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## G = Dic10.Q8order 320 = 26·5

### 1st non-split extension by Dic10 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Dic10.Q8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — Dic5⋊3Q8 — Dic10.Q8
 Lower central C5 — C10 — C2×C20 — Dic10.Q8
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for Dic10.Q8
G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, cac-1=a11, dad-1=a9, cbc-1=a15b, bd=db, dcd-1=a10c-1 >

Subgroups: 310 in 90 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C52C8, C40, Dic10, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, Q8.Q8, C2×C52C8, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×Dic10, C10.D8, C10.Q16, C20.8Q8, C20.44D4, C5×C4.Q8, Dic53Q8, C4.Dic10, Dic10.Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C4○D8, C8.C22, C22×D5, Q8.Q8, C4○D20, D4×D5, Q8×D5, D10⋊Q8, SD16⋊D5, SD163D5, Dic10.Q8

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

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H order 1 2 2 2 4 4 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 4 4 8 10 10 20 20 20 40 2 2 4 4 20 20 2 ··· 2 4 4 4 4 8 ··· 8 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + - + + + + - - + - image C1 C2 C2 C2 C2 C2 C2 C2 Q8 D4 D5 C4○D4 D10 D10 C4○D8 C4○D20 C8.C22 Q8×D5 D4×D5 SD16⋊D5 SD16⋊3D5 kernel Dic10.Q8 C10.D8 C10.Q16 C20.8Q8 C20.44D4 C5×C4.Q8 Dic5⋊3Q8 C4.Dic10 Dic10 C2×Dic5 C4.Q8 C20 C4⋊C4 C2×C8 C10 C4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 4 2 4 8 1 2 2 4 4

Matrix representation of Dic10.Q8 in GL6(𝔽41)

 0 40 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 1 0 0 0 0 5 35
,
 14 34 0 0 0 0 34 27 0 0 0 0 0 0 33 24 0 0 0 0 23 8 0 0 0 0 0 0 2 30 0 0 0 0 4 39
,
 6 2 0 0 0 0 2 35 0 0 0 0 0 0 40 40 0 0 0 0 2 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 31 30 0 0 0 0 39 10 0 0 0 0 0 0 39 11 0 0 0 0 37 2

G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,5,0,0,0,0,1,35],[14,34,0,0,0,0,34,27,0,0,0,0,0,0,33,23,0,0,0,0,24,8,0,0,0,0,0,0,2,4,0,0,0,0,30,39],[6,2,0,0,0,0,2,35,0,0,0,0,0,0,40,2,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,31,39,0,0,0,0,30,10,0,0,0,0,0,0,39,37,0,0,0,0,11,2] >;

Dic10.Q8 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,484);
// by ID

G=gap.SmallGroup(320,484);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,1094,135,100,570,297,136,12550]);
// Polycyclic

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

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