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G = Dic10.Q8order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic10.1Q8, C4.2(Q8×D5), C54(Q8.Q8), C4⋊C4.34D10, C4.Q8.6D5, C20.11(C2×Q8), (C2×C8).136D10, C10.52(C4○D8), C4.71(C4○D20), (C2×Dic5).47D4, C10.Q16.6C2, C10.D8.5C2, C22.212(D4×D5), C4.Dic10.6C2, C20.167(C4○D4), (C2×C20).273C23, (C2×C40).283C22, Dic53Q8.6C2, C10.36(C22⋊Q8), C20.8Q8.14C2, C2.13(D10⋊Q8), C20.44D4.14C2, C2.22(SD16⋊D5), C10.40(C8.C22), C4⋊Dic5.105C22, (C4×Dic5).36C22, C2.20(SD163D5), (C2×Dic10).85C22, (C5×C4.Q8).10C2, (C2×C10).278(C2×D4), (C5×C4⋊C4).66C22, (C2×C52C8).54C22, (C2×C4).376(C22×D5), SmallGroup(320,484)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic10.Q8
C1C5C10C20C2×C20C4×Dic5Dic53Q8 — Dic10.Q8
C5C10C2×C20 — Dic10.Q8
C1C22C2×C4C4.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 [×3], C4 [×2], C4 [×7], C22, C5, C8 [×2], C2×C4, C2×C4 [×6], Q8 [×3], C10 [×3], C42 [×2], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8, C2×C8, C2×Q8, Dic5 [×5], C20 [×2], C20 [×2], C2×C10, Q8⋊C4 [×2], C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C52C8, C40, Dic10 [×2], Dic10, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], Q8.Q8, C2×C52C8, C4×Dic5, C4×Dic5, C10.D4 [×2], C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C10.D8, C10.Q16, C20.8Q8, C20.44D4, C5×C4.Q8, Dic53Q8, C4.Dic10, Dic10.Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C4○D8, C8.C22, 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)
(1 146 11 156)(2 145 12 155)(3 144 13 154)(4 143 14 153)(5 142 15 152)(6 141 16 151)(7 160 17 150)(8 159 18 149)(9 158 19 148)(10 157 20 147)(21 87 31 97)(22 86 32 96)(23 85 33 95)(24 84 34 94)(25 83 35 93)(26 82 36 92)(27 81 37 91)(28 100 38 90)(29 99 39 89)(30 98 40 88)(41 71 51 61)(42 70 52 80)(43 69 53 79)(44 68 54 78)(45 67 55 77)(46 66 56 76)(47 65 57 75)(48 64 58 74)(49 63 59 73)(50 62 60 72)(101 171 111 161)(102 170 112 180)(103 169 113 179)(104 168 114 178)(105 167 115 177)(106 166 116 176)(107 165 117 175)(108 164 118 174)(109 163 119 173)(110 162 120 172)(121 259 131 249)(122 258 132 248)(123 257 133 247)(124 256 134 246)(125 255 135 245)(126 254 136 244)(127 253 137 243)(128 252 138 242)(129 251 139 241)(130 250 140 260)(181 216 191 206)(182 215 192 205)(183 214 193 204)(184 213 194 203)(185 212 195 202)(186 211 196 201)(187 210 197 220)(188 209 198 219)(189 208 199 218)(190 207 200 217)(221 275 231 265)(222 274 232 264)(223 273 233 263)(224 272 234 262)(225 271 235 261)(226 270 236 280)(227 269 237 279)(228 268 238 278)(229 267 239 277)(230 266 240 276)(281 302 291 312)(282 301 292 311)(283 320 293 310)(284 319 294 309)(285 318 295 308)(286 317 296 307)(287 316 297 306)(288 315 298 305)(289 314 299 304)(290 313 300 303)
