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G = (C2×C20).288D4order 320 = 26·5

262nd non-split extension by C2×C20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20).288D4, (C2×Dic5).6Q8, C22.47(Q8×D5), (C22×C4).99D10, C2.6(D103Q8), C10.71(C22⋊Q8), C2.7(C20.17D4), C10.40(C4.4D4), C10.21(C42.C2), C23.376(C22×D5), C10.26(C422C2), C22.104(C4○D20), (C22×C20).392C22, (C22×C10).346C23, C54(C23.83C23), C22.48(Q82D5), C22.100(D42D5), C10.10C42.19C2, C2.12(Dic5.Q8), C10.63(C22.D4), (C22×Dic5).54C22, C2.13(C23.23D10), (C2×C4⋊C4).19D5, (C10×C4⋊C4).32C2, (C2×C10).81(C2×Q8), (C2×C10).449(C2×D4), (C2×C4).37(C5⋊D4), C2.12(C4⋊C4⋊D5), C22.136(C2×C5⋊D4), (C2×C10).155(C4○D4), (C2×C10.D4).33C2, SmallGroup(320,609)

Series: Derived Chief Lower central Upper central

C1C22×C10 — (C2×C20).288D4
C1C5C10C2×C10C22×C10C22×Dic5C2×C10.D4 — (C2×C20).288D4
C5C22×C10 — (C2×C20).288D4
C1C23C2×C4⋊C4

Generators and relations for (C2×C20).288D4
 G = < a,b,c,d | a2=b20=c4=1, d2=b10, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=ab9, dcd-1=ac-1 >

Subgroups: 438 in 134 conjugacy classes, 57 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C4 [×9], C22 [×3], C22 [×4], C5, C2×C4 [×2], C2×C4 [×21], C23, C10 [×3], C10 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], Dic5 [×5], C20 [×4], C2×C10 [×3], C2×C10 [×4], C2.C42 [×5], C2×C4⋊C4, C2×C4⋊C4, C2×Dic5 [×2], C2×Dic5 [×11], C2×C20 [×2], C2×C20 [×8], C22×C10, C23.83C23, C10.D4 [×2], C5×C4⋊C4 [×2], C22×Dic5 [×2], C22×Dic5 [×2], C22×C20, C22×C20 [×2], C10.10C42, C10.10C42 [×4], C2×C10.D4, C10×C4⋊C4, (C2×C20).288D4
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], C5⋊D4 [×2], C22×D5, C23.83C23, C4○D20 [×2], D42D5 [×2], Q8×D5, Q82D5, C2×C5⋊D4, Dic5.Q8 [×2], C4⋊C4⋊D5 [×2], C23.23D10, C20.17D4, D103Q8, (C2×C20).288D4

