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G = Dic108Q8order 320 = 26·5

6th semidirect product of Dic10 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic108Q8, C42.169D10, C10.332- 1+4, C4⋊Q8.14D5, C4.17(Q8×D5), C55(Q83Q8), C20.52(C2×Q8), C4⋊C4.121D10, (C2×C20).98C23, (C2×Q8).141D10, (Q8×Dic5).13C2, Dic5.26(C2×Q8), C20.135(C4○D4), C4.18(D42D5), C10.44(C22×Q8), (C2×C10).265C24, (C4×C20).206C22, (C4×Dic10).25C2, C4.Dic10.15C2, Dic53Q8.12C2, Dic5⋊Q8.10C2, C4⋊Dic5.382C22, Dic5.Q8.4C2, (Q8×C10).132C22, C22.286(C23×D5), (C2×Dic5).280C23, (C4×Dic5).165C22, C10.D4.57C22, C2.34(Q8.10D10), (C2×Dic10).310C22, C2.27(C2×Q8×D5), (C5×C4⋊Q8).14C2, C10.99(C2×C4○D4), C2.63(C2×D42D5), (C2×C4).90(C22×D5), (C5×C4⋊C4).208C22, SmallGroup(320,1393)

Series: Derived Chief Lower central Upper central

C1C2×C10 — Dic108Q8
C1C5C10C2×C10C2×Dic5C4×Dic5Dic53Q8 — Dic108Q8
C5C2×C10 — Dic108Q8
C1C22C4⋊Q8

Generators and relations for Dic108Q8
 G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, ac=ca, dad-1=a11, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 534 in 200 conjugacy classes, 107 normal (27 characteristic)
C1, C2 [×3], C4 [×4], C4 [×15], C22, C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×10], C10 [×3], C42, C42 [×8], C4⋊C4 [×4], C4⋊C4 [×18], C2×Q8 [×2], C2×Q8 [×2], Dic5 [×4], Dic5 [×6], C20 [×4], C20 [×5], C2×C10, C4×Q8 [×6], C42.C2 [×6], C4⋊Q8, C4⋊Q8 [×2], Dic10 [×4], Dic10 [×2], C2×Dic5 [×8], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×4], Q83Q8, C4×Dic5 [×8], C10.D4 [×12], C4⋊Dic5 [×2], C4⋊Dic5 [×4], C4×C20, C5×C4⋊C4 [×4], C2×Dic10 [×2], Q8×C10 [×2], C4×Dic10 [×2], Dic53Q8 [×2], Dic5.Q8 [×4], C4.Dic10 [×2], Dic5⋊Q8 [×2], Q8×Dic5 [×2], C5×C4⋊Q8, Dic108Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22×Q8, C2×C4○D4, 2- 1+4, C22×D5 [×7], Q83Q8, D42D5 [×2], Q8×D5 [×2], C23×D5, C2×D42D5, C2×Q8×D5, Q8.10D10, Dic108Q8

