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G = D208Dic3order 480 = 25·3·5

5th semidirect product of D20 and Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D208Dic3, Dic157D4, C123(C4×D5), C1513(C4×D4), C53(D4×Dic3), C6020(C2×C4), C42(D5×Dic3), C6.40(D4×D5), (C3×D20)⋊12C4, C4⋊Dic313D5, (C6×D20).9C2, C205(C2×Dic3), C10.41(S3×D4), C30.52(C2×D4), C35(D208C4), D103(C2×Dic3), (C2×D20).10S3, (C2×C20).126D6, (C4×Dic15)⋊25C2, C30.73(C4○D4), (C2×C12).127D10, C2.1(C20⋊D6), (C22×D5).54D6, D10⋊Dic311C2, C2.5(D20⋊S3), (C2×C30).124C23, (C2×C60).200C22, C30.130(C22×C4), C6.36(Q82D5), C10.14(D42S3), (C2×Dic3).107D10, C10.26(C22×Dic3), (C10×Dic3).78C22, (C2×Dic15).206C22, C6.89(C2×C4×D5), (C6×D5)⋊4(C2×C4), (C2×D5×Dic3)⋊9C2, C2.14(C2×D5×Dic3), C22.58(C2×S3×D5), (C5×C4⋊Dic3)⋊10C2, (C2×C4).210(S3×D5), (D5×C2×C6).24C22, (C2×C6).136(C22×D5), (C2×C10).136(C22×S3), SmallGroup(480,510)

Series: Derived Chief Lower central Upper central

C1C30 — D208Dic3
C1C5C15C30C2×C30D5×C2×C6C2×D5×Dic3 — D208Dic3
C15C30 — D208Dic3
C1C22C2×C4

Generators and relations for D208Dic3
 G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, ac=ca, dad=a19, bc=cb, bd=db, dcd=c-1 >

Subgroups: 892 in 188 conjugacy classes, 70 normal (32 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×8], C5, C6 [×3], C6 [×4], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×4], C10 [×3], Dic3 [×5], C12 [×2], C2×C6, C2×C6 [×8], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×3], C20 [×2], C20 [×2], D10 [×4], D10 [×4], C2×C10, C2×Dic3 [×2], C2×Dic3 [×6], C2×C12, C3×D4 [×4], C22×C6 [×2], C3×D5 [×4], C30 [×3], C4×D4, C4×D5 [×4], D20 [×4], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C4×Dic3, C4⋊Dic3, C6.D4 [×2], C22×Dic3 [×2], C6×D4, C5×Dic3 [×2], Dic15 [×2], Dic15, C60 [×2], C6×D5 [×4], C6×D5 [×4], C2×C30, C4×Dic5, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5 [×2], C2×D20, D4×Dic3, D5×Dic3 [×4], C3×D20 [×4], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6 [×2], D208C4, D10⋊Dic3 [×2], C5×C4⋊Dic3, C4×Dic15, C2×D5×Dic3 [×2], C6×D20, D208Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D5, Dic3 [×4], D6 [×3], C22×C4, C2×D4, C4○D4, D10 [×3], C2×Dic3 [×6], C22×S3, C4×D4, C4×D5 [×2], C22×D5, S3×D4, D42S3, C22×Dic3, S3×D5, C2×C4×D5, D4×D5, Q82D5, D4×Dic3, D5×Dic3 [×2], C2×S3×D5, D208C4, D20⋊S3, C20⋊D6, C2×D5×Dic3, D208Dic3

