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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 [×3], C4 [×8], C4 [×13], C22, C5, C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×4], Q8 [×10], C10 [×3], C42 [×3], C42 [×6], C4⋊C4 [×3], C4⋊C4 [×15], C2×Q8, C2×Q8 [×7], Dic5 [×4], Dic5 [×6], C20 [×8], C20 [×3], C2×C10, C4×Q8, C4×Q8 [×5], C4⋊Q8 [×9], Dic10 [×4], Dic10 [×6], C2×Dic5 [×8], C2×C20, C2×C20 [×6], C5×Q8 [×4], Q82, C4×Dic5 [×6], C10.D4 [×6], C4⋊Dic5 [×9], C4×C20 [×3], C5×C4⋊C4 [×3], C2×Dic10, C2×Dic10 [×6], Q8×C10, C4×Dic10 [×3], C202Q8 [×3], C20⋊Q8 [×6], Q8×Dic5 [×2], Q8×C20, Q8×Dic10
Quotients: C1, C2 [×15], C22 [×35], Q8 [×8], C23 [×15], D5, C2×Q8 [×12], C24, D10 [×7], C22×Q8 [×2], 2+ 1+4, Dic10 [×4], C22×D5 [×7], Q82, C2×Dic10 [×6], Q8×D5 [×2], C23×D5, C22×Dic10, C2×Q8×D5, D48D10, Q8×Dic10

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

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

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

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

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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