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## G = C2×C4×Dic10order 320 = 26·5

### Direct product of C2×C4 and Dic10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C4×Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C22×Dic10 — C2×C4×Dic10
 Lower central C5 — C10 — C2×C4×Dic10
 Upper central C1 — C22×C4 — C2×C42

Generators and relations for C2×C4×Dic10
G = < a,b,c,d | a2=b4=c20=1, d2=c10, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 750 in 298 conjugacy classes, 183 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×14], C22, C22 [×6], C5, C2×C4 [×14], C2×C4 [×22], Q8 [×16], C23, C10 [×3], C10 [×4], C42 [×4], C42 [×8], C4⋊C4 [×12], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×12], Dic5 [×8], Dic5 [×4], C20 [×8], C20 [×2], C2×C10, C2×C10 [×6], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×3], C4×Q8 [×8], C22×Q8, Dic10 [×16], C2×Dic5 [×16], C2×Dic5 [×4], C2×C20 [×14], C2×C20 [×2], C22×C10, C2×C4×Q8, C4×Dic5 [×8], C10.D4 [×8], C4⋊Dic5 [×4], C4×C20 [×4], C2×Dic10 [×12], C22×Dic5 [×4], C22×C20 [×3], C4×Dic10 [×8], C2×C4×Dic5 [×2], C2×C10.D4 [×2], C2×C4⋊Dic5, C2×C4×C20, C22×Dic10, C2×C4×Dic10
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×4], C23 [×15], D5, C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C4×Q8 [×4], C23×C4, C22×Q8, C2×C4○D4, Dic10 [×4], C4×D5 [×4], C22×D5 [×7], C2×C4×Q8, C2×Dic10 [×6], C2×C4×D5 [×6], C4○D20 [×2], C23×D5, C4×Dic10 [×4], C22×Dic10, D5×C22×C4, C2×C4○D20, C2×C4×Dic10

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

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

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

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

104 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4P 4Q ··· 4AF 5A 5B 10A ··· 10N 20A ··· 20AV order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 1 ··· 1 2 ··· 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2

104 irreducible representations

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

Matrix representation of C2×C4×Dic10 in GL4(𝔽41) generated by

 1 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 32 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 40 0 0 0 0 32 30 0 0 11 27
,
 40 0 0 0 0 1 0 0 0 0 3 38 0 0 17 38
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[32,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,40,0,0,0,0,32,11,0,0,30,27],[40,0,0,0,0,1,0,0,0,0,3,17,0,0,38,38] >;

C2×C4×Dic10 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC4xDic10");
// GroupNames label

G:=SmallGroup(320,1139);
// by ID

G=gap.SmallGroup(320,1139);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,184,80,12550]);
// Polycyclic

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

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