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G = Dic109Q8order 320 = 26·5

7th semidirect product of Dic10 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic109Q8, C42.170D10, C10.812+ 1+4, C52Q82, C4⋊Q8.15D5, C4.18(Q8×D5), C20⋊Q8.14C2, C20.53(C2×Q8), C4⋊C4.215D10, (C2×Q8).85D10, (C2×C20).99C23, Dic5.27(C2×Q8), C10.45(C22×Q8), (C4×C20).207C22, (C2×C10).266C24, (C4×Dic10).26C2, C2.85(D46D10), Dic5⋊Q8.11C2, Dic53Q8.13C2, C4⋊Dic5.383C22, (Q8×C10).133C22, C22.287(C23×D5), (C2×Dic5).281C23, (C4×Dic5).166C22, C10.D4.58C22, (C2×Dic10).192C22, C2.28(C2×Q8×D5), (C5×C4⋊Q8).15C2, (C2×C4).91(C22×D5), (C5×C4⋊C4).209C22, SmallGroup(320,1394)

Series: Derived Chief Lower central Upper central

C1C2×C10 — Dic109Q8
C1C5C10C2×C10C2×Dic5C2×Dic10C4×Dic10 — Dic109Q8
C5C2×C10 — Dic109Q8
C1C22C4⋊Q8

Generators and relations for Dic109Q8
 G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, cac-1=a11, ad=da, bc=cb, dbd-1=a10b, dcd-1=c-1 >

Subgroups: 630 in 212 conjugacy classes, 115 normal (13 characteristic)
C1, C2, C2 [×2], C4 [×4], C4 [×17], C22, C5, C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×14], C10, C10 [×2], C42, C42 [×8], C4⋊C4 [×4], C4⋊C4 [×14], C2×Q8 [×2], C2×Q8 [×6], Dic5 [×8], Dic5 [×4], C20 [×4], C20 [×5], C2×C10, C4×Q8 [×6], C4⋊Q8, C4⋊Q8 [×8], Dic10 [×8], Dic10 [×4], C2×Dic5 [×8], C2×C20, C2×C20 [×6], C5×Q8 [×2], Q82, C4×Dic5 [×8], C10.D4 [×12], C4⋊Dic5 [×2], C4×C20, C5×C4⋊C4 [×4], C2×Dic10 [×6], Q8×C10 [×2], C4×Dic10 [×2], Dic53Q8 [×4], C20⋊Q8 [×4], Dic5⋊Q8 [×4], C5×C4⋊Q8, Dic109Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×8], C23 [×15], D5, C2×Q8 [×12], C24, D10 [×7], C22×Q8 [×2], 2+ 1+4, C22×D5 [×7], Q82, Q8×D5 [×4], C23×D5, D46D10, C2×Q8×D5 [×2], Dic109Q8

