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G = (C2×Dic5).Q8  order 320 = 26·5

2nd non-split extension by C2×Dic5 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×Dic5).2Q8, C22.40(Q8×D5), (C2×Dic5).17D4, C22.152(D4×D5), (C22×C4).11D10, C2.9(D10⋊Q8), C2.4(C42⋊2D5), C10.22(C22⋊Q8), (C22×C20).8C22, C2.C42.4D5, C10.1(C42⋊2C2), C10.16(C4.4D4), C22.85(C4○D20), C10.12(C42.C2), C23.354(C22×D5), C2.9(D10.12D4), C22.83(D4⋊2D5), C10.10C42.6C2, (C22×C10).283C23, C2.8(Dic5.5D4), C5⋊1(C23.83C23), C2.9(Dic5.Q8), C10.5(C22.D4), C2.8(C23.D10), (C22×Dic5).8C22, (C2×C10).64(C2×Q8), (C2×C10).195(C2×D4), (C2×C10.D4).9C2, (C2×C10).127(C4○D4), (C5×C2.C42).1C2, SmallGroup(320,285)

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 — C10.10C42 — (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=b5d2, ab=ba, ece-1=ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, dcd-1=ab5c, ede-1=ad-1 >

Subgroups: 454 in 134 conjugacy classes, 55 normal (51 characteristic)
C1, C2, C4, C22, C5, C2×C4, C23, C10, C4⋊C4, C22×C4, C22×C4, Dic5, C20, C2×C10, C2.C42, C2.C42, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C23.83C23, C10.D4, C22×Dic5, C22×C20, C10.10C42, C5×C2.C42, C2×C10.D4, (C2×Dic5).Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22×D5, C23.83C23, C4○D20, D4×D5, D4⋊2D5, Q8×D5, C42⋊2D5, C23.D10, D10.12D4, Dic5.5D4, Dic5.Q8, D10⋊Q8, (C2×Dic5).Q8

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

(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)

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

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

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

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

dim1111222222444
type+++++-+++--
imageC1C2C2C2D4Q8D5C4○D4D10C4○D20D4×D5D4⋊2D5Q8×D5
kernel(C2×Dic5).Q8C10.10C42C5×C2.C42C2×C10.D4C2×Dic5C2×Dic5C2.C42C2×C10C22×C4C22C22C22C22
# reps141222210624242

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

4000000
0400000
001000
000100
000010
000001
,
3510000
4000000
0040000
0004000
000010
000001
,
18350000
20230000
00253400
00251600
000010
000001
,
2130000
25390000
00253900
00251600
0000189
00001423
,
2130000
28390000
009000
000900
00003327
0000318

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],[35,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,20,0,0,0,0,35,23,0,0,0,0,0,0,25,25,0,0,0,0,34,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,25,0,0,0,0,13,39,0,0,0,0,0,0,25,25,0,0,0,0,39,16,0,0,0,0,0,0,18,14,0,0,0,0,9,23],[2,28,0,0,0,0,13,39,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,33,31,0,0,0,0,27,8] >;
 

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

(C_2\times {\rm Dic}_5).Q_8
 
% in TeX
 
G:=Group("(C2xDic5).Q8");
 
// GroupNames label
 
G:=SmallGroup(320,285);
 
// by ID
 
G=gap.SmallGroup(320,285);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,64,590,387,100,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^10=d^4=1,c^2=b^5,e^2=b^5*d^2,a*b=b*a,e*c*e^-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,d*c*d^-1=a*b^5*c,e*d*e^-1=a*d^-1>;
 
// generators/relations
 

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