(1 31 253 313)(2 22 254 304)(3 33 255 315)(4 24 256 306)(5 35 257 317)(6 26 258 308)(7 37 259 319)(8 28 260 310)(9 39 241 301)(10 30 242 312)(11 21 243 303)(12 32 244 314)(13 23 245 305)(14 34 246 316)(15 25 247 307)(16 36 248 318)(17 27 249 309)(18 38 250 320)(19 29 251 311)(20 40 252 302)(41 192 115 273)(42 183 116 264)(43 194 117 275)(44 185 118 266)(45 196 119 277)(46 187 120 268)(47 198 101 279)(48 189 102 270)(49 200 103 261)(50 191 104 272)(51 182 105 263)(52 193 106 274)(53 184 107 265)(54 195 108 276)(55 186 109 267)(56 197 110 278)(57 188 111 269)(58 199 112 280)(59 190 113 271)(60 181 114 262)(61 220 167 228)(62 211 168 239)(63 202 169 230)(64 213 170 221)(65 204 171 232)(66 215 172 223)(67 206 173 234)(68 217 174 225)(69 208 175 236)(70 219 176 227)(71 210 177 238)(72 201 178 229)(73 212 179 240)(74 203 180 231)(75 214 161 222)(76 205 162 233)(77 216 163 224)(78 207 164 235)(79 218 165 226)(80 209 166 237)(81 126 284 155)(82 137 285 146)(83 128 286 157)(84 139 287 148)(85 130 288 159)(86 121 289 150)(87 132 290 141)(88 123 291 152)(89 134 292 143)(90 125 293 154)(91 136 294 145)(92 127 295 156)(93 138 296 147)(94 129 297 158)(95 140 298 149)(96 131 299 160)(97 122 300 151)(98 133 281 142)(99 124 282 153)(100 135 283 144)
(1 216 253 224)(2 205 254 233)(3 214 255 222)(4 203 256 231)(5 212 257 240)(6 201 258 229)(7 210 259 238)(8 219 260 227)(9 208 241 236)(10 217 242 225)(11 206 243 234)(12 215 244 223)(13 204 245 232)(14 213 246 221)(15 202 247 230)(16 211 248 239)(17 220 249 228)(18 209 250 237)(19 218 251 226)(20 207 252 235)(21 77 303 163)(22 66 304 172)(23 75 305 161)(24 64 306 170)(25 73 307 179)(26 62 308 168)(27 71 309 177)(28 80 310 166)(29 69 311 175)(30 78 312 164)(31 67 313 173)(32 76 314 162)(33 65 315 171)(34 74 316 180)(35 63 317 169)(36 72 318 178)(37 61 319 167)(38 70 320 176)(39 79 301 165)(40 68 302 174)(41 294 115 91)(42 283 116 100)(43 292 117 89)(44 281 118 98)(45 290 119 87)(46 299 120 96)(47 288 101 85)(48 297 102 94)(49 286 103 83)(50 295 104 92)(51 284 105 81)(52 293 106 90)(53 282 107 99)(54 291 108 88)(55 300 109 97)(56 289 110 86)(57 298 111 95)(58 287 112 84)(59 296 113 93)(60 285 114 82)(121 268 150 187)(122 277 151 196)(123 266 152 185)(124 275 153 194)(125 264 154 183)(126 273 155 192)(127 262 156 181)(128 271 157 190)(129 280 158 199)(130 269 159 188)(131 278 160 197)(132 267 141 186)(133 276 142 195)(134 265 143 184)(135 274 144 193)(136 263 145 182)(137 272 146 191)(138 261 147 200)(139 270 148 189)(140 279 149 198)

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

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

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

47 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12224444444444455888810···102020202020···2040···40
size111122448101020202040224420202···244448···84···4

47 irreducible representations

dim111111112222222244444
type++++++++-++++--+-
imageC1C2C2C2C2C2C2C2Q8D4D5C4○D4D10D10C4○D8C4○D20C8.C22Q8×D5D4×D5SD16⋊D5SD163D5
kernelDic10.Q8C10.D8C10.Q16C20.8Q8C20.44D4C5×C4.Q8Dic53Q8C4.Dic10Dic10C2×Dic5C4.Q8C20C4⋊C4C2×C8C10C4C10C4C22C2C2
# reps111111112222424812244

Matrix representation of Dic10.Q8 in GL6(𝔽41)

0400000
100000
0040000
0004000
0000401
0000535
,
14340000
34270000
00332400
0023800
0000230
0000439
,
620000
2350000
00404000
002100
000010
000001
,
3200000
0320000
00313000
00391000
00003911
0000372

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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