Smallest permutation representation of D208Dic3
On 240 points
Generators in S240
(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)
(1 168 16 153)(2 167 17 152)(3 166 18 151)(4 165 19 180)(5 164 20 179)(6 163 21 178)(7 162 22 177)(8 161 23 176)(9 160 24 175)(10 159 25 174)(11 158 26 173)(12 157 27 172)(13 156 28 171)(14 155 29 170)(15 154 30 169)(31 74 46 89)(32 73 47 88)(33 72 48 87)(34 71 49 86)(35 70 50 85)(36 69 51 84)(37 68 52 83)(38 67 53 82)(39 66 54 81)(40 65 55 80)(41 64 56 79)(42 63 57 78)(43 62 58 77)(44 61 59 76)(45 90 60 75)(91 223 106 238)(92 222 107 237)(93 221 108 236)(94 220 109 235)(95 219 110 234)(96 218 111 233)(97 217 112 232)(98 216 113 231)(99 215 114 230)(100 214 115 229)(101 213 116 228)(102 212 117 227)(103 211 118 226)(104 240 119 225)(105 239 120 224)(121 205 136 190)(122 204 137 189)(123 203 138 188)(124 202 139 187)(125 201 140 186)(126 200 141 185)(127 199 142 184)(128 198 143 183)(129 197 144 182)(130 196 145 181)(131 195 146 210)(132 194 147 209)(133 193 148 208)(134 192 149 207)(135 191 150 206)
(1 129 82 115)(2 130 83 116)(3 131 84 117)(4 132 85 118)(5 133 86 119)(6 134 87 120)(7 135 88 91)(8 136 89 92)(9 137 90 93)(10 138 61 94)(11 139 62 95)(12 140 63 96)(13 141 64 97)(14 142 65 98)(15 143 66 99)(16 144 67 100)(17 145 68 101)(18 146 69 102)(19 147 70 103)(20 148 71 104)(21 149 72 105)(22 150 73 106)(23 121 74 107)(24 122 75 108)(25 123 76 109)(26 124 77 110)(27 125 78 111)(28 126 79 112)(29 127 80 113)(30 128 81 114)(31 222 161 190)(32 223 162 191)(33 224 163 192)(34 225 164 193)(35 226 165 194)(36 227 166 195)(37 228 167 196)(38 229 168 197)(39 230 169 198)(40 231 170 199)(41 232 171 200)(42 233 172 201)(43 234 173 202)(44 235 174 203)(45 236 175 204)(46 237 176 205)(47 238 177 206)(48 239 178 207)(49 240 179 208)(50 211 180 209)(51 212 151 210)(52 213 152 181)(53 214 153 182)(54 215 154 183)(55 216 155 184)(56 217 156 185)(57 218 157 186)(58 219 158 187)(59 220 159 188)(60 221 160 189)
(1 129)(2 148)(3 137)(4 126)(5 145)(6 134)(7 123)(8 142)(9 131)(10 150)(11 139)(12 128)(13 147)(14 136)(15 125)(16 144)(17 133)(18 122)(19 141)(20 130)(21 149)(22 138)(23 127)(24 146)(25 135)(26 124)(27 143)(28 132)(29 121)(30 140)(31 216)(32 235)(33 224)(34 213)(35 232)(36 221)(37 240)(38 229)(39 218)(40 237)(41 226)(42 215)(43 234)(44 223)(45 212)(46 231)(47 220)(48 239)(49 228)(50 217)(51 236)(52 225)(53 214)(54 233)(55 222)(56 211)(57 230)(58 219)(59 238)(60 227)(61 106)(62 95)(63 114)(64 103)(65 92)(66 111)(67 100)(68 119)(69 108)(70 97)(71 116)(72 105)(73 94)(74 113)(75 102)(76 91)(77 110)(78 99)(79 118)(80 107)(81 96)(82 115)(83 104)(84 93)(85 112)(86 101)(87 120)(88 109)(89 98)(90 117)(151 204)(152 193)(153 182)(154 201)(155 190)(156 209)(157 198)(158 187)(159 206)(160 195)(161 184)(162 203)(163 192)(164 181)(165 200)(166 189)(167 208)(168 197)(169 186)(170 205)(171 194)(172 183)(173 202)(174 191)(175 210)(176 199)(177 188)(178 207)(179 196)(180 185)

