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## G = Q8×Dic10order 320 = 26·5

### Direct product of Q8 and Dic10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Q8×Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — Q8×Dic5 — Q8×Dic10
 Lower central C5 — C2×C10 — Q8×Dic10
 Upper central C1 — C22 — C4×Q8

Generators and relations for Q8×Dic10
G = < a,b,c,d | a4=c20=1, b2=a2, d2=c10, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 646 in 212 conjugacy classes, 123 normal (18 characteristic)
C1, C2, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, Q8, Q8, C10, C42, C42, C4⋊C4, C4⋊C4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4×Q8, C4×Q8, C4⋊Q8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, Q82, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C5×C4⋊C4, C2×Dic10, C2×Dic10, Q8×C10, C4×Dic10, C202Q8, C20⋊Q8, Q8×Dic5, Q8×C20, Q8×Dic10
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C24, D10, C22×Q8, 2+ 1+4, Dic10, C22×D5, Q82, C2×Dic10, Q8×D5, C23×D5, C22×Dic10, C2×Q8×D5, D48D10, Q8×Dic10

Smallest permutation representation of Q8×Dic10
Regular action on 320 points
Generators in S320
(1 320 177 145)(2 301 178 146)(3 302 179 147)(4 303 180 148)(5 304 161 149)(6 305 162 150)(7 306 163 151)(8 307 164 152)(9 308 165 153)(10 309 166 154)(11 310 167 155)(12 311 168 156)(13 312 169 157)(14 313 170 158)(15 314 171 159)(16 315 172 160)(17 316 173 141)(18 317 174 142)(19 318 175 143)(20 319 176 144)(21 89 242 71)(22 90 243 72)(23 91 244 73)(24 92 245 74)(25 93 246 75)(26 94 247 76)(27 95 248 77)(28 96 249 78)(29 97 250 79)(30 98 251 80)(31 99 252 61)(32 100 253 62)(33 81 254 63)(34 82 255 64)(35 83 256 65)(36 84 257 66)(37 85 258 67)(38 86 259 68)(39 87 260 69)(40 88 241 70)(41 276 282 133)(42 277 283 134)(43 278 284 135)(44 279 285 136)(45 280 286 137)(46 261 287 138)(47 262 288 139)(48 263 289 140)(49 264 290 121)(50 265 291 122)(51 266 292 123)(52 267 293 124)(53 268 294 125)(54 269 295 126)(55 270 296 127)(56 271 297 128)(57 272 298 129)(58 273 299 130)(59 274 300 131)(60 275 281 132)(101 197 223 213)(102 198 224 214)(103 199 225 215)(104 200 226 216)(105 181 227 217)(106 182 228 218)(107 183 229 219)(108 184 230 220)(109 185 231 201)(110 186 232 202)(111 187 233 203)(112 188 234 204)(113 189 235 205)(114 190 236 206)(115 191 237 207)(116 192 238 208)(117 193 239 209)(118 194 240 210)(119 195 221 211)(120 196 222 212)
(1 40 177 241)(2 21 178 242)(3 22 179 243)(4 23 180 244)(5 24 161 245)(6 25 162 246)(7 26 163 247)(8 27 164 248)(9 28 165 249)(10 29 166 250)(11 30 167 251)(12 31 168 252)(13 32 169 253)(14 33 170 254)(15 34 171 255)(16 35 172 256)(17 36 173 257)(18 37 174 258)(19 38 175 259)(20 39 176 260)(41 116 282 238)(42 117 283 239)(43 118 284 240)(44 119 285 221)(45 120 286 222)(46 101 287 223)(47 102 288 224)(48 103 289 225)(49 104 290 226)(50 105 291 227)(51 106 292 228)(52 107 293 229)(53 108 294 230)(54 109 295 231)(55 110 296 232)(56 111 297 233)(57 112 298 234)(58 113 299 235)(59 114 300 236)(60 115 281 237)(61 156 99 311)(62 157 100 312)(63 158 81 313)(64 159 82 314)(65 160 83 315)(66 141 84 316)(67 142 85 317)(68 143 86 318)(69 144 87 319)(70 145 88 320)(71 146 89 301)(72 147 90 302)(73 148 91 303)(74 149 92 304)(75 150 93 305)(76 151 94 306)(77 152 95 307)(78 153 96 308)(79 154 97 309)(80 155 98 310)(121 200 264 216)(122 181 265 217)(123 182 266 218)(124 183 267 219)(125 184 268 220)(126 185 269 201)(127 186 270 202)(128 187 271 203)(129 188 272 204)(130 189 273 205)(131 190 274 206)(132 191 275 207)(133 192 276 208)(134 193 277 209)(135 194 278 210)(136 195 279 211)(137 196 280 212)(138 197 261 213)(139 198 262 214)(140 199 263 215)
(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 58 11 48)(2 57 12 47)(3 56 13 46)(4 55 14 45)(5 54 15 44)(6 53 16 43)(7 52 17 42)(8 51 18 41)(9 50 19 60)(10 49 20 59)(21 112 31 102)(22 111 32 101)(23 110 33 120)(24 109 34 119)(25 108 35 118)(26 107 36 117)(27 106 37 116)(28 105 38 115)(29 104 39 114)(30 103 40 113)(61 214 71 204)(62 213 72 203)(63 212 73 202)(64 211 74 201)(65 210 75 220)(66 209 76 219)(67 208 77 218)(68 207 78 217)(69 206 79 216)(70 205 80 215)(81 196 91 186)(82 195 92 185)(83 194 93 184)(84 193 94 183)(85 192 95 182)(86 191 96 181)(87 190 97 200)(88 189 98 199)(89 188 99 198)(90 187 100 197)(121 144 131 154)(122 143 132 153)(123 142 133 152)(124 141 134 151)(125 160 135 150)(126 159 136 149)(127 158 137 148)(128 157 138 147)(129 156 139 146)(130 155 140 145)(161 295 171 285)(162 294 172 284)(163 293 173 283)(164 292 174 282)(165 291 175 281)(166 290 176 300)(167 289 177 299)(168 288 178 298)(169 287 179 297)(170 286 180 296)(221 245 231 255)(222 244 232 254)(223 243 233 253)(224 242 234 252)(225 241 235 251)(226 260 236 250)(227 259 237 249)(228 258 238 248)(229 257 239 247)(230 256 240 246)(261 302 271 312)(262 301 272 311)(263 320 273 310)(264 319 274 309)(265 318 275 308)(266 317 276 307)(267 316 277 306)(268 315 278 305)(269 314 279 304)(270 313 280 303)