Smallest permutation representation of (C2×C20).288D4
Regular action on 320 points
Generators in S320
(1 184)(2 185)(3 186)(4 187)(5 188)(6 189)(7 190)(8 191)(9 192)(10 193)(11 194)(12 195)(13 196)(14 197)(15 198)(16 199)(17 200)(18 181)(19 182)(20 183)(21 307)(22 308)(23 309)(24 310)(25 311)(26 312)(27 313)(28 314)(29 315)(30 316)(31 317)(32 318)(33 319)(34 320)(35 301)(36 302)(37 303)(38 304)(39 305)(40 306)(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 97)(62 98)(63 99)(64 100)(65 81)(66 82)(67 83)(68 84)(69 85)(70 86)(71 87)(72 88)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(121 148)(122 149)(123 150)(124 151)(125 152)(126 153)(127 154)(128 155)(129 156)(130 157)(131 158)(132 159)(133 160)(134 141)(135 142)(136 143)(137 144)(138 145)(139 146)(140 147)(161 250)(162 251)(163 252)(164 253)(165 254)(166 255)(167 256)(168 257)(169 258)(170 259)(171 260)(172 241)(173 242)(174 243)(175 244)(176 245)(177 246)(178 247)(179 248)(180 249)(201 231)(202 232)(203 233)(204 234)(205 235)(206 236)(207 237)(208 238)(209 239)(210 240)(211 221)(212 222)(213 223)(214 224)(215 225)(216 226)(217 227)(218 228)(219 229)(220 230)(261 298)(262 299)(263 300)(264 281)(265 282)(266 283)(267 284)(268 285)(269 286)(270 287)(271 288)(272 289)(273 290)(274 291)(275 292)(276 293)(277 294)(278 295)(279 296)(280 297)
(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 80 258 308)(2 95 259 21)(3 78 260 306)(4 93 241 39)(5 76 242 304)(6 91 243 37)(7 74 244 302)(8 89 245 35)(9 72 246 320)(10 87 247 33)(11 70 248 318)(12 85 249 31)(13 68 250 316)(14 83 251 29)(15 66 252 314)(16 81 253 27)(17 64 254 312)(18 99 255 25)(19 62 256 310)(20 97 257 23)(22 184 96 169)(24 182 98 167)(26 200 100 165)(28 198 82 163)(30 196 84 161)(32 194 86 179)(34 192 88 177)(36 190 90 175)(38 188 92 173)(40 186 94 171)(41 138 272 223)(42 144 273 212)(43 136 274 221)(44 142 275 210)(45 134 276 239)(46 160 277 208)(47 132 278 237)(48 158 279 206)(49 130 280 235)(50 156 261 204)(51 128 262 233)(52 154 263 202)(53 126 264 231)(54 152 265 220)(55 124 266 229)(56 150 267 218)(57 122 268 227)(58 148 269 216)(59 140 270 225)(60 146 271 214)(61 168 309 183)(63 166 311 181)(65 164 313 199)(67 162 315 197)(69 180 317 195)(71 178 319 193)(73 176 301 191)(75 174 303 189)(77 172 305 187)(79 170 307 185)(101 123 284 228)(102 149 285 217)(103 121 286 226)(104 147 287 215)(105 139 288 224)(106 145 289 213)(107 137 290 222)(108 143 291 211)(109 135 292 240)(110 141 293 209)(111 133 294 238)(112 159 295 207)(113 131 296 236)(114 157 297 205)(115 129 298 234)(116 155 299 203)(117 127 300 232)(118 153 281 201)(119 125 282 230)(120 151 283 219)
(1 287 11 297)(2 279 12 269)(3 285 13 295)(4 277 14 267)(5 283 15 293)(6 275 16 265)(7 281 17 291)(8 273 18 263)(9 299 19 289)(10 271 20 261)(21 236 31 226)(22 215 32 205)(23 234 33 224)(24 213 34 203)(25 232 35 222)(26 211 36 201)(27 230 37 240)(28 209 38 219)(29 228 39 238)(30 207 40 217)(41 177 51 167)(42 255 52 245)(43 175 53 165)(44 253 54 243)(45 173 55 163)(46 251 56 241)(47 171 57 161)(48 249 58 259)(49 169 59 179)(50 247 60 257)(61 156 71 146)(62 138 72 128)(63 154 73 144)(64 136 74 126)(65 152 75 142)(66 134 76 124)(67 150 77 160)(68 132 78 122)(69 148 79 158)(70 130 80 140)(81 125 91 135)(82 141 92 151)(83 123 93 133)(84 159 94 149)(85 121 95 131)(86 157 96 147)(87 139 97 129)(88 155 98 145)(89 137 99 127)(90 153 100 143)(101 172 111 162)(102 250 112 260)(103 170 113 180)(104 248 114 258)(105 168 115 178)(106 246 116 256)(107 166 117 176)(108 244 118 254)(109 164 119 174)(110 242 120 252)(181 300 191 290)(182 272 192 262)(183 298 193 288)(184 270 194 280)(185 296 195 286)(186 268 196 278)(187 294 197 284)(188 266 198 276)(189 292 199 282)(190 264 200 274)(202 301 212 311)(204 319 214 309)(206 317 216 307)(208 315 218 305)(210 313 220 303)(221 302 231 312)(223 320 233 310)(225 318 235 308)(227 316 237 306)(229 314 239 304)

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

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

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

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

dim11112222222444
type++++-+++--+
imageC1C2C2C2Q8D4D5C4○D4D10C5⋊D4C4○D20D42D5Q8×D5Q82D5
kernel(C2×C20).288D4C10.10C42C2×C10.D4C10×C4⋊C4C2×Dic5C2×C20C2×C4⋊C4C2×C10C22×C4C2×C4C22C22C22C22
# reps1511222106816422

Matrix representation of (C2×C20).288D4 in GL6(𝔽41)

4000000
0400000
001000
000100
0000400
0000040
,
3240000
17400000
007100
0040000
0000163
00001025
,
25290000
18160000
00193200
00222200
0000320
0000032
,
20150000
39210000
00143000
00142700
00003237
000009

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[3,17,0,0,0,0,24,40,0,0,0,0,0,0,7,40,0,0,0,0,1,0,0,0,0,0,0,0,16,10,0,0,0,0,3,25],[25,18,0,0,0,0,29,16,0,0,0,0,0,0,19,22,0,0,0,0,32,22,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[20,39,0,0,0,0,15,21,0,0,0,0,0,0,14,14,0,0,0,0,30,27,0,0,0,0,0,0,32,0,0,0,0,0,37,9] >;

(C2×C20).288D4 in GAP, Magma, Sage, TeX

(C_2\times C_{20})._{288}D_4
% in TeX

G:=Group("(C2xC20).288D4");
// GroupNames label

G:=SmallGroup(320,609);
// by ID

G=gap.SmallGroup(320,609);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,232,254,387,268,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=a*b^9,d*c*d^-1=a*c^-1>;
// generators/relations

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