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G = (C2×C4)⋊Dic10order 320 = 26·5

2nd semidirect product of C2×C4 and Dic10 acting via Dic10/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20)⋊7Q8, (C2×Dic5)⋊2Q8, (C2×C4)⋊2Dic10, (C2×C20).52D4, C2.14(C20⋊Q8), C10.21(C4⋊Q8), C10.40C22≀C2, C22.45(Q8×D5), (C2×Dic5).66D4, C22.244(D4×D5), (C22×C4).98D10, C2.9(C23⋊D10), C10.58(C22⋊Q8), C2.20(D10⋊Q8), C2.5(Dic5⋊Q8), C2.9(C20.48D4), (C22×C20).63C22, (C22×Dic10).5C2, C22.47(C2×Dic10), C23.374(C22×D5), C22.102(C4○D20), (C22×C10).343C23, C52(C23.78C23), C10.10C42.37C2, (C22×Dic5).52C22, (C2×C4⋊C4).16D5, (C10×C4⋊C4).21C2, (C2×C10).78(C2×Q8), (C2×C10).330(C2×D4), (C2×C4).35(C5⋊D4), (C2×C10).82(C4○D4), C22.134(C2×C5⋊D4), (C2×C10.D4).15C2, SmallGroup(320,606)

Series: Derived Chief Lower central Upper central

C1C22×C10 — (C2×C4)⋊Dic10
C1C5C10C2×C10C22×C10C22×Dic5C22×Dic10 — (C2×C4)⋊Dic10
C5C22×C10 — (C2×C4)⋊Dic10
C1C23C2×C4⋊C4

Generators and relations for (C2×C4)⋊Dic10
 G = < a,b,c,d | a2=b4=c20=1, d2=c10, dbd-1=ab=ba, ac=ca, ad=da, cbc-1=b-1, dcd-1=c-1 >

Subgroups: 630 in 182 conjugacy classes, 67 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×13], C22 [×3], C22 [×4], C5, C2×C4 [×4], C2×C4 [×23], Q8 [×8], C23, C10 [×3], C10 [×4], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×6], Dic5 [×8], C20 [×5], C2×C10 [×3], C2×C10 [×4], C2.C42 [×3], C2×C4⋊C4, C2×C4⋊C4 [×2], C22×Q8, Dic10 [×8], C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×4], C2×C20 [×7], C22×C10, C23.78C23, C10.D4 [×4], C5×C4⋊C4 [×2], C2×Dic10 [×6], C22×Dic5 [×4], C22×C20 [×3], C10.10C42, C10.10C42 [×2], C2×C10.D4 [×2], C10×C4⋊C4, C22×Dic10, (C2×C4)⋊Dic10
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×6], C23, D5, C2×D4 [×3], C2×Q8 [×3], C4○D4, D10 [×3], C22≀C2, C22⋊Q8 [×3], C4⋊Q8 [×3], Dic10 [×2], C5⋊D4 [×2], C22×D5, C23.78C23, C2×Dic10, C4○D20, D4×D5 [×2], Q8×D5 [×2], C2×C5⋊D4, C20⋊Q8 [×2], D10⋊Q8 [×2], C20.48D4, C23⋊D10, Dic5⋊Q8, (C2×C4)⋊Dic10

