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## G = Dic10⋊10Q8order 320 = 26·5

### The semidirect product of Dic10 and Q8 acting through Inn(Dic10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Dic10⋊10Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C2×Dic10 — C4×Dic10 — Dic10⋊10Q8
 Lower central C5 — C2×C10 — Dic10⋊10Q8
 Upper central C1 — C22 — C4×Q8

Generators and relations for Dic1010Q8
G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, ac=ca, ad=da, cbc-1=a10b, bd=db, dcd-1=c-1 >

Subgroups: 550 in 200 conjugacy classes, 107 normal (29 characteristic)
C1, C2, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C10, C42, C42, C42, C4⋊C4, C4⋊C4, C4⋊C4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4×Q8, C4×Q8, C42.C2, C4⋊Q8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, Q83Q8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, Q8×C10, C4×Dic10, C4×Dic10, C202Q8, C20.6Q8, Dic53Q8, Dic5.Q8, Dic5⋊Q8, Q8×C20, Dic1010Q8
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, C24, D10, C22×Q8, C2×C4○D4, 2- 1+4, C22×D5, Q83Q8, C4○D20, Q8×D5, C23×D5, C2×C4○D20, C2×Q8×D5, D4.10D10, Dic1010Q8

Smallest permutation representation of Dic1010Q8
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)
(1 247 11 257)(2 246 12 256)(3 245 13 255)(4 244 14 254)(5 243 15 253)(6 242 16 252)(7 241 17 251)(8 260 18 250)(9 259 19 249)(10 258 20 248)(21 317 31 307)(22 316 32 306)(23 315 33 305)(24 314 34 304)(25 313 35 303)(26 312 36 302)(27 311 37 301)(28 310 38 320)(29 309 39 319)(30 308 40 318)(41 267 51 277)(42 266 52 276)(43 265 53 275)(44 264 54 274)(45 263 55 273)(46 262 56 272)(47 261 57 271)(48 280 58 270)(49 279 59 269)(50 278 60 268)(61 110 71 120)(62 109 72 119)(63 108 73 118)(64 107 74 117)(65 106 75 116)(66 105 76 115)(67 104 77 114)(68 103 78 113)(69 102 79 112)(70 101 80 111)(81 215 91 205)(82 214 92 204)(83 213 93 203)(84 212 94 202)(85 211 95 201)(86 210 96 220)(87 209 97 219)(88 208 98 218)(89 207 99 217)(90 206 100 216)(121 237 131 227)(122 236 132 226)(123 235 133 225)(124 234 134 224)(125 233 135 223)(126 232 136 222)(127 231 137 221)(128 230 138 240)(129 229 139 239)(130 228 140 238)(141 165 151 175)(142 164 152 174)(143 163 153 173)(144 162 154 172)(145 161 155 171)(146 180 156 170)(147 179 157 169)(148 178 158 168)(149 177 159 167)(150 176 160 166)(181 281 191 291)(182 300 192 290)(183 299 193 289)(184 298 194 288)(185 297 195 287)(186 296 196 286)(187 295 197 285)(188 294 198 284)(189 293 199 283)(190 292 200 282)
(1 59 189 156)(2 60 190 157)(3 41 191 158)(4 42 192 159)(5 43 193 160)(6 44 194 141)(7 45 195 142)(8 46 196 143)(9 47 197 144)(10 48 198 145)(11 49 199 146)(12 50 200 147)(13 51 181 148)(14 52 182 149)(15 53 183 150)(16 54 184 151)(17 55 185 152)(18 56 186 153)(19 57 187 154)(20 58 188 155)(21 240 88 77)(22 221 89 78)(23 222 90 79)(24 223 91 80)(25 224 92 61)(26 225 93 62)(27 226 94 63)(28 227 95 64)(29 228 96 65)(30 229 97 66)(31 230 98 67)(32 231 99 68)(33 232 100 69)(34 233 81 70)(35 234 82 71)(36 235 83 72)(37 236 84 73)(38 237 85 74)(39 238 86 75)(40 239 87 76)(101 314 135 205)(102 315 136 206)(103 316 137 207)(104 317 138 208)(105 318 139 209)(106 319 140 210)(107 320 121 211)(108 301 122 212)(109 302 123 213)(110 303 124 214)(111 304 125 215)(112 305 126 216)(113 306 127 217)(114 307 128 218)(115 308 129 219)(116 309 130 220)(117 310 131 201)(118 311 132 202)(119 312 133 203)(120 313 134 204)(161 248 280 294)(162 249 261 295)(163 250 262 296)(164 251 263 297)(165 252 264 298)(166 253 265 299)(167 254 266 300)(168 255 267 281)(169 256 268 282)(170 257 269 283)(171 258 270 284)(172 259 271 285)(173 260 272 286)(174 241 273 287)(175 242 274 288)(176 243 275 289)(177 244 276 290)(178 245 277 291)(179 246 278 292)(180 247 279 293)
(1 114 189 128)(2 115 190 129)(3 116 191 130)(4 117 192 131)(5 118 193 132)(6 119 194 133)(7 120 195 134)(8 101 196 135)(9 102 197 136)(10 103 198 137)(11 104 199 138)(12 105 200 139)(13 106 181 140)(14 107 182 121)(15 108 183 122)(16 109 184 123)(17 110 185 124)(18 111 186 125)(19 112 187 126)(20 113 188 127)(21 269 88 170)(22 270 89 171)(23 271 90 172)(24 272 91 173)(25 273 92 174)(26 274 93 175)(27 275 94 176)(28 276 95 177)(29 277 96 178)(30 278 97 179)(31 279 98 180)(32 280 99 161)(33 261 100 162)(34 262 81 163)(35 263 82 164)(36 264 83 165)(37 265 84 166)(38 266 85 167)(39 267 86 168)(40 268 87 169)(41 220 158 309)(42 201 159 310)(43 202 160 311)(44 203 141 312)(45 204 142 313)(46 205 143 314)(47 206 144 315)(48 207 145 316)(49 208 146 317)(50 209 147 318)(51 210 148 319)(52 211 149 320)(53 212 150 301)(54 213 151 302)(55 214 152 303)(56 215 153 304)(57 216 154 305)(58 217 155 306)(59 218 156 307)(60 219 157 308)(61 287 224 241)(62 288 225 242)(63 289 226 243)(64 290 227 244)(65 291 228 245)(66 292 229 246)(67 293 230 247)(68 294 231 248)(69 295 232 249)(70 296 233 250)(71 297 234 251)(72 298 235 252)(73 299 236 253)(74 300 237 254)(75 281 238 255)(76 282 239 256)(77 283 240 257)(78 284 221 258)(79 285 222 259)(80 286 223 260)

