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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, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C10, C42, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C2×C42, C2×C42, C2×C4⋊C4, C4×Q8, C22×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C2×C4×Q8, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×Dic10, C22×Dic5, C22×C20, C4×Dic10, C2×C4×Dic5, C2×C10.D4, C2×C4⋊Dic5, C2×C4×C20, C22×Dic10, C2×C4×Dic10
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D5, C22×C4, C2×Q8, C4○D4, C24, D10, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, Dic10, C4×D5, C22×D5, C2×C4×Q8, C2×Dic10, C2×C4×D5, C4○D20, C23×D5, C4×Dic10, 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
(1 280)(2 261)(3 262)(4 263)(5 264)(6 265)(7 266)(8 267)(9 268)(10 269)(11 270)(12 271)(13 272)(14 273)(15 274)(16 275)(17 276)(18 277)(19 278)(20 279)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 99)(30 100)(31 81)(32 82)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 101)(50 102)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 109)(58 110)(59 111)(60 112)(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)(121 216)(122 217)(123 218)(124 219)(125 220)(126 201)(127 202)(128 203)(129 204)(130 205)(131 206)(132 207)(133 208)(134 209)(135 210)(136 211)(137 212)(138 213)(139 214)(140 215)(141 314)(142 315)(143 316)(144 317)(145 318)(146 319)(147 320)(148 301)(149 302)(150 303)(151 304)(152 305)(153 306)(154 307)(155 308)(156 309)(157 310)(158 311)(159 312)(160 313)(161 247)(162 248)(163 249)(164 250)(165 251)(166 252)(167 253)(168 254)(169 255)(170 256)(171 257)(172 258)(173 259)(174 260)(175 241)(176 242)(177 243)(178 244)(179 245)(180 246)(181 225)(182 226)(183 227)(184 228)(185 229)(186 230)(187 231)(188 232)(189 233)(190 234)(191 235)(192 236)(193 237)(194 238)(195 239)(196 240)(197 221)(198 222)(199 223)(200 224)
(1 28 108 255)(2 29 109 256)(3 30 110 257)(4 31 111 258)(5 32 112 259)(6 33 113 260)(7 34 114 241)(8 35 115 242)(9 36 116 243)(10 37 117 244)(11 38 118 245)(12 39 119 246)(13 40 120 247)(14 21 101 248)(15 22 102 249)(16 23 103 250)(17 24 104 251)(18 25 105 252)(19 26 106 253)(20 27 107 254)(41 174 265 83)(42 175 266 84)(43 176 267 85)(44 177 268 86)(45 178 269 87)(46 179 270 88)(47 180 271 89)(48 161 272 90)(49 162 273 91)(50 163 274 92)(51 164 275 93)(52 165 276 94)(53 166 277 95)(54 167 278 96)(55 168 279 97)(56 169 280 98)(57 170 261 99)(58 171 262 100)(59 172 263 81)(60 173 264 82)(61 238 202 317)(62 239 203 318)(63 240 204 319)(64 221 205 320)(65 222 206 301)(66 223 207 302)(67 224 208 303)(68 225 209 304)(69 226 210 305)(70 227 211 306)(71 228 212 307)(72 229 213 308)(73 230 214 309)(74 231 215 310)(75 232 216 311)(76 233 217 312)(77 234 218 313)(78 235 219 314)(79 236 220 315)(80 237 201 316)(121 158 295 188)(122 159 296 189)(123 160 297 190)(124 141 298 191)(125 142 299 192)(126 143 300 193)(127 144 281 194)(128 145 282 195)(129 146 283 196)(130 147 284 197)(131 148 285 198)(132 149 286 199)(133 150 287 200)(134 151 288 181)(135 152 289 182)(136 153 290 183)(137 154 291 184)(138 155 292 185)(139 156 293 186)(140 157 294 187)
(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)
(1 139 11 129)(2 138 12 128)(3 137 13 127)(4 136 14 126)(5 135 15 125)(6 134 16 124)(7 133 17 123)(8 132 18 122)(9 131 19 121)(10 130 20 140)(21 143 31 153)(22 142 32 152)(23 141 33 151)(24 160 34 150)(25 159 35 149)(26 158 36 148)(27 157 37 147)(28 156 38 146)(29 155 39 145)(30 154 40 144)(41 68 51 78)(42 67 52 77)(43 66 53 76)(44 65 54 75)(45 64 55 74)(46 63 56 73)(47 62 57 72)(48 61 58 71)(49 80 59 70)(50 79 60 69)(81 306 91 316)(82 305 92 315)(83 304 93 314)(84 303 94 313)(85 302 95 312)(86 301 96 311)(87 320 97 310)(88 319 98 309)(89 318 99 308)(90 317 100 307)(101 300 111 290)(102 299 112 289)(103 298 113 288)(104 297 114 287)(105 296 115 286)(106 295 116 285)(107 294 117 284)(108 293 118 283)(109 292 119 282)(110 291 120 281)(161 238 171 228)(162 237 172 227)(163 236 173 226)(164 235 174 225)(165 234 175 224)(166 233 176 223)(167 232 177 222)(168 231 178 221)(169 230 179 240)(170 229 180 239)(181 250 191 260)(182 249 192 259)(183 248 193 258)(184 247 194 257)(185 246 195 256)(186 245 196 255)(187 244 197 254)(188 243 198 253)(189 242 199 252)(190 241 200 251)(201 263 211 273)(202 262 212 272)(203 261 213 271)(204 280 214 270)(205 279 215 269)(206 278 216 268)(207 277 217 267)(208 276 218 266)(209 275 219 265)(210 274 220 264)

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

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

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

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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