Smallest permutation representation of (C2×C4)⋊Dic10
Regular action on 320 points
Generators in S320
(1 237)(2 238)(3 239)(4 240)(5 221)(6 222)(7 223)(8 224)(9 225)(10 226)(11 227)(12 228)(13 229)(14 230)(15 231)(16 232)(17 233)(18 234)(19 235)(20 236)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)(33 58)(34 59)(35 60)(36 41)(37 42)(38 43)(39 44)(40 45)(61 279)(62 280)(63 261)(64 262)(65 263)(66 264)(67 265)(68 266)(69 267)(70 268)(71 269)(72 270)(73 271)(74 272)(75 273)(76 274)(77 275)(78 276)(79 277)(80 278)(81 102)(82 103)(83 104)(84 105)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 113)(93 114)(94 115)(95 116)(96 117)(97 118)(98 119)(99 120)(100 101)(121 142)(122 143)(123 144)(124 145)(125 146)(126 147)(127 148)(128 149)(129 150)(130 151)(131 152)(132 153)(133 154)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 141)(161 286)(162 287)(163 288)(164 289)(165 290)(166 291)(167 292)(168 293)(169 294)(170 295)(171 296)(172 297)(173 298)(174 299)(175 300)(176 281)(177 282)(178 283)(179 284)(180 285)(181 314)(182 315)(183 316)(184 317)(185 318)(186 319)(187 320)(188 301)(189 302)(190 303)(191 304)(192 305)(193 306)(194 307)(195 308)(196 309)(197 310)(198 311)(199 312)(200 313)(201 250)(202 251)(203 252)(204 253)(205 254)(206 255)(207 256)(208 257)(209 258)(210 259)(211 260)(212 241)(213 242)(214 243)(215 244)(216 245)(217 246)(218 247)(219 248)(220 249)
(1 282 182 213)(2 214 183 283)(3 284 184 215)(4 216 185 285)(5 286 186 217)(6 218 187 287)(7 288 188 219)(8 220 189 289)(9 290 190 201)(10 202 191 291)(11 292 192 203)(12 204 193 293)(13 294 194 205)(14 206 195 295)(15 296 196 207)(16 208 197 297)(17 298 198 209)(18 210 199 299)(19 300 200 211)(20 212 181 281)(21 143 108 271)(22 272 109 144)(23 145 110 273)(24 274 111 146)(25 147 112 275)(26 276 113 148)(27 149 114 277)(28 278 115 150)(29 151 116 279)(30 280 117 152)(31 153 118 261)(32 262 119 154)(33 155 120 263)(34 264 101 156)(35 157 102 265)(36 266 103 158)(37 159 104 267)(38 268 105 160)(39 141 106 269)(40 270 107 142)(41 68 82 137)(42 138 83 69)(43 70 84 139)(44 140 85 71)(45 72 86 121)(46 122 87 73)(47 74 88 123)(48 124 89 75)(49 76 90 125)(50 126 91 77)(51 78 92 127)(52 128 93 79)(53 80 94 129)(54 130 95 61)(55 62 96 131)(56 132 97 63)(57 64 98 133)(58 134 99 65)(59 66 100 135)(60 136 81 67)(161 319 246 221)(162 222 247 320)(163 301 248 223)(164 224 249 302)(165 303 250 225)(166 226 251 304)(167 305 252 227)(168 228 253 306)(169 307 254 229)(170 230 255 308)(171 309 256 231)(172 232 257 310)(173 311 258 233)(174 234 259 312)(175 313 260 235)(176 236 241 314)(177 315 242 237)(178 238 243 316)(179 317 244 239)(180 240 245 318)
(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)
(1 140 11 130)(2 139 12 129)(3 138 13 128)(4 137 14 127)(5 136 15 126)(6 135 16 125)(7 134 17 124)(8 133 18 123)(9 132 19 122)(10 131 20 121)(21 201 31 211)(22 220 32 210)(23 219 33 209)(24 218 34 208)(25 217 35 207)(26 216 36 206)(27 215 37 205)(28 214 38 204)(29 213 39 203)(30 212 40 202)(41 255 51 245)(42 254 52 244)(43 253 53 243)(44 252 54 242)(45 251 55 241)(46 250 56 260)(47 249 57 259)(48 248 58 258)(49 247 59 257)(50 246 60 256)(61 182 71 192)(62 181 72 191)(63 200 73 190)(64 199 74 189)(65 198 75 188)(66 197 76 187)(67 196 77 186)(68 195 78 185)(69 194 79 184)(70 193 80 183)(81 171 91 161)(82 170 92 180)(83 169 93 179)(84 168 94 178)(85 167 95 177)(86 166 96 176)(87 165 97 175)(88 164 98 174)(89 163 99 173)(90 162 100 172)(101 297 111 287)(102 296 112 286)(103 295 113 285)(104 294 114 284)(105 293 115 283)(106 292 116 282)(107 291 117 281)(108 290 118 300)(109 289 119 299)(110 288 120 298)(141 227 151 237)(142 226 152 236)(143 225 153 235)(144 224 154 234)(145 223 155 233)(146 222 156 232)(147 221 157 231)(148 240 158 230)(149 239 159 229)(150 238 160 228)(261 313 271 303)(262 312 272 302)(263 311 273 301)(264 310 274 320)(265 309 275 319)(266 308 276 318)(267 307 277 317)(268 306 278 316)(269 305 279 315)(270 304 280 314)

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

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

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

62 conjugacy classes

class 1 2A···2G4A···4F4G···4N5A5B10A···10N20A···20X
order12···24···44···45510···1020···20
size11···14···420···20222···24···4

62 irreducible representations

dim11111222222222244
type++++++-+-++-+-
imageC1C2C2C2C2D4Q8D4Q8D5C4○D4D10Dic10C5⋊D4C4○D20D4×D5Q8×D5
kernel(C2×C4)⋊Dic10C10.10C42C2×C10.D4C10×C4⋊C4C22×Dic10C2×Dic5C2×Dic5C2×C20C2×C20C2×C4⋊C4C2×C10C22×C4C2×C4C2×C4C22C22C22
# reps13211442222688844

Matrix representation of (C2×C4)⋊Dic10 in GL6(𝔽41)

100000
010000
0040000
0004000
000010
000001
,
100000
010000
00174000
0012400
00001237
00002629
,
9110000
30140000
0004000
001700
0000404
0000201
,
40170000
2410000
00212600
0022000
0000294
00001512

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,0,0,0,0,12,26,0,0,0,0,37,29],[9,30,0,0,0,0,11,14,0,0,0,0,0,0,0,1,0,0,0,0,40,7,0,0,0,0,0,0,40,20,0,0,0,0,4,1],[40,24,0,0,0,0,17,1,0,0,0,0,0,0,21,2,0,0,0,0,26,20,0,0,0,0,0,0,29,15,0,0,0,0,4,12] >;

(C2×C4)⋊Dic10 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes {\rm Dic}_{10}
% in TeX

G:=Group("(C2xC4):Dic10");
// GroupNames label

G:=SmallGroup(320,606);
// by ID

G=gap.SmallGroup(320,606);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,254,387,184,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^20=1,d^2=c^10,d*b*d^-1=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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