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G = Dic10.4Q8order 320 = 26·5

2nd non-split extension by Dic10 of Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic10.4Q8, C42.67D10, C4.9(Q8×D5), C57(Q8.Q8), C4⋊C4.72D10, C20.31(C2×Q8), (C2×C20).273D4, C203C8.20C2, C42.C2.2D5, C20.69(C4○D4), C10.107(C4○D8), (C2×C20).381C23, (C4×C20).111C22, C4.32(Q82D5), (C4×Dic10).16C2, C10.Q16.12C2, C10.D8.12C2, C10.73(C22⋊Q8), C20.Q8.13C2, C2.10(D103Q8), C4⋊Dic5.342C22, C2.20(D4.9D10), C2.26(D4.8D10), C10.121(C8.C22), (C2×Dic10).281C22, (C2×C10).512(C2×D4), (C2×C4).64(C5⋊D4), (C5×C42.C2).1C2, (C5×C4⋊C4).119C22, (C2×C4).479(C22×D5), C22.185(C2×C5⋊D4), (C2×C52C8).124C22, SmallGroup(320,690)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic10.4Q8
C1C5C10C20C2×C20C4⋊Dic5C4×Dic10 — Dic10.4Q8
C5C10C2×C20 — Dic10.4Q8
C1C22C42C42.C2

Generators and relations for Dic10.4Q8
 G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=dad-1=a-1, cac-1=a11, cbc-1=a5b, bd=db, dcd-1=a15c-1 >

Subgroups: 278 in 90 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C4 [×2], C4 [×7], C22, C5, C8 [×2], C2×C4 [×3], C2×C4 [×4], Q8 [×3], C10 [×3], C42, C42, C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×Q8, Dic5 [×3], C20 [×2], C20 [×4], C2×C10, Q8⋊C4 [×2], C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C52C8 [×2], Dic10 [×2], Dic10, C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×2], Q8.Q8, C2×C52C8 [×2], C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C203C8, C10.D8, C20.Q8, C10.Q16 [×2], C4×Dic10, C5×C42.C2, Dic10.4Q8
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, C5⋊D4 [×2], C22×D5, Q8.Q8, Q8×D5, Q82D5, C2×C5⋊D4, D103Q8, D4.8D10, D4.9D10, Dic10.4Q8

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

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

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

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

47 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K5A5B8A8B8C8D10A···10F20A···20L20M···20T
order12224444444444455888810···1020···2020···20
size111122224882020202022202020202···24···48···8

47 irreducible representations

dim11111112222222244444
type+++++++-++++--+-
imageC1C2C2C2C2C2C2Q8D4D5C4○D4D10D10C4○D8C5⋊D4C8.C22Q8×D5Q82D5D4.8D10D4.9D10
kernelDic10.4Q8C203C8C10.D8C20.Q8C10.Q16C4×Dic10C5×C42.C2Dic10C2×C20C42.C2C20C42C4⋊C4C10C2×C4C10C4C4C2C2
# reps11112112222244812244

Matrix representation of Dic10.4Q8 in GL6(𝔽41)

1390000
1400000
000100
0040600
000010
000001
,
1860000
21230000
00251600
0021600
0000400
0000040
,
15340000
32260000
0040000
0004000
000001
0000400
,
2280000
16390000
00251600
0021600
000026
0000639

G:=sub<GL(6,GF(41))| [1,1,0,0,0,0,39,40,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,21,0,0,0,0,6,23,0,0,0,0,0,0,25,2,0,0,0,0,16,16,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[15,32,0,0,0,0,34,26,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[2,16,0,0,0,0,28,39,0,0,0,0,0,0,25,2,0,0,0,0,16,16,0,0,0,0,0,0,2,6,0,0,0,0,6,39] >;

Dic10.4Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}._4Q_8
% in TeX

G:=Group("Dic10.4Q8");
// GroupNames label

G:=SmallGroup(320,690);
// by ID

G=gap.SmallGroup(320,690);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,254,219,100,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=1,b^2=a^10,d^2=c^2,b*a*b^-1=d*a*d^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=a^15*c^-1>;
// generators/relations

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