Smallest permutation representation of Dic109Q8
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)
(1 136 11 126)(2 135 12 125)(3 134 13 124)(4 133 14 123)(5 132 15 122)(6 131 16 121)(7 130 17 140)(8 129 18 139)(9 128 19 138)(10 127 20 137)(21 223 31 233)(22 222 32 232)(23 221 33 231)(24 240 34 230)(25 239 35 229)(26 238 36 228)(27 237 37 227)(28 236 38 226)(29 235 39 225)(30 234 40 224)(41 106 51 116)(42 105 52 115)(43 104 53 114)(44 103 54 113)(45 102 55 112)(46 101 56 111)(47 120 57 110)(48 119 58 109)(49 118 59 108)(50 117 60 107)(61 190 71 200)(62 189 72 199)(63 188 73 198)(64 187 74 197)(65 186 75 196)(66 185 76 195)(67 184 77 194)(68 183 78 193)(69 182 79 192)(70 181 80 191)(81 259 91 249)(82 258 92 248)(83 257 93 247)(84 256 94 246)(85 255 95 245)(86 254 96 244)(87 253 97 243)(88 252 98 242)(89 251 99 241)(90 250 100 260)(141 310 151 320)(142 309 152 319)(143 308 153 318)(144 307 154 317)(145 306 155 316)(146 305 156 315)(147 304 157 314)(148 303 158 313)(149 302 159 312)(150 301 160 311)(161 262 171 272)(162 261 172 271)(163 280 173 270)(164 279 174 269)(165 278 175 268)(166 277 176 267)(167 276 177 266)(168 275 178 265)(169 274 179 264)(170 273 180 263)(201 286 211 296)(202 285 212 295)(203 284 213 294)(204 283 214 293)(205 282 215 292)(206 281 216 291)(207 300 217 290)(208 299 218 289)(209 298 219 288)(210 297 220 287)
(1 55 272 95)(2 46 273 86)(3 57 274 97)(4 48 275 88)(5 59 276 99)(6 50 277 90)(7 41 278 81)(8 52 279 92)(9 43 280 83)(10 54 261 94)(11 45 262 85)(12 56 263 96)(13 47 264 87)(14 58 265 98)(15 49 266 89)(16 60 267 100)(17 51 268 91)(18 42 269 82)(19 53 270 93)(20 44 271 84)(21 63 208 145)(22 74 209 156)(23 65 210 147)(24 76 211 158)(25 67 212 149)(26 78 213 160)(27 69 214 151)(28 80 215 142)(29 71 216 153)(30 62 217 144)(31 73 218 155)(32 64 219 146)(33 75 220 157)(34 66 201 148)(35 77 202 159)(36 68 203 150)(37 79 204 141)(38 70 205 152)(39 61 206 143)(40 72 207 154)(101 180 254 135)(102 171 255 126)(103 162 256 137)(104 173 257 128)(105 164 258 139)(106 175 259 130)(107 166 260 121)(108 177 241 132)(109 168 242 123)(110 179 243 134)(111 170 244 125)(112 161 245 136)(113 172 246 127)(114 163 247 138)(115 174 248 129)(116 165 249 140)(117 176 250 131)(118 167 251 122)(119 178 252 133)(120 169 253 124)(181 282 319 226)(182 293 320 237)(183 284 301 228)(184 295 302 239)(185 286 303 230)(186 297 304 221)(187 288 305 232)(188 299 306 223)(189 290 307 234)(190 281 308 225)(191 292 309 236)(192 283 310 227)(193 294 311 238)(194 285 312 229)(195 296 313 240)(196 287 314 231)(197 298 315 222)(198 289 316 233)(199 300 317 224)(200 291 318 235)
(1 66 272 148)(2 67 273 149)(3 68 274 150)(4 69 275 151)(5 70 276 152)(6 71 277 153)(7 72 278 154)(8 73 279 155)(9 74 280 156)(10 75 261 157)(11 76 262 158)(12 77 263 159)(13 78 264 160)(14 79 265 141)(15 80 266 142)(16 61 267 143)(17 62 268 144)(18 63 269 145)(19 64 270 146)(20 65 271 147)(21 82 208 42)(22 83 209 43)(23 84 210 44)(24 85 211 45)(25 86 212 46)(26 87 213 47)(27 88 214 48)(28 89 215 49)(29 90 216 50)(30 91 217 51)(31 92 218 52)(32 93 219 53)(33 94 220 54)(34 95 201 55)(35 96 202 56)(36 97 203 57)(37 98 204 58)(38 99 205 59)(39 100 206 60)(40 81 207 41)(101 229 254 285)(102 230 255 286)(103 231 256 287)(104 232 257 288)(105 233 258 289)(106 234 259 290)(107 235 260 291)(108 236 241 292)(109 237 242 293)(110 238 243 294)(111 239 244 295)(112 240 245 296)(113 221 246 297)(114 222 247 298)(115 223 248 299)(116 224 249 300)(117 225 250 281)(118 226 251 282)(119 227 252 283)(120 228 253 284)(121 200 166 318)(122 181 167 319)(123 182 168 320)(124 183 169 301)(125 184 170 302)(126 185 171 303)(127 186 172 304)(128 187 173 305)(129 188 174 306)(130 189 175 307)(131 190 176 308)(132 191 177 309)(133 192 178 310)(134 193 179 311)(135 194 180 312)(136 195 161 313)(137 196 162 314)(138 197 163 315)(139 198 164 316)(140 199 165 317)

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

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

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

53 conjugacy classes

class 1 2A2B2C4A4B4C4D4E···4I4J···4Q4R4S4T4U5A5B10A···10F20A···20L20M···20T
order122244444···44···444445510···1020···2020···20
size111122224···410···1020202020222···24···48···8

53 irreducible representations

dim11111122222444
type++++++-+++++-
imageC1C2C2C2C2C2Q8D5D10D10D102+ 1+4Q8×D5D46D10
kernelDic109Q8C4×Dic10Dic53Q8C20⋊Q8Dic5⋊Q8C5×C4⋊Q8Dic10C4⋊Q8C42C4⋊C4C2×Q8C10C4C2
# reps12444182284184

Matrix representation of Dic109Q8 in GL6(𝔽41)

10340000
32310000
006100
0040000
000010
000001
,
1180000
9400000
006100
0063500
000010
000001
,
40230000
3210000
0040000
0004000
000001
0000400
,
3170000
9100000
001000
000100
0000111
00001140

G:=sub<GL(6,GF(41))| [10,32,0,0,0,0,34,31,0,0,0,0,0,0,6,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,9,0,0,0,0,18,40,0,0,0,0,0,0,6,6,0,0,0,0,1,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,32,0,0,0,0,23,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[31,9,0,0,0,0,7,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,11,0,0,0,0,11,40] >;

Dic109Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_9Q_8
% in TeX

G:=Group("Dic10:9Q8");
// GroupNames label

G:=SmallGroup(320,1394);
// by ID

G=gap.SmallGroup(320,1394);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,100,570,185,192,12550]);
// 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,c*a*c^-1=a^11,a*d=d*a,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations

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