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## G = (C2×C4)⋊Dic10order 320 = 26·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — (C2×C4)⋊Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C22×Dic10 — (C2×C4)⋊Dic10
 Lower central C5 — C22×C10 — (C2×C4)⋊Dic10
 Upper central C1 — C23 — C2×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, C2, C4, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C10, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C2.C42, C2×C4⋊C4, C2×C4⋊C4, C22×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.78C23, C10.D4, C5×C4⋊C4, C2×Dic10, C22×Dic5, C22×C20, C10.10C42, C10.10C42, C2×C10.D4, C10×C4⋊C4, C22×Dic10, (C2×C4)⋊Dic10
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22≀C2, C22⋊Q8, C4⋊Q8, Dic10, C5⋊D4, C22×D5, C23.78C23, C2×Dic10, C4○D20, D4×D5, Q8×D5, C2×C5⋊D4, C20⋊Q8, D10⋊Q8, C20.48D4, C23⋊D10, Dic5⋊Q8, (C2×C4)⋊Dic10

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

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

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

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

62 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4F 4G ··· 4N 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 4 ··· 4 20 ··· 20 2 2 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + - + - + + - + - image C1 C2 C2 C2 C2 D4 Q8 D4 Q8 D5 C4○D4 D10 Dic10 C5⋊D4 C4○D20 D4×D5 Q8×D5 kernel (C2×C4)⋊Dic10 C10.10C42 C2×C10.D4 C10×C4⋊C4 C22×Dic10 C2×Dic5 C2×Dic5 C2×C20 C2×C20 C2×C4⋊C4 C2×C10 C22×C4 C2×C4 C2×C4 C22 C22 C22 # reps 1 3 2 1 1 4 4 2 2 2 2 6 8 8 8 4 4

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

 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 40 0 0 0 0 1 24 0 0 0 0 0 0 12 37 0 0 0 0 26 29
,
 9 11 0 0 0 0 30 14 0 0 0 0 0 0 0 40 0 0 0 0 1 7 0 0 0 0 0 0 40 4 0 0 0 0 20 1
,
 40 17 0 0 0 0 24 1 0 0 0 0 0 0 21 26 0 0 0 0 2 20 0 0 0 0 0 0 29 4 0 0 0 0 15 12

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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