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## G = (C2×C4).Dic10order 320 = 26·5

### 9th non-split extension by C2×C4 of 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 — C2×C10.D4 — (C2×C4).Dic10
 Lower central C5 — C22×C10 — (C2×C4).Dic10
 Upper central C1 — C23 — C2.C42

Generators and relations for (C2×C4).Dic10
G = < a,b,c,d | a2=b4=c20=1, d2=b2c10, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 454 in 134 conjugacy classes, 57 normal (51 characteristic)
C1, C2, C4, C22, C5, C2×C4, C2×C4, C23, C10, C4⋊C4, C22×C4, C22×C4, Dic5, C20, C2×C10, C2.C42, C2.C42, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.83C23, C10.D4, C4⋊Dic5, C22×Dic5, C22×C20, C10.10C42, C5×C2.C42, C2×C10.D4, C2×C4⋊Dic5, (C2×C4).Dic10
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, Dic10, C22×D5, C23.83C23, C2×Dic10, C4○D20, D4×D5, D42D5, Q82D5, C20.6Q8, Dic5.14D4, C23.D10, Dic5.5D4, C4.Dic10, D10.13D4, C4⋊C4⋊D5, (C2×C4).Dic10

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

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

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

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

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 4 4 4 type + + + + + + - + + - + - + image C1 C2 C2 C2 C2 D4 Q8 D5 C4○D4 D10 Dic10 C4○D20 D4×D5 D4⋊2D5 Q8⋊2D5 kernel (C2×C4).Dic10 C10.10C42 C5×C2.C42 C2×C10.D4 C2×C4⋊Dic5 C2×Dic5 C2×C20 C2.C42 C2×C10 C22×C4 C2×C4 C22 C22 C22 C22 # reps 1 4 1 1 1 2 2 2 10 6 8 16 2 4 2

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

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 11 9 0 0 0 0 32 30 0 0 0 0 0 0 0 26 0 0 0 0 30 0
,
 32 2 0 0 0 0 0 9 0 0 0 0 0 0 0 9 0 0 0 0 32 19 0 0 0 0 0 0 17 26 0 0 0 0 11 24
,
 6 35 0 0 0 0 13 35 0 0 0 0 0 0 38 35 0 0 0 0 15 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,0,0,0,0,0,30,0,0,0,0,26,0],[32,0,0,0,0,0,2,9,0,0,0,0,0,0,0,32,0,0,0,0,9,19,0,0,0,0,0,0,17,11,0,0,0,0,26,24],[6,13,0,0,0,0,35,35,0,0,0,0,0,0,38,15,0,0,0,0,35,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

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

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

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

G:=SmallGroup(320,287);
// by ID

G=gap.SmallGroup(320,287);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,64,254,387,268,12550]);
// Polycyclic

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

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