Dic5.D8 - GroupNames

G = Dic₅.D₈ order 320 = 2⁶·5

1^st 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

Derived series

C₁ — C₂×C₂₀ — Dic₅.D₈

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×Dic₅ — C₄×Dic₅ — Dic₅⋊C₈ — Dic₅.D₈

Lower central

C₅ — C₂×C₁₀ — C₂×C₂₀ — Dic₅.D₈

Upper central

C₁ — C₂² — C₂×C₄ — C₄⋊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, C₄, C₂², C₅, C₈, C₂×C₄, C₂×C₄, Q₈, C₁₀, C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×Q₈, Dic₅, Dic₅, C₂₀, C₂×C₁₀, C₄⋊C₈, C₄⋊Q₈, C₅⋊C₈, Dic₁₀, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₄.₁₀D₈, C₄×Dic₅, C₁₀.D₄, C₄⋊Dic₅, C₅×C₄⋊C₄, C₂×C₅⋊C₈, C₂×Dic₁₀, Dic₅⋊C₈, C₂₀⋊Q₈, Dic₅.D₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂²⋊C₄, D₈, SD₁₆, 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	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	4G	5	8A	8B	8C	8D	8E	8F	8G	8H	10A	10B	10C	20A	20B	20C	20D	20E	20F
size	1	1	1	1	4	8	10	10	10	10	40	4	20	20	20	20	20	20	20	20	4	4	4	8	8	8	8	8	8
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	-1	1	1	1	1	-1	1	1	-1	-1	-1	-1	1	1	1	1	1	1	-1	1	1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	1	1	-1	1	1	1	1	-1	1	-1	1	1	1	1	-1	-1	-1	1	1	1	-1	1	1	-1	-1	-1	linear of order 2
ρ₄	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-1	-1	1	1	1	1	1	1	1	1	1	linear of order 2
ρ₅	1	1	1	1	1	-1	-1	-1	-1	-1	1	1	-i	-i	-i	i	i	i	i	-i	1	1	1	-1	1	1	-1	-1	-1	linear of order 4
ρ₆	1	1	1	1	1	1	-1	-1	-1	-1	-1	1	-i	i	i	-i	-i	i	i	-i	1	1	1	1	1	1	1	1	1	linear of order 4
ρ₇	1	1	1	1	1	1	-1	-1	-1	-1	-1	1	i	-i	-i	i	i	-i	-i	i	1	1	1	1	1	1	1	1	1	linear of order 4
ρ₈	1	1	1	1	1	-1	-1	-1	-1	-1	1	1	i	i	i	-i	-i	-i	-i	i	1	1	1	-1	1	1	-1	-1	-1	linear of order 4
ρ₉	2	2	2	2	-2	0	2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	2	2	2	0	-2	-2	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	-2	0	-2	-2	2	2	0	2	0	0	0	0	0	0	0	0	2	2	2	0	-2	-2	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	-2	2	-2	0	0	0	0	2	-2	0	2	-√2	0	0	0	0	√2	-√2	√2	2	-2	-2	0	0	0	0	0	0	orthogonal lifted from D₈
ρ₁₂	2	-2	2	-2	0	0	0	0	2	-2	0	2	√2	0	0	0	0	-√2	√2	-√2	2	-2	-2	0	0	0	0	0	0	orthogonal lifted from D₈
ρ₁₃	2	2	-2	-2	0	0	2	-2	0	0	0	2	0	√2	-√2	-√2	√2	0	0	0	-2	-2	2	0	0	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₄	2	2	-2	-2	0	0	2	-2	0	0	0	2	0	-√2	√2	√2	-√2	0	0	0	-2	-2	2	0	0	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₅	2	2	-2	-2	0	0	-2	2	0	0	0	2	0	√-2	-√-2	√-2	-√-2	0	0	0	-2	-2	2	0	0	0	0	0	0	complex lifted from SD₁₆
ρ₁₆	2	2	-2	-2	0	0	-2	2	0	0	0	2	0	-√-2	√-2	-√-2	√-2	0	0	0	-2	-2	2	0	0	0	0	0	0	complex lifted from SD₁₆
ρ₁₇	2	-2	2	-2	0	0	0	0	-2	2	0	2	√-2	0	0	0	0	√-2	-√-2	-√-2	2	-2	-2	0	0	0	0	0	0	complex lifted from SD₁₆
ρ₁₈	2	-2	2	-2	0	0	0	0	-2	2	0	2	-√-2	0	0	0	0	-√-2	√-2	√-2	2	-2	-2	0	0	0	0	0	0	complex lifted from SD₁₆
ρ₁₉	4	4	4	4	4	4	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	-1	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from F₅
ρ₂₀	4	4	4	4	4	-4	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	-1	-1	-1	1	-1	-1	1	1	1	orthogonal lifted from C₂×F₅
ρ₂₁	4	4	4	4	-4	0	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	-1	-1	-1	√5	1	1	√5	-√5	-√5	orthogonal lifted from C₂²⋊F₅
ρ₂₂	4	4	4	4	-4	0	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	-1	-1	-1	-√5	1	1	-√5	√5	√5	orthogonal lifted from C₂²⋊F₅
ρ₂₃	4	-4	-4	4	0	0	0	0	0	0	0	4	0	0	0	0	0	0	0	0	-4	4	-4	0	0	0	0	0	0	symplectic lifted from C₄.₁₀D₄, Schur index 2
ρ₂₄	4	-4	-4	4	0	0	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	1	-1	1	2ζ₄ζ₅²+2ζ₄ζ₅+ζ₄	-√5	√5	2ζ₄ζ₅⁴+2ζ₄ζ₅³+ζ₄	2ζ₄³ζ₅⁴+2ζ₄³ζ₅²+ζ₄³	2ζ₄³ζ₅³+2ζ₄³ζ₅+ζ₄³	symplectic lifted from Dic₅.D₄, Schur index 2
ρ₂₅	4	-4	-4	4	0	0	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	1	-1	1	2ζ₄³ζ₅⁴+2ζ₄³ζ₅²+ζ₄³	√5	-√5	2ζ₄³ζ₅³+2ζ₄³ζ₅+ζ₄³	2ζ₄ζ₅⁴+2ζ₄ζ₅³+ζ₄	2ζ₄ζ₅²+2ζ₄ζ₅+ζ₄	symplectic lifted from Dic₅.D₄, Schur index 2
ρ₂₆	4	-4	-4	4	0	0	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	1	-1	1	2ζ₄³ζ₅³+2ζ₄³ζ₅+ζ₄³	√5	-√5	2ζ₄³ζ₅⁴+2ζ₄³ζ₅²+ζ₄³	2ζ₄ζ₅²+2ζ₄ζ₅+ζ₄	2ζ₄ζ₅⁴+2ζ₄ζ₅³+ζ₄	symplectic lifted from Dic₅.D₄, Schur index 2
ρ₂₇	4	-4	-4	4	0	0	0	0	0	0	0	-1	0	0	0	0	0	0	0	0	1	-1	1	2ζ₄ζ₅⁴+2ζ₄ζ₅³+ζ₄	-√5	√5	2ζ₄ζ₅²+2ζ₄ζ₅+ζ₄	2ζ₄³ζ₅³+2ζ₄³ζ₅+ζ₄³	2ζ₄³ζ₅⁴+2ζ₄³ζ₅²+ζ₄³	symplectic lifted from Dic₅.D₄, Schur index 2
ρ₂₈	8	-8	8	-8	0	0	0	0	0	0	0	-2	0	0	0	0	0	0	0	0	-2	2	2	0	0	0	0	0	0	orthogonal lifted from D₂₀⋊C₄, Schur index 2
ρ₂₉	8	8	-8	-8	0	0	0	0	0	0	0	-2	0	0	0	0	0	0	0	0	2	2	-2	0	0	0	0	0	0	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 53 6 58)(2 52 7 57)(3 51 8 56)(4 60 9 55)(5 59 10 54)(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 72 36 77)(32 71 37 76)(33 80 38 75)(34 79 39 74)(35 78 40 73)(41 89 46 84)(42 88 47 83)(43 87 48 82)(44 86 49 81)(45 85 50 90)(91 137 96 132)(92 136 97 131)(93 135 98 140)(94 134 99 139)(95 133 100 138)(101 144 106 149)(102 143 107 148)(103 142 108 147)(104 141 109 146)(105 150 110 145)(111 157 116 152)(112 156 117 151)(113 155 118 160)(114 154 119 159)(115 153 120 158)(121 163 126 168)(122 162 127 167)(123 161 128 166)(124 170 129 165)(125 169 130 164)(171 218 176 213)(172 217 177 212)(173 216 178 211)(174 215 179 220)(175 214 180 219)(181 229 186 224)(182 228 187 223)(183 227 188 222)(184 226 189 221)(185 225 190 230)(191 238 196 233)(192 237 197 232)(193 236 198 231)(194 235 199 240)(195 234 200 239)(201 249 206 244)(202 248 207 243)(203 247 208 242)(204 246 209 241)(205 245 210 250)(251 293 256 298)(252 292 257 297)(253 291 258 296)(254 300 259 295)(255 299 260 294)(261 304 266 309)(262 303 267 308)(263 302 268 307)(264 301 269 306)(265 310 270 305)(271 313 276 318)(272 312 277 317)(273 311 278 316)(274 320 279 315)(275 319 280 314)