Smallest permutation representation of Dic108Q8
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)
(1 246 11 256)(2 245 12 255)(3 244 13 254)(4 243 14 253)(5 242 15 252)(6 241 16 251)(7 260 17 250)(8 259 18 249)(9 258 19 248)(10 257 20 247)(21 172 31 162)(22 171 32 161)(23 170 33 180)(24 169 34 179)(25 168 35 178)(26 167 36 177)(27 166 37 176)(28 165 38 175)(29 164 39 174)(30 163 40 173)(41 189 51 199)(42 188 52 198)(43 187 53 197)(44 186 54 196)(45 185 55 195)(46 184 56 194)(47 183 57 193)(48 182 58 192)(49 181 59 191)(50 200 60 190)(61 100 71 90)(62 99 72 89)(63 98 73 88)(64 97 74 87)(65 96 75 86)(66 95 76 85)(67 94 77 84)(68 93 78 83)(69 92 79 82)(70 91 80 81)(101 270 111 280)(102 269 112 279)(103 268 113 278)(104 267 114 277)(105 266 115 276)(106 265 116 275)(107 264 117 274)(108 263 118 273)(109 262 119 272)(110 261 120 271)(121 203 131 213)(122 202 132 212)(123 201 133 211)(124 220 134 210)(125 219 135 209)(126 218 136 208)(127 217 137 207)(128 216 138 206)(129 215 139 205)(130 214 140 204)(141 289 151 299)(142 288 152 298)(143 287 153 297)(144 286 154 296)(145 285 155 295)(146 284 156 294)(147 283 157 293)(148 282 158 292)(149 281 159 291)(150 300 160 290)(221 319 231 309)(222 318 232 308)(223 317 233 307)(224 316 234 306)(225 315 235 305)(226 314 236 304)(227 313 237 303)(228 312 238 302)(229 311 239 301)(230 310 240 320)
(1 177 116 91)(2 178 117 92)(3 179 118 93)(4 180 119 94)(5 161 120 95)(6 162 101 96)(7 163 102 97)(8 164 103 98)(9 165 104 99)(10 166 105 100)(11 167 106 81)(12 168 107 82)(13 169 108 83)(14 170 109 84)(15 171 110 85)(16 172 111 86)(17 173 112 87)(18 174 113 88)(19 175 114 89)(20 176 115 90)(21 270 75 241)(22 271 76 242)(23 272 77 243)(24 273 78 244)(25 274 79 245)(26 275 80 246)(27 276 61 247)(28 277 62 248)(29 278 63 249)(30 279 64 250)(31 280 65 251)(32 261 66 252)(33 262 67 253)(34 263 68 254)(35 264 69 255)(36 265 70 256)(37 266 71 257)(38 267 72 258)(39 268 73 259)(40 269 74 260)(41 231 217 157)(42 232 218 158)(43 233 219 159)(44 234 220 160)(45 235 201 141)(46 236 202 142)(47 237 203 143)(48 238 204 144)(49 239 205 145)(50 240 206 146)(51 221 207 147)(52 222 208 148)(53 223 209 149)(54 224 210 150)(55 225 211 151)(56 226 212 152)(57 227 213 153)(58 228 214 154)(59 229 215 155)(60 230 216 156)(121 297 193 313)(122 298 194 314)(123 299 195 315)(124 300 196 316)(125 281 197 317)(126 282 198 318)(127 283 199 319)(128 284 200 320)(129 285 181 301)(130 286 182 302)(131 287 183 303)(132 288 184 304)(133 289 185 305)(134 290 186 306)(135 291 187 307)(136 292 188 308)(137 293 189 309)(138 294 190 310)(139 295 191 311)(140 296 192 312)
(1 43 116 219)(2 54 117 210)(3 45 118 201)(4 56 119 212)(5 47 120 203)(6 58 101 214)(7 49 102 205)(8 60 103 216)(9 51 104 207)(10 42 105 218)(11 53 106 209)(12 44 107 220)(13 55 108 211)(14 46 109 202)(15 57 110 213)(16 48 111 204)(17 59 112 215)(18 50 113 206)(19 41 114 217)(20 52 115 208)(21 296 75 312)(22 287 76 303)(23 298 77 314)(24 289 78 305)(25 300 79 316)(26 291 80 307)(27 282 61 318)(28 293 62 309)(29 284 63 320)(30 295 64 311)(31 286 65 302)(32 297 66 313)(33 288 67 304)(34 299 68 315)(35 290 69 306)(36 281 70 317)(37 292 71 308)(38 283 72 319)(39 294 73 310)(40 285 74 301)(81 223 167 149)(82 234 168 160)(83 225 169 151)(84 236 170 142)(85 227 171 153)(86 238 172 144)(87 229 173 155)(88 240 174 146)(89 231 175 157)(90 222 176 148)(91 233 177 159)(92 224 178 150)(93 235 179 141)(94 226 180 152)(95 237 161 143)(96 228 162 154)(97 239 163 145)(98 230 164 156)(99 221 165 147)(100 232 166 158)(121 252 193 261)(122 243 194 272)(123 254 195 263)(124 245 196 274)(125 256 197 265)(126 247 198 276)(127 258 199 267)(128 249 200 278)(129 260 181 269)(130 251 182 280)(131 242 183 271)(132 253 184 262)(133 244 185 273)(134 255 186 264)(135 246 187 275)(136 257 188 266)(137 248 189 277)(138 259 190 268)(139 250 191 279)(140 241 192 270)

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

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

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

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

dim111111112222224444
type++++++++-++++---
imageC1C2C2C2C2C2C2C2Q8D5C4○D4D10D10D102- 1+4D42D5Q8×D5Q8.10D10
kernelDic108Q8C4×Dic10Dic53Q8Dic5.Q8C4.Dic10Dic5⋊Q8Q8×Dic5C5×C4⋊Q8Dic10C4⋊Q8C20C42C4⋊C4C2×Q8C10C4C4C2
# reps122422214242841444

Matrix representation of Dic108Q8 in GL6(𝔽41)

1040000
26310000
000100
00403400
0000400
0000040
,
24150000
8170000
00213900
00152000
000010
000001
,
100000
010000
001000
000100
00002236
00001519
,
11120000
31300000
0040000
0004000
0000177
00002324

G:=sub<GL(6,GF(41))| [10,26,0,0,0,0,4,31,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[24,8,0,0,0,0,15,17,0,0,0,0,0,0,21,15,0,0,0,0,39,20,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,15,0,0,0,0,36,19],[11,31,0,0,0,0,12,30,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,17,23,0,0,0,0,7,24] >;

Dic108Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_8Q_8
% in TeX

G:=Group("Dic10:8Q8");
// GroupNames label

G:=SmallGroup(320,1393);
// by ID

G=gap.SmallGroup(320,1393);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,219,268,1571,297,136,12550]);
// 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,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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