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G = Dic10⋊Q8order 320 = 26·5

1st semidirect product of Dic10 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic101Q8, Dic5.20SD16, C4.1(Q8×D5), C20⋊Q8.5C2, C20.9(C2×Q8), C4⋊C4.31D10, C53(Q8⋊Q8), C4.Q8.4D5, (C2×C8).135D10, C2.20(D5×SD16), C4.70(C4○D20), C10.35(C2×SD16), C10.Q16.5C2, C22.209(D4×D5), C20.Q8.4C2, C20.166(C4○D4), (C2×C20).270C23, (C2×C40).282C22, (C2×Dic5).217D4, Dic53Q8.5C2, C10.35(C22⋊Q8), C20.8Q8.13C2, C2.12(D10⋊Q8), C20.44D4.13C2, C2.20(SD16⋊D5), C10.38(C8.C22), C4⋊Dic5.102C22, (C4×Dic5).34C22, (C2×Dic10).84C22, (C5×C4.Q8).9C2, (C2×C10).275(C2×D4), (C5×C4⋊C4).63C22, (C2×C52C8).52C22, (C2×C4).373(C22×D5), SmallGroup(320,481)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic10⋊Q8
C1C5C10C20C2×C20C4×Dic5C20⋊Q8 — Dic10⋊Q8
C5C10C2×C20 — Dic10⋊Q8
C1C22C2×C4C4.Q8

Generators and relations for Dic10⋊Q8
 G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, cac-1=a9, dad-1=a11, bc=cb, dbd-1=a15b, dcd-1=c-1 >

Subgroups: 358 in 96 conjugacy classes, 41 normal (37 characteristic)
C1, C2 [×3], C4 [×2], C4 [×8], C22, C5, C8 [×2], C2×C4, C2×C4 [×6], Q8 [×5], C10 [×3], C42 [×2], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8, C2×C8, C2×Q8 [×2], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×2], C2×C10, Q8⋊C4 [×2], C4⋊C8, C4.Q8, C4.Q8, C4×Q8, C4⋊Q8, C52C8, C40, Dic10 [×2], Dic10 [×3], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], Q8⋊Q8, C2×C52C8, C4×Dic5, C4×Dic5, C10.D4 [×2], C4⋊Dic5, C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C2×Dic10, C20.Q8, C10.Q16, C20.8Q8, C20.44D4, C5×C4.Q8, Dic53Q8, C20⋊Q8, Dic10⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, SD16 [×2], C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C2×SD16, C8.C22, C22×D5, Q8⋊Q8, C4○D20, D4×D5, Q8×D5, D10⋊Q8, D5×SD16, SD16⋊D5, Dic10⋊Q8

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

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

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

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

47 conjugacy classes

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

47 irreducible representations

dim111111112222222244444
type++++++++-++++--+-
imageC1C2C2C2C2C2C2C2Q8D4D5SD16C4○D4D10D10C4○D20C8.C22Q8×D5D4×D5D5×SD16SD16⋊D5
kernelDic10⋊Q8C20.Q8C10.Q16C20.8Q8C20.44D4C5×C4.Q8Dic53Q8C20⋊Q8Dic10C2×Dic5C4.Q8Dic5C20C4⋊C4C2×C8C4C10C4C22C2C2
# reps111111112224242812244

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

04000
13500
00139
00140
,
5800
383600
002335
002018
,
43100
143700
0010
0001
,
21300
283900
00818
001733
G:=sub<GL(4,GF(41))| [0,1,0,0,40,35,0,0,0,0,1,1,0,0,39,40],[5,38,0,0,8,36,0,0,0,0,23,20,0,0,35,18],[4,14,0,0,31,37,0,0,0,0,1,0,0,0,0,1],[2,28,0,0,13,39,0,0,0,0,8,17,0,0,18,33] >;

Dic10⋊Q8 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,481);
// by ID

G=gap.SmallGroup(320,481);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,135,268,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=1,b^2=a^10,d^2=c^2,b*a*b^-1=a^-1,c*a*c^-1=a^9,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;
// generators/relations

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