Copied to
clipboard

G = Q8⋊Dic10order 320 = 26·5

1st semidirect product of Q8 and Dic10 acting via Dic10/Dic5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q81Dic10, Dic5.19SD16, (C5×Q8)⋊1Q8, C20⋊Q8.2C2, C20.5(C2×Q8), C4⋊C4.15D10, C51(Q8⋊Q8), C406C4.6C2, (C2×C8).119D10, (C2×Q8).99D10, Q8⋊C4.5D5, (Q8×Dic5).3C2, C4.5(C2×Dic10), C2.14(D5×SD16), C10.27(C2×SD16), Q8⋊Dic5.1C2, C22.185(D4×D5), C20.8Q8.5C2, C20.Q8.2C2, C20.157(C4○D4), C4.82(D42D5), (C2×C40).130C22, (C2×C20).231C23, C2.8(Q16⋊D5), (C2×Dic5).203D4, C10.11(C22⋊Q8), C4⋊Dic5.81C22, (Q8×C10).14C22, C10.53(C8.C22), (C4×Dic5).22C22, C2.16(Dic5.14D4), (C2×C10).244(C2×D4), (C5×C4⋊C4).32C22, (C5×Q8⋊C4).5C2, (C2×C52C8).27C22, (C2×C4).338(C22×D5), SmallGroup(320,418)

Series: Derived Chief Lower central Upper central

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

Generators and relations for Q8⋊Dic10
 G = < a,b,c,d | a4=c20=1, b2=a2, d2=c10, bab-1=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd-1=c-1 >

Subgroups: 342 in 96 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C4 [×2], C4 [×8], C22, C5, C8 [×2], C2×C4, C2×C4 [×6], Q8 [×2], Q8 [×3], C10 [×3], C42 [×2], C4⋊C4, C4⋊C4 [×4], C2×C8, C2×C8, C2×Q8, C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×3], C2×C10, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8 [×2], C4×Q8, C4⋊Q8, C52C8, C40, Dic10 [×2], 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, C5×C4⋊C4, C2×C40, C2×Dic10, Q8×C10, C20.Q8, C20.8Q8, C406C4, Q8⋊Dic5, C5×Q8⋊C4, C20⋊Q8, Q8×Dic5, Q8⋊Dic10
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, Dic10 [×2], C22×D5, Q8⋊Q8, C2×Dic10, D4×D5, D42D5, Dic5.14D4, D5×SD16, Q16⋊D5, Q8⋊Dic10

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

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

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

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

47 conjugacy classes

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

47 irreducible representations

dim1111111122222222244444
type+++++++++-++++---+
imageC1C2C2C2C2C2C2C2D4Q8D5SD16C4○D4D10D10D10Dic10C8.C22D42D5D4×D5D5×SD16Q16⋊D5
kernelQ8⋊Dic10C20.Q8C20.8Q8C406C4Q8⋊Dic5C5×Q8⋊C4C20⋊Q8Q8×Dic5C2×Dic5C5×Q8Q8⋊C4Dic5C20C4⋊C4C2×C8C2×Q8Q8C10C4C22C2C2
# reps1111111122242222812244

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

1000
0100
0015
001640
,
40000
04000
002012
001121
,
27200
251100
00517
00136
,
241800
341700
0010
0001
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,16,0,0,5,40],[40,0,0,0,0,40,0,0,0,0,20,11,0,0,12,21],[27,25,0,0,2,11,0,0,0,0,5,1,0,0,17,36],[24,34,0,0,18,17,0,0,0,0,1,0,0,0,0,1] >;

Q8⋊Dic10 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,418);
// by ID

G=gap.SmallGroup(320,418);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,254,219,58,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^20=1,b^2=a^2,d^2=c^10,b*a*b^-1=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

׿
×
𝔽