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4H 4I 4J 4K 4L 4M 4N 4O 4P ··· 4U 5A 5B 10A ··· 10F 20A ··· 20H 20I ··· 20AF order 1 2 2 2 4 ··· 4 4 4 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 ··· 2 4 4 4 10 10 10 10 20 ··· 20 2 2 2 ··· 2 2 ··· 2 4 ··· 4

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + - + + + + - - - image C1 C2 C2 C2 C2 C2 C2 C2 Q8 D5 C4○D4 D10 D10 D10 C4○D20 2- 1+4 Q8×D5 D4.10D10 kernel Dic10⋊10Q8 C4×Dic10 C20⋊2Q8 C20.6Q8 Dic5⋊3Q8 Dic5.Q8 Dic5⋊Q8 Q8×C20 Dic10 C4×Q8 C20 C42 C4⋊C4 C2×Q8 C4 C10 C4 C2 # reps 1 3 1 2 2 4 2 1 4 2 4 6 6 2 16 1 4 4

Matrix representation of Dic1010Q8 in GL4(𝔽41) generated by

 25 2 0 0 39 13 0 0 0 0 1 0 0 0 0 1
,
 23 35 0 0 20 18 0 0 0 0 1 0 0 0 0 1
,
 18 6 0 0 35 23 0 0 0 0 32 23 0 0 0 9
,
 40 0 0 0 0 40 0 0 0 0 7 13 0 0 34 34
G:=sub<GL(4,GF(41))| [25,39,0,0,2,13,0,0,0,0,1,0,0,0,0,1],[23,20,0,0,35,18,0,0,0,0,1,0,0,0,0,1],[18,35,0,0,6,23,0,0,0,0,32,0,0,0,23,9],[40,0,0,0,0,40,0,0,0,0,7,34,0,0,13,34] >;

Dic1010Q8 in GAP, Magma, Sage, TeX

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

G:=Group("Dic10:10Q8");
// GroupNames label

G:=SmallGroup(320,1239);
// by ID

G=gap.SmallGroup(320,1239);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,100,185,192,12550]);
// Polycyclic

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

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