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G = Q8.2Dic10order 320 = 26·5

2nd non-split extension by Q8 of Dic10 acting via Dic10/Dic5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8.2Dic10, C52(Q8.Q8), (C5×Q8).2Q8, C20.8(C2×Q8), C4⋊C4.21D10, C406C4.7C2, (C2×C8).121D10, Q8⋊C4.6D5, (Q8×Dic5).6C2, C4.8(C2×Dic10), C10.46(C4○D8), (C2×Q8).102D10, Q8⋊Dic5.6C2, C10.D8.3C2, C22.193(D4×D5), C4.Dic10.4C2, C20.8Q8.6C2, C20.160(C4○D4), C4.85(D42D5), (C2×C20).239C23, (C2×C40).132C22, (C2×Dic5).207D4, C10.14(C22⋊Q8), C4⋊Dic5.87C22, (Q8×C10).22C22, C2.12(Q16⋊D5), C10.57(C8.C22), (C4×Dic5).29C22, C2.15(SD163D5), C2.19(Dic5.14D4), (C2×C10).252(C2×D4), (C5×C4⋊C4).40C22, (C5×Q8⋊C4).6C2, (C2×C52C8).34C22, (C2×C4).346(C22×D5), SmallGroup(320,426)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Q8.2Dic10
C1C5C10C2×C10C2×C20C4×Dic5Q8×Dic5 — Q8.2Dic10
C5C10C2×C20 — Q8.2Dic10
C1C22C2×C4Q8⋊C4

Generators and relations for Q8.2Dic10
 G = < a,b,c,d | a4=c20=1, b2=a2, d2=a2c10, bab-1=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd-1=c-1 >

Subgroups: 294 in 90 conjugacy classes, 41 normal (37 characteristic)
C1, C2 [×3], C4 [×2], C4 [×7], C22, C5, C8 [×2], C2×C4, C2×C4 [×6], Q8 [×2], Q8, C10 [×3], C42 [×2], C4⋊C4, C4⋊C4 [×5], C2×C8, C2×C8, C2×Q8, Dic5 [×4], C20 [×2], C20 [×3], C2×C10, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C52C8, C40, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8, Q8.Q8, C2×C52C8, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5 [×2], C4⋊Dic5 [×2], C5×C4⋊C4, C2×C40, Q8×C10, C10.D8, C20.8Q8, C406C4, Q8⋊Dic5, C5×Q8⋊C4, C4.Dic10, Q8×Dic5, Q8.2Dic10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C4○D8, C8.C22, Dic10 [×2], C22×D5, Q8.Q8, C2×Dic10, D4×D5, D42D5, Dic5.14D4, SD163D5, Q16⋊D5, Q8.2Dic10

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

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

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

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

47 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12224444444444455888810···102020202020···2040···40
size111122448101020202040224420202···244448···84···4

47 irreducible representations

dim1111111122222222244444
type+++++++++-++++---+
imageC1C2C2C2C2C2C2C2D4Q8D5C4○D4D10D10D10C4○D8Dic10C8.C22D42D5D4×D5SD163D5Q16⋊D5
kernelQ8.2Dic10C10.D8C20.8Q8C406C4Q8⋊Dic5C5×Q8⋊C4C4.Dic10Q8×Dic5C2×Dic5C5×Q8Q8⋊C4C20C4⋊C4C2×C8C2×Q8C10Q8C10C4C22C2C2
# reps1111111122222224812244

Matrix representation of Q8.2Dic10 in GL4(𝔽41) generated by

1900
184000
0010
0001
,
81300
363300
0010
0001
,
191400
272200
00911
003014
,
32000
03200
00320
00229
G:=sub<GL(4,GF(41))| [1,18,0,0,9,40,0,0,0,0,1,0,0,0,0,1],[8,36,0,0,13,33,0,0,0,0,1,0,0,0,0,1],[19,27,0,0,14,22,0,0,0,0,9,30,0,0,11,14],[32,0,0,0,0,32,0,0,0,0,32,22,0,0,0,9] >;

Q8.2Dic10 in GAP, Magma, Sage, TeX

Q_8._2{\rm Dic}_{10}
% in TeX

G:=Group("Q8.2Dic10");
// GroupNames label

G:=SmallGroup(320,426);
// by ID

G=gap.SmallGroup(320,426);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,926,219,226,851,438,102,12550]);
// Polycyclic

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

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