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G = Dic10.2Q8order 320 = 26·5

2nd non-split extension by Dic10 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic10.2Q8, C4.6(Q8×D5), C55(Q8.Q8), C4⋊C4.45D10, (C2×C8).26D10, C2.D8.6D5, C20.19(C2×Q8), C4.79(C4○D20), C10.27(C4○D8), (C2×Dic5).55D4, C10.Q16.9C2, C22.226(D4×D5), C4.Dic10.9C2, C20.Q8.8C2, C20.171(C4○D4), C2.12(D83D5), (C2×C20).293C23, (C2×C40).240C22, Dic53Q8.9C2, C10.40(C22⋊Q8), C20.8Q8.11C2, C2.17(D10⋊Q8), C2.21(Q16⋊D5), C20.44D4.11C2, C10.68(C8.C22), C4⋊Dic5.119C22, (C4×Dic5).42C22, (C2×Dic10).92C22, (C5×C2.D8).12C2, (C2×C10).298(C2×D4), (C5×C4⋊C4).86C22, (C2×C52C8).67C22, (C2×C4).396(C22×D5), SmallGroup(320,504)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic10.2Q8
C1C5C10C20C2×C20C4×Dic5Dic53Q8 — Dic10.2Q8
C5C10C2×C20 — Dic10.2Q8
C1C22C2×C4C2.D8

Generators and relations for Dic10.2Q8
 G = < a,b,c,d | a20=c4=1, b2=a10, d2=a10c2, bab-1=a-1, cac-1=a11, dad-1=a9, cbc-1=a15b, bd=db, dcd-1=c-1 >

Subgroups: 310 in 90 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C4 [×2], C4 [×7], C22, C5, C8 [×2], C2×C4, C2×C4 [×6], Q8 [×3], C10 [×3], C42 [×2], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8, C2×C8, C2×Q8, Dic5 [×5], C20 [×2], C20 [×2], C2×C10, Q8⋊C4 [×2], C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C52C8, C40, Dic10 [×2], Dic10, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], Q8.Q8, C2×C52C8, C4×Dic5, C4×Dic5, C10.D4 [×2], C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C20.Q8, C10.Q16, C20.8Q8, C20.44D4, C5×C2.D8, Dic53Q8, C4.Dic10, Dic10.2Q8
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, C22×D5, Q8.Q8, C4○D20, D4×D5, Q8×D5, D10⋊Q8, D83D5, Q16⋊D5, Dic10.2Q8

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

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

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

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

47 conjugacy classes

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

47 irreducible representations

dim111111112222222244444
type++++++++-++++--+-
imageC1C2C2C2C2C2C2C2Q8D4D5C4○D4D10D10C4○D8C4○D20C8.C22Q8×D5D4×D5D83D5Q16⋊D5
kernelDic10.2Q8C20.Q8C10.Q16C20.8Q8C20.44D4C5×C2.D8Dic53Q8C4.Dic10Dic10C2×Dic5C2.D8C20C4⋊C4C2×C8C10C4C10C4C22C2C2
# reps111111112222424812244

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

14000
83400
004023
00321
,
352900
20600
00216
002220
,
112800
223000
00512
003936
,
281500
161300
00320
00032
G:=sub<GL(4,GF(41))| [1,8,0,0,40,34,0,0,0,0,40,32,0,0,23,1],[35,20,0,0,29,6,0,0,0,0,21,22,0,0,6,20],[11,22,0,0,28,30,0,0,0,0,5,39,0,0,12,36],[28,16,0,0,15,13,0,0,0,0,32,0,0,0,0,32] >;

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

{\rm Dic}_{10}._2Q_8
% in TeX

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

G:=SmallGroup(320,504);
// by ID

G=gap.SmallGroup(320,504);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,1094,135,268,570,297,136,12550]);
// Polycyclic

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

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