136)(97,160,120,137)(98,151,111,138)(99,152,112,139)(100,153,113,140)(101,164,127,141)(102,165,128,142)(103,166,129,143)(104,167,130,144)(105,168,121,145)(106,169,122,146)(107,170,123,147)(108,161,124,148)(109,162,125,149)(110,163,126,150)(171,230,193,249)(172,221,194,250)(173,222,195,241)(174,223,196,242)(175,224,197,243)(176,225,198,244)(177,226,199,245)(178,227,200,246)(179,228,191,247)(180,229,192,248)(181,212,210,234)(182,213,201,235)(183,214,202,236)(184,215,203,237)(185,216,204,238)(186,217,205,239)(187,218,206,240)(188,219,207,231)(189,220,208,232)(190,211,209,233)(261,292,290,314)(262,293,281,315)(263,294,282,316)(264,295,283,317)(265,296,284,318)(266,297,285,319)(267,298,286,320)(268,299,287,311)(269,300,288,312)(270,291,289,313), (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), (1,174,6,179)(2,173,7,178)(3,172,8,177)(4,171,9,176)(5,180,10,175)(11,190,16,185)(12,189,17,184)(13,188,18,183)(14,187,19,182)(15,186,20,181)(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([(1,102,18,96),(2,103,19,97),(3,104,20,98),(4,105,11,99),(5,106,12,100),(6,107,13,91),(7,108,14,92),(8,109,15,93),(9,110,16,94),(10,101,17,95),(21,223,316,214),(22,224,317,215),(23,225,318,216),(24,226,319,217),(25,227,320,218),(26,228,311,219),(27,229,312,220),(28,230,313,211),(29,221,314,212),(30,222,315,213),(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),(51,147,67,131),(52,148,68,132),(53,149,69,133),(54,150,70,134),(55,141,61,135),(56,142,62,136),(57,143,63,137),(58,144,64,138),(59,145,65,139),(60,146,66,140),(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),(171,289,190,273),(172,290,181,274),(173,281,182,275),(174,282,183,276),(175,283,184,277),(176,284,185,278),(177,285,186,279),(178,286,187,280),(179,287,188,271),(180,288,189,272),(191,268,207,259),(192,269,208,260),(193,270,209,251),(194,261,210,252),(195,262,201,253),(196,263,202,254),(197,264,203,255),(198,265,204,256),(199,266,205,257),(200,267,206,258),(231,308,247,299),(232,309,248,300),(233,310,249,291),(234,301,250,292),(235,302,241,293),(236,303,242,294),(237,304,243,295),(238,305,244,296),(239,306,245,297),(240,307,246,298)], [(1,79,39,56),(2,80,40,57),(3,71,31,58),(4,72,32,59),(5,73,33,60),(6,74,34,51),(7,75,35,52),(8,76,36,53),(9,77,37,54),(10,78,38,55),(11,81,41,65),(12,82,42,66),(13,83,43,67),(14,84,44,68),(15,85,45,69),(16,86,46,70),(17,87,47,61),(18,88,48,62),(19,89,49,63),(20,90,50,64),(21,254,303,276),(22,255,304,277),(23,256,305,278),(24,257,306,279),(25,258,307,280),(26,259,308,271),(27,260,309,272),(28,251,310,273),(29,252,301,274),(30,253,302,275),(91,154,114,131),(92,155,115,132),(93,156,116,133),(94,157,117,134),(95,158,118,135),(96,159,119,136),(97,160,120,137),(98,151,111,138),(99,152,112,139),(100,153,113,140),(101,164,127,141),(102,165,128,142),(103,166,129,143),(104,167,130,144),(105,168,121,145),(106,169,122,146),(107,170,123,147),(108,161,124,148),(109,162,125,149),(110,163,126,150),(171,230,193,249),(172,221,194,250),(173,222,195,241),(174,223,196,242),(175,224,197,243),(176,225,198,244),(177,226,199,245),(178,227,200,246),(179,228,191,247),(180,229,192,248),(181,212,210,234),(182,213,201,235),(183,214,202,236),(184,215,203,237),(185,216,204,238),(186,217,205,239),(187,218,206,240),(188,219,207,231),(189,220,208,232),(190,211,209,233),(261,292,290,314),(262,293,281,315),(263,294,282,316),(264,295,283,317),(265,296,284,318),(266,297,285,319),(267,298,286,320),(268,299,287,311),(269,300,288,312),(270,291,289,313)], [(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)], [(1,174,6,179),(2,173,7,178),(3,172,8,177),(4,171,9,176),(5,180,10,175),(11,190,16,185),(12,189,17,184),(13,188,18,183),(14,187,19,182),(15,186,20,181),(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)])

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

dim111122222
type++++-+
imageC1C2C2C4D5C4○D4Dic5D10C4○D20
kernelC425Dic5C10.10C42C2×C4×C20C4×C20C2×C42C2×C10C42C22×C4C22
# reps16182128648

Matrix representation of C425Dic5 in GL5(𝔽41)

400000
0392800
013200
0001835
000623
,
400000
032000
003200
000213
0002839
,
400000
00100
040600
000040
000135
,
90000
0383600
018300
000314
0002610

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,39,13,0,0,0,28,2,0,0,0,0,0,18,6,0,0,0,35,23],[40,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,2,28,0,0,0,13,39],[40,0,0,0,0,0,0,40,0,0,0,1,6,0,0,0,0,0,0,1,0,0,0,40,35],[9,0,0,0,0,0,38,18,0,0,0,36,3,0,0,0,0,0,31,26,0,0,0,4,10] >;

C425Dic5 in GAP, Magma, Sage, TeX

C_4^2\rtimes_5{\rm Dic}_5
% in TeX

G:=Group("C4^2:5Dic5");
// GroupNames label

G:=SmallGroup(320,564);
// by ID

G=gap.SmallGroup(320,564);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,120,1094,184,12550]);
// 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^-1,d*c*d^-1=c^-1>;
// generators/relations

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