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G = Q8.Dic10order 320 = 26·5

1st non-split extension by Q8 of Dic10 acting via Dic10/Dic5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8.1Dic10, C51(Q8.Q8), (C5×Q8).1Q8, C20.7(C2×Q8), C4⋊C4.19D10, (C2×C8).15D10, C405C4.7C2, Q8⋊C4.4D5, (Q8×Dic5).5C2, C4.7(C2×Dic10), C10.67(C4○D8), (C2×C40).15C22, (C2×Q8).101D10, Q8⋊Dic5.4C2, C22.190(D4×D5), C4.Dic10.2C2, C20.Q8.3C2, C20.8Q8.4C2, C20.159(C4○D4), C2.6(Q8.D10), C4.84(D42D5), (C2×C20).236C23, (C2×Dic5).206D4, C10.13(C22⋊Q8), C4⋊Dic5.85C22, (Q8×C10).19C22, C2.15(SD16⋊D5), C10.33(C8.C22), (C4×Dic5).26C22, C2.18(Dic5.14D4), (C2×C10).249(C2×D4), (C5×C4⋊C4).37C22, (C5×Q8⋊C4).4C2, (C2×C52C8).31C22, (C2×C4).343(C22×D5), SmallGroup(320,423)

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=a2c-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, C20.Q8, C20.8Q8, C405C4, Q8⋊Dic5, C5×Q8⋊C4, C4.Dic10, Q8×Dic5, Q8.Dic10
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, SD16⋊D5, Q8.D10, Q8.Dic10

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

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

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

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

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×D5SD16⋊D5Q8.D10
kernelQ8.Dic10C20.Q8C20.8Q8C405C4Q8⋊Dic5C5×Q8⋊C4C4.Dic10Q8×Dic5C2×Dic5C5×Q8Q8⋊C4C20C4⋊C4C2×C8C2×Q8C10Q8C10C4C22C2C2
# reps1111111122222224812244

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

0100
40000
0010
0001
,
382100
21300
0010
0001
,
401100
11100
001639
00228
,
9000
0900
001510
00226
G:=sub<GL(4,GF(41))| [0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[38,21,0,0,21,3,0,0,0,0,1,0,0,0,0,1],[40,11,0,0,11,1,0,0,0,0,16,2,0,0,39,28],[9,0,0,0,0,9,0,0,0,0,15,2,0,0,10,26] >;

Q8.Dic10 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,423);
// by ID

G=gap.SmallGroup(320,423);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,926,219,58,851,438,102,12550]);
// 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=a^2*c^-1>;
// generators/relations

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