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G = Dic5.D8order 320 = 26·5

1st non-split extension by Dic5 of D8 acting via D8/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic5.1D8, Dic5.1Q16, Dic5.3SD16, C4⋊C4.2F5, C5⋊(C4.10D8), C20⋊Q8.1C2, C4⋊Dic5.5C4, C2.5(D20⋊C4), C2.5(Q8⋊F5), (C2×Dic5).97D4, C10.3(D4⋊C4), C10.3(Q8⋊C4), Dic5⋊C8.1C2, (C4×Dic5).2C22, C10.2(C4.10D4), C22.58(C22⋊F5), C2.5(Dic5.D4), (C5×C4⋊C4).2C4, (C2×C20).8(C2×C4), (C2×C4).11(C2×F5), (C2×C10).23(C22⋊C4), SmallGroup(320,211)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic5.D8
C1C5C10C2×C10C2×Dic5C4×Dic5Dic5⋊C8 — Dic5.D8
C5C2×C10C2×C20 — Dic5.D8
C1C22C2×C4C4⋊C4

Generators and relations for Dic5.D8
 G = < a,b,c,d | a10=c8=1, b2=a5, d2=b, bab-1=a-1, cac-1=dad-1=a3, cbc-1=a5b, bd=db, dcd-1=a5bc-1 >

Subgroups: 282 in 64 conjugacy classes, 26 normal (all characteristic)
C1, C2 [×3], C4 [×7], C22, C5, C8 [×2], C2×C4, C2×C4 [×4], Q8 [×2], C10 [×3], C42, C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×Q8, Dic5 [×4], Dic5, C20 [×2], C2×C10, C4⋊C8 [×2], C4⋊Q8, C5⋊C8 [×2], Dic10 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20, C4.10D8, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×C5⋊C8 [×2], C2×Dic10, Dic5⋊C8 [×2], C20⋊Q8, Dic5.D8
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8, SD16 [×2], Q16, F5, C4.10D4, D4⋊C4, Q8⋊C4, C2×F5, C4.10D8, C22⋊F5, Dic5.D4, D20⋊C4, Q8⋊F5, Dic5.D8

Character table of Dic5.D8

 class 12A2B2C4A4B4C4D4E4F4G58A8B8C8D8E8F8G8H10A10B10C20A20B20C20D20E20F
 size 111148101010104042020202020202020444888888
ρ111111111111111111111111111111    trivial
ρ211111-11111-111-1-1-1-1111111-111-1-1-1    linear of order 2
ρ311111-11111-11-11111-1-1-1111-111-1-1-1    linear of order 2
ρ4111111111111-1-1-1-1-1-1-1-1111111111    linear of order 2
ρ511111-1-1-1-1-111-i-i-iiiii-i111-111-1-1-1    linear of order 4
ρ6111111-1-1-1-1-11-iii-i-iii-i111111111    linear of order 4
ρ7111111-1-1-1-1-11i-i-iii-i-ii111111111    linear of order 4
ρ811111-1-1-1-1-111iii-i-i-i-ii111-111-1-1-1    linear of order 4
ρ92222-2022-2-202000000002220-2-2000    orthogonal lifted from D4
ρ102222-20-2-22202000000002220-2-2000    orthogonal lifted from D4
ρ112-22-200002-202-200002-222-2-2000000    orthogonal lifted from D8
ρ122-22-200002-20220000-22-22-2-2000000    orthogonal lifted from D8
ρ1322-2-2002-2000202-2-22000-2-22000000    symplectic lifted from Q16, Schur index 2
ρ1422-2-2002-200020-222-2000-2-22000000    symplectic lifted from Q16, Schur index 2
ρ1522-2-200-2200020-2--2-2--2000-2-22000000    complex lifted from SD16
ρ1622-2-200-2200020--2-2--2-2000-2-22000000    complex lifted from SD16
ρ172-22-20000-2202-20000-2--2--22-2-2000000    complex lifted from SD16
ρ182-22-20000-2202--20000--2-2-22-2-2000000    complex lifted from SD16
ρ1944444400000-100000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ2044444-400000-100000000-1-1-11-1-1111    orthogonal lifted from C2×F5
ρ214444-4000000-100000000-1-1-15115-5-5    orthogonal lifted from C22⋊F5
ρ224444-4000000-100000000-1-1-1-511-555    orthogonal lifted from C22⋊F5
ρ234-4-440000000400000000-44-4000000    symplectic lifted from C4.10D4, Schur index 2
ρ244-4-440000000-1000000001-114ζ52+2ζ4ζ54-554ζ54+2ζ4ζ53443ζ54+2ζ43ζ524343ζ53+2ζ43ζ543    symplectic lifted from Dic5.D4, Schur index 2
ρ254-4-440000000-1000000001-1143ζ54+2ζ43ζ52435-543ζ53+2ζ43ζ5434ζ54+2ζ4ζ5344ζ52+2ζ4ζ54    symplectic lifted from Dic5.D4, Schur index 2
ρ264-4-440000000-1000000001-1143ζ53+2ζ43ζ5435-543ζ54+2ζ43ζ52434ζ52+2ζ4ζ544ζ54+2ζ4ζ534    symplectic lifted from Dic5.D4, Schur index 2
ρ274-4-440000000-1000000001-114ζ54+2ζ4ζ534-554ζ52+2ζ4ζ5443ζ53+2ζ43ζ54343ζ54+2ζ43ζ5243    symplectic lifted from Dic5.D4, Schur index 2
ρ288-88-80000000-200000000-222000000    orthogonal lifted from D20⋊C4, Schur index 2
ρ2988-8-80000000-20000000022-2000000    symplectic lifted from Q8⋊F5, Schur index 2

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

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

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

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

Matrix representation of Dic5.D8 in GL6(𝔽41)

100000
010000
0014000
0010400
0010040
001000
,
4000000
0400000
001061213
0016182513
0028312523
000313529
,
15260000
15150000
005163928
00333333
00388831
001363625
,
2350000
35390000
0028333326
0020183613
00235285
001538138

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,1,1,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,10,16,28,0,0,0,6,18,31,31,0,0,12,25,25,35,0,0,13,13,23,29],[15,15,0,0,0,0,26,15,0,0,0,0,0,0,5,3,38,13,0,0,16,3,8,6,0,0,39,33,8,36,0,0,28,33,31,25],[2,35,0,0,0,0,35,39,0,0,0,0,0,0,28,20,23,15,0,0,33,18,5,38,0,0,33,36,28,13,0,0,26,13,5,8] >;

Dic5.D8 in GAP, Magma, Sage, TeX

{\rm Dic}_5.D_8
% in TeX

G:=Group("Dic5.D8");
// GroupNames label

G:=SmallGroup(320,211);
// by ID

G=gap.SmallGroup(320,211);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,219,100,1571,570,136,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^8=1,b^2=a^5,d^2=b,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=a^5*b*c^-1>;
// generators/relations

Export

Character table of Dic5.D8 in TeX

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