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G = (C2×Dic5)⋊Q8  order 320 = 26·5

1st semidirect product of C2×Dic5 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20)⋊1Q8, (C2×Dic5)⋊1Q8, (C2×C4)⋊1Dic10, (C2×C20).27D4, (C2×C4).16D20, C2.6(C20⋊Q8), C10.1C22≀C2, C10.11(C4⋊Q8), C22.38(Q8×D5), (C2×Dic5).15D4, C2.4(C20⋊2Q8), C22.150(D4×D5), (C22×C4).67D10, C22.77(C2×D20), C2.6(C22⋊D20), C2.6(D10⋊2Q8), C10.20(C22⋊Q8), (C22×C20).44C22, (C22×Dic10).2C2, C22.40(C2×Dic10), C2.C42.12D5, C23.352(C22×D5), C22.81(D4⋊2D5), (C22×C10).281C23, C5⋊1(C23.78C23), (C22×Dic5).6C22, C10.10C42.24C2, C2.8(Dic5.14D4), (C2×C10).93(C2×D4), (C2×C4⋊Dic5).7C2, (C2×C10).62(C2×Q8), (C2×C10).126(C4○D4), (C2×C10.D4).20C2, (C5×C2.C42).8C2, SmallGroup(320,283)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C10 — (C2×Dic5)⋊Q8
Chief series
C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C22×Dic10 — (C2×Dic5)⋊Q8
Lower central
C5 — C22×C10 — (C2×Dic5)⋊Q8
Upper central
C1 — C23 — C2.C42

Generators and relations for (C2×Dic5)⋊Q8
 G = < a,b,c,d,e | a2=b10=d4=1, c2=b5, e2=d2, ab=ba, dcd-1=ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, ece-1=b5c, ede-1=d-1 >

Subgroups: 646 in 182 conjugacy classes, 69 normal (25 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C10, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C2.C42, C2.C42, C2×C4⋊C4, C22×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.78C23, C10.D4, C4⋊Dic5, C2×Dic10, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C10.10C42, C5×C2.C42, C2×C10.D4, C2×C4⋊Dic5, C22×Dic10, (C2×Dic5)⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22≀C2, C22⋊Q8, C4⋊Q8, Dic10, D20, C22×D5, C23.78C23, C2×Dic10, C2×D20, D4×D5, D4⋊2D5, Q8×D5, C20⋊2Q8, Dic5.14D4, C22⋊D20, C20⋊Q8, D10⋊2Q8, (C2×Dic5)⋊Q8

Smallest permutation representation of (C2×Dic5)⋊Q8
►Regular action on 320 points
Generators in S320
(1 36)(2 37)(3 38)(4 39)(5 40)(6 31)(7 32)(8 33)(9 34)(10 35)(11 299)(12 300)(13 291)(14 292)(15 293)(16 294)(17 295)(18 296)(19 297)(20 298)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 41)(30 42)(51 87)(52 88)(53 89)(54 90)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 71)(68 72)(69 73)(70 74)(91 127)(92 128)(93 129)(94 130)(95 121)(96 122)(97 123)(98 124)(99 125)(100 126)(101 115)(102 116)(103 117)(104 118)(105 119)(106 120)(107 111)(108 112)(109 113)(110 114)(131 170)(132 161)(133 162)(134 163)(135 164)(136 165)(137 166)(138 167)(139 168)(140 169)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 151)(148 152)(149 153)(150 154)(171 210)(172 201)(173 202)(174 203)(175 204)(176 205)(177 206)(178 207)(179 208)(180 209)(181 192)(182 193)(183 194)(184 195)(185 196)(186 197)(187 198)(188 199)(189 200)(190 191)(211 250)(212 241)(213 242)(214 243)(215 244)(216 245)(217 246)(218 247)(219 248)(220 249)(221 232)(222 233)(223 234)(224 235)(225 236)(226 237)(227 238)(228 239)(229 240)(230 231)(251 290)(252 281)(253 282)(254 283)(255 284)(256 285)(257 286)(258 287)(259 288)(260 289)(261 272)(262 273)(263 274)(264 275)(265 276)(266 277)(267 278)(268 279)(269 280)(270 271)(301 312)(302 313)(303 314)(304 315)(305 316)(306 317)(307 318)(308 319)(309 320)(310 311)

(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)

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

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

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

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

dim111111222222222444
type+++++++-+-++-++--
imageC1C2C2C2C2C2D4Q8D4Q8D5C4○D4D10Dic10D20D4×D5D4⋊2D5Q8×D5
kernel(C2×Dic5)⋊Q8C10.10C42C5×C2.C42C2×C10.D4C2×C4⋊Dic5C22×Dic10C2×Dic5C2×Dic5C2×C20C2×C20C2.C42C2×C10C22×C4C2×C4C2×C4C22C22C22
# reps1211214224226168422

Matrix representation of (C2×Dic5)⋊Q8 ►in GL6(𝔽41)

4000000
0400000
0040000
0004000
0000400
0000040
,
34400000
100000
0040000
0004000
0000400
0000040
,
30140000
9110000
00401400
0035100
0000040
000010
,
170000
0400000
00342500
0039700
0000400
000001
,
100000
010000
00203200
00402100
000010
0000040

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[34,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[30,9,0,0,0,0,14,11,0,0,0,0,0,0,40,35,0,0,0,0,14,1,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[1,0,0,0,0,0,7,40,0,0,0,0,0,0,34,39,0,0,0,0,25,7,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,20,40,0,0,0,0,32,21,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
 

(C2×Dic5)⋊Q8 in GAP, Magma, Sage, TeX 

(C_2\times {\rm Dic}_5)\rtimes Q_8
 
% in TeX
 
G:=Group("(C2xDic5):Q8");
 
// GroupNames label
 
G:=SmallGroup(320,283);
 
// by ID
 
G=gap.SmallGroup(320,283);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,254,387,58,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^10=d^4=1,c^2=b^5,e^2=d^2,a*b=b*a,d*c*d^-1=a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,e*c*e^-1=b^5*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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