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## G = C2×C4.Dic10order 320 = 26·5

### Direct product of C2 and C4.Dic10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×C4.Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C4×Dic5 — C2×C4.Dic10
 Lower central C5 — C2×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, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b2c-1 >

Subgroups: 606 in 226 conjugacy classes, 127 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×12], C22, C22 [×6], C5, C2×C4 [×10], C2×C4 [×20], C23, C10 [×3], C10 [×4], C42 [×4], C4⋊C4 [×4], C4⋊C4 [×20], C22×C4, C22×C4 [×2], C22×C4 [×4], Dic5 [×8], C20 [×4], C20 [×4], C2×C10, C2×C10 [×6], C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×5], C42.C2 [×8], C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×10], C2×C20 [×4], C22×C10, C2×C42.C2, C4×Dic5 [×4], C10.D4 [×8], C4⋊Dic5 [×12], C5×C4⋊C4 [×4], C22×Dic5 [×2], C22×Dic5 [×2], C22×C20, C22×C20 [×2], C4.Dic10 [×8], C2×C4×Dic5, C2×C10.D4 [×2], C2×C4⋊Dic5, C2×C4⋊Dic5 [×2], C10×C4⋊C4, C2×C4.Dic10
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×4], C24, D10 [×7], C42.C2 [×4], C22×Q8, C2×C4○D4 [×2], Dic10 [×4], C22×D5 [×7], C2×C42.C2, C2×Dic10 [×6], D42D5 [×2], Q82D5 [×2], C23×D5, C4.Dic10 [×4], C22×Dic10, C2×D42D5, C2×Q82D5, C2×C4.Dic10

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

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 4S 4T 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 4 4 4 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + - + + + - - + image C1 C2 C2 C2 C2 C2 Q8 D5 C4○D4 D10 D10 Dic10 D4⋊2D5 Q8⋊2D5 kernel C2×C4.Dic10 C4.Dic10 C2×C4×Dic5 C2×C10.D4 C2×C4⋊Dic5 C10×C4⋊C4 C2×C20 C2×C4⋊C4 C2×C10 C4⋊C4 C22×C4 C2×C4 C22 C22 # reps 1 8 1 2 3 1 4 2 8 8 6 16 4 4

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

 40 0 0 0 0 0 0 40 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 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 13 37 0 0 0 0 22 28
,
 35 1 0 0 0 0 40 0 0 0 0 0 0 0 14 30 0 0 0 0 11 9 0 0 0 0 0 0 1 0 0 0 0 0 27 40
,
 2 13 0 0 0 0 25 39 0 0 0 0 0 0 16 12 0 0 0 0 23 25 0 0 0 0 0 0 6 36 0 0 0 0 7 35

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,22,0,0,0,0,37,28],[35,40,0,0,0,0,1,0,0,0,0,0,0,0,14,11,0,0,0,0,30,9,0,0,0,0,0,0,1,27,0,0,0,0,0,40],[2,25,0,0,0,0,13,39,0,0,0,0,0,0,16,23,0,0,0,0,12,25,0,0,0,0,0,0,6,7,0,0,0,0,36,35] >;

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(320,1171);
// by ID

G=gap.SmallGroup(320,1171);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,1571,297,80,12550]);
// 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,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c^-1>;
// generators/relations

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