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4H 4I 4J 4K 4L 4M 4N 4O 4P ··· 4U 5A 5B 10A ··· 10F 20A ··· 20H 20I ··· 20AF order 1 2 2 2 4 ··· 4 4 4 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 ··· 2 4 4 4 10 10 10 10 20 ··· 20 2 2 2 ··· 2 2 ··· 2 4 ··· 4

65 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + - - + + + + - + - + image C1 C2 C2 C2 C2 C2 Q8 Q8 D5 D10 D10 D10 Dic10 2+ 1+4 Q8×D5 D4⋊8D10 kernel Q8×Dic10 C4×Dic10 C20⋊2Q8 C20⋊Q8 Q8×Dic5 Q8×C20 Dic10 C5×Q8 C4×Q8 C42 C4⋊C4 C2×Q8 Q8 C10 C4 C2 # reps 1 3 3 6 2 1 4 4 2 6 6 2 16 1 4 4

Matrix representation of Q8×Dic10 in GL4(𝔽41) generated by

 0 1 0 0 40 0 0 0 0 0 1 0 0 0 0 1
,
 1 11 0 0 11 40 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 16 39 0 0 2 28
,
 40 0 0 0 0 40 0 0 0 0 31 37 0 0 15 10
G:=sub<GL(4,GF(41))| [0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,11,0,0,11,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,16,2,0,0,39,28],[40,0,0,0,0,40,0,0,0,0,31,15,0,0,37,10] >;

Q8×Dic10 in GAP, Magma, Sage, TeX

Q_8\times {\rm Dic}_{10}
% in TeX

G:=Group("Q8xDic10");
// GroupNames label

G:=SmallGroup(320,1238);
// by ID

G=gap.SmallGroup(320,1238);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,387,184,675,80,12550]);
// Polycyclic

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

׿
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