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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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20).2Q8, (C2×C4).9Dic10, (C2×Dic5).18D4, (C22×C4).13D10, C22.153(D4×D5), C10.5(C22⋊Q8), C2.5(C20.6Q8), C10.1(C42.C2), C2.6(C4.Dic10), C10.17(C4.4D4), C22.87(C4○D20), (C22×C20).10C22, C22.42(C2×Dic10), C2.C42.15D5, C23.356(C22×D5), C10.18(C422C2), C2.8(D10.13D4), C22.85(D42D5), (C22×C10).285C23, C2.9(Dic5.5D4), C53(C23.83C23), C22.41(Q82D5), C10.10C42.25C2, C2.10(C23.D10), C10.37(C22.D4), (C22×Dic5).10C22, C2.10(Dic5.14D4), (C2×C10).21(C2×Q8), C2.8(C4⋊C4⋊D5), (C2×C10).196(C2×D4), (C2×C4⋊Dic5).10C2, (C2×C10).129(C4○D4), (C2×C10.D4).10C2, (C5×C2.C42).11C2, SmallGroup(320,287)

Series: Derived Chief Lower central Upper central

C1C22×C10 — (C2×C4).Dic10
C1C5C10C2×C10C22×C10C22×Dic5C2×C10.D4 — (C2×C4).Dic10
C5C22×C10 — (C2×C4).Dic10
C1C23C2.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 [×7], C4 [×9], C22 [×7], C5, C2×C4 [×2], C2×C4 [×21], C23, C10 [×7], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×4], Dic5 [×5], C20 [×4], C2×C10 [×7], C2.C42, C2.C42 [×4], C2×C4⋊C4 [×2], C2×Dic5 [×2], C2×Dic5 [×11], C2×C20 [×2], C2×C20 [×8], C22×C10, C23.83C23, C10.D4 [×2], C4⋊Dic5 [×2], C22×Dic5 [×4], C22×C20 [×3], C10.10C42 [×4], C5×C2.C42, C2×C10.D4, C2×C4⋊Dic5, (C2×C4).Dic10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, C2×D4, C2×Q8, C4○D4 [×5], D10 [×3], C22⋊Q8, C22.D4, C4.4D4, C42.C2 [×2], C422C2 [×2], Dic10 [×2], C22×D5, C23.83C23, C2×Dic10, C4○D20 [×2], D4×D5, D42D5 [×2], 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
(1 218)(2 219)(3 220)(4 201)(5 202)(6 203)(7 204)(8 205)(9 206)(10 207)(11 208)(12 209)(13 210)(14 211)(15 212)(16 213)(17 214)(18 215)(19 216)(20 217)(21 255)(22 256)(23 257)(24 258)(25 259)(26 260)(27 241)(28 242)(29 243)(30 244)(31 245)(32 246)(33 247)(34 248)(35 249)(36 250)(37 251)(38 252)(39 253)(40 254)(41 106)(42 107)(43 108)(44 109)(45 110)(46 111)(47 112)(48 113)(49 114)(50 115)(51 116)(52 117)(53 118)(54 119)(55 120)(56 101)(57 102)(58 103)(59 104)(60 105)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 97)(69 98)(70 99)(71 100)(72 81)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)(121 180)(122 161)(123 162)(124 163)(125 164)(126 165)(127 166)(128 167)(129 168)(130 169)(131 170)(132 171)(133 172)(134 173)(135 174)(136 175)(137 176)(138 177)(139 178)(140 179)(141 307)(142 308)(143 309)(144 310)(145 311)(146 312)(147 313)(148 314)(149 315)(150 316)(151 317)(152 318)(153 319)(154 320)(155 301)(156 302)(157 303)(158 304)(159 305)(160 306)(181 239)(182 240)(183 221)(184 222)(185 223)(186 224)(187 225)(188 226)(189 227)(190 228)(191 229)(192 230)(193 231)(194 232)(195 233)(196 234)(197 235)(198 236)(199 237)(200 238)(261 297)(262 298)(263 299)(264 300)(265 281)(266 282)(267 283)(268 284)(269 285)(270 286)(271 287)(272 288)(273 289)(274 290)(275 291)(276 292)(277 293)(278 294)(279 295)(280 296)