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

(1 219 53 175 6 214 58 180)(2 216 52 178 7 211 57 173)(3 213 51 171 8 218 56 176)(4 220 60 174 9 215 55 179)(5 217 59 177 10 212 54 172)(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)(31 236 72 198 36 231 77 193)(32 233 71 191 37 238 76 196)(33 240 80 194 38 235 75 199)(34 237 79 197 39 232 74 192)(35 234 78 200 40 239 73 195)(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)(91 291 137 258 96 296 132 253)(92 298 136 251 97 293 131 256)(93 295 135 254 98 300 140 259)(94 292 134 257 99 297 139 252)(95 299 133 260 100 294 138 255)(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)(111 311 157 278 116 316 152 273)(112 318 156 271 117 313 151 276)(113 315 155 274 118 320 160 279)(114 312 154 277 119 317 159 272)(115 319 153 280 120 314 158 275)

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

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

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

Matrix representation of Dic₅.D₈ ►in GL₆(𝔽₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	40	0	0
0	0	1	0	40	0
0	0	1	0	0	40
0	0	1	0	0	0

40	0	0	0	0	0
0	40	0	0	0	0
0	0	10	6	12	13
0	0	16	18	25	13
0	0	28	31	25	23
0	0	0	31	35	29

15	26	0	0	0	0
15	15	0	0	0	0
0	0	5	16	39	28
0	0	3	3	33	33
0	0	38	8	8	31
0	0	13	6	36	25

2	35	0	0	0	0
35	39	0	0	0	0
0	0	28	33	33	26
0	0	20	18	36	13
0	0	23	5	28	5
0	0	15	38	13	8

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

G = Dic5.D8 order 320 = 26·5

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

G = Dic₅.D₈ order 320 = 2⁶·5

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