(1 96 117 248)(2 68 118 35)(3 98 119 250)(4 70 120 37)(5 100 101 252)(6 72 102 39)(7 82 103 254)(8 74 104 21)(9 84 105 256)(10 76 106 23)(11 86 107 258)(12 78 108 25)(13 88 109 260)(14 80 110 27)(15 90 111 242)(16 62 112 29)(17 92 113 244)(18 64 114 31)(19 94 115 246)(20 66 116 33)(22 206 75 60)(24 208 77 42)(26 210 79 44)(28 212 61 46)(30 214 63 48)(32 216 65 50)(34 218 67 52)(36 220 69 54)(38 202 71 56)(40 204 73 58)(41 257 207 85)(43 259 209 87)(45 241 211 89)(47 243 213 91)(49 245 215 93)(51 247 217 95)(53 249 219 97)(55 251 201 99)(57 253 203 81)(59 255 205 83)(121 184 160 261)(122 223 141 298)(123 186 142 263)(124 225 143 300)(125 188 144 265)(126 227 145 282)(127 190 146 267)(128 229 147 284)(129 192 148 269)(130 231 149 286)(131 194 150 271)(132 233 151 288)(133 196 152 273)(134 235 153 290)(135 198 154 275)(136 237 155 292)(137 200 156 277)(138 239 157 294)(139 182 158 279)(140 221 159 296)(161 185 307 262)(162 224 308 299)(163 187 309 264)(164 226 310 281)(165 189 311 266)(166 228 312 283)(167 191 313 268)(168 230 314 285)(169 193 315 270)(170 232 316 287)(171 195 317 272)(172 234 318 289)(173 197 319 274)(174 236 320 291)(175 199 301 276)(176 238 302 293)(177 181 303 278)(178 240 304 295)(179 183 305 280)(180 222 306 297)
(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)
(1 266 107 199)(2 188 108 275)(3 264 109 197)(4 186 110 273)(5 262 111 195)(6 184 112 271)(7 280 113 193)(8 182 114 269)(9 278 115 191)(10 200 116 267)(11 276 117 189)(12 198 118 265)(13 274 119 187)(14 196 120 263)(15 272 101 185)(16 194 102 261)(17 270 103 183)(18 192 104 279)(19 268 105 181)(20 190 106 277)(21 158 64 129)(22 138 65 147)(23 156 66 127)(24 136 67 145)(25 154 68 125)(26 134 69 143)(27 152 70 123)(28 132 71 141)(29 150 72 121)(30 130 73 159)(31 148 74 139)(32 128 75 157)(33 146 76 137)(34 126 77 155)(35 144 78 135)(36 124 79 153)(37 142 80 133)(38 122 61 151)(39 160 62 131)(40 140 63 149)(41 293 217 228)(42 237 218 282)(43 291 219 226)(44 235 220 300)(45 289 201 224)(46 233 202 298)(47 287 203 222)(48 231 204 296)(49 285 205 240)(50 229 206 294)(51 283 207 238)(52 227 208 292)(53 281 209 236)(54 225 210 290)(55 299 211 234)(56 223 212 288)(57 297 213 232)(58 221 214 286)(59 295 215 230)(60 239 216 284)(81 180 243 316)(82 305 244 169)(83 178 245 314)(84 303 246 167)(85 176 247 312)(86 301 248 165)(87 174 249 310)(88 319 250 163)(89 172 251 308)(90 317 252 161)(91 170 253 306)(92 315 254 179)(93 168 255 304)(94 313 256 177)(95 166 257 302)(96 311 258 175)(97 164 259 320)(98 309 260 173)(99 162 241 318)(100 307 242 171)

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

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

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

62 conjugacy classes

class 1 2A···2G4A···4F4G···4N5A5B10A···10N20A···20X
order12···24···44···45510···1020···20
size11···14···420···20222···24···4

62 irreducible representations

dim111112222222444
type++++++-++-+-+
imageC1C2C2C2C2D4Q8D5C4○D4D10Dic10C4○D20D4×D5D42D5Q82D5
kernel(C2×C4).Dic10C10.10C42C5×C2.C42C2×C10.D4C2×C4⋊Dic5C2×Dic5C2×C20C2.C42C2×C10C22×C4C2×C4C22C22C22C22
# reps14111222106816242

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

100000
010000
001000
000100
0000400
0000040
,
4000000
0400000
0011900
00323000
0000026
0000300
,
3220000
090000
000900
00321900
00001726
00001124
,
6350000
13350000
00383500
0015300
000010
000001

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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