G:=sub<Sym(240)| (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), (1,168,16,153)(2,167,17,152)(3,166,18,151)(4,165,19,180)(5,164,20,179)(6,163,21,178)(7,162,22,177)(8,161,23,176)(9,160,24,175)(10,159,25,174)(11,158,26,173)(12,157,27,172)(13,156,28,171)(14,155,29,170)(15,154,30,169)(31,74,46,89)(32,73,47,88)(33,72,48,87)(34,71,49,86)(35,70,50,85)(36,69,51,84)(37,68,52,83)(38,67,53,82)(39,66,54,81)(40,65,55,80)(41,64,56,79)(42,63,57,78)(43,62,58,77)(44,61,59,76)(45,90,60,75)(91,223,106,238)(92,222,107,237)(93,221,108,236)(94,220,109,235)(95,219,110,234)(96,218,111,233)(97,217,112,232)(98,216,113,231)(99,215,114,230)(100,214,115,229)(101,213,116,228)(102,212,117,227)(103,211,118,226)(104,240,119,225)(105,239,120,224)(121,205,136,190)(122,204,137,189)(123,203,138,188)(124,202,139,187)(125,201,140,186)(126,200,141,185)(127,199,142,184)(128,198,143,183)(129,197,144,182)(130,196,145,181)(131,195,146,210)(132,194,147,209)(133,193,148,208)(134,192,149,207)(135,191,150,206), (1,129,82,115)(2,130,83,116)(3,131,84,117)(4,132,85,118)(5,133,86,119)(6,134,87,120)(7,135,88,91)(8,136,89,92)(9,137,90,93)(10,138,61,94)(11,139,62,95)(12,140,63,96)(13,141,64,97)(14,142,65,98)(15,143,66,99)(16,144,67,100)(17,145,68,101)(18,146,69,102)(19,147,70,103)(20,148,71,104)(21,149,72,105)(22,150,73,106)(23,121,74,107)(24,122,75,108)(25,123,76,109)(26,124,77,110)(27,125,78,111)(28,126,79,112)(29,127,80,113)(30,128,81,114)(31,222,161,190)(32,223,162,191)(33,224,163,192)(34,225,164,193)(35,226,165,194)(36,227,166,195)(37,228,167,196)(38,229,168,197)(39,230,169,198)(40,231,170,199)(41,232,171,200)(42,233,172,201)(43,234,173,202)(44,235,174,203)(45,236,175,204)(46,237,176,205)(47,238,177,206)(48,239,178,207)(49,240,179,208)(50,211,180,209)(51,212,151,210)(52,213,152,181)(53,214,153,182)(54,215,154,183)(55,216,155,184)(56,217,156,185)(57,218,157,186)(58,219,158,187)(59,220,159,188)(60,221,160,189), (1,129)(2,148)(3,137)(4,126)(5,145)(6,134)(7,123)(8,142)(9,131)(10,150)(11,139)(12,128)(13,147)(14,136)(15,125)(16,144)(17,133)(18,122)(19,141)(20,130)(21,149)(22,138)(23,127)(24,146)(25,135)(26,124)(27,143)(28,132)(29,121)(30,140)(31,216)(32,235)(33,224)(34,213)(35,232)(36,221)(37,240)(38,229)(39,218)(40,237)(41,226)(42,215)(43,234)(44,223)(45,212)(46,231)(47,220)(48,239)(49,228)(50,217)(51,236)(52,225)(53,214)(54,233)(55,222)(56,211)(57,230)(58,219)(59,238)(60,227)(61,106)(62,95)(63,114)(64,103)(65,92)(66,111)(67,100)(68,119)(69,108)(70,97)(71,116)(72,105)(73,94)(74,113)(75,102)(76,91)(77,110)(78,99)(79,118)(80,107)(81,96)(82,115)(83,104)(84,93)(85,112)(86,101)(87,120)(88,109)(89,98)(90,117)(151,204)(152,193)(153,182)(154,201)(155,190)(156,209)(157,198)(158,187)(159,206)(160,195)(161,184)(162,203)(163,192)(164,181)(165,200)(166,189)(167,208)(168,197)(169,186)(170,205)(171,194)(172,183)(173,202)(174,191)(175,210)(176,199)(177,188)(178,207)(179,196)(180,185)>;

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), (1,168,16,153)(2,167,17,152)(3,166,18,151)(4,165,19,180)(5,164,20,179)(6,163,21,178)(7,162,22,177)(8,161,23,176)(9,160,24,175)(10,159,25,174)(11,158,26,173)(12,157,27,172)(13,156,28,171)(14,155,29,170)(15,154,30,169)(31,74,46,89)(32,73,47,88)(33,72,48,87)(34,71,49,86)(35,70,50,85)(36,69,51,84)(37,68,52,83)(38,67,53,82)(39,66,54,81)(40,65,55,80)(41,64,56,79)(42,63,57,78)(43,62,58,77)(44,61,59,76)(45,90,60,75)(91,223,106,238)(92,222,107,237)(93,221,108,236)(94,220,109,235)(95,219,110,234)(96,218,111,233)(97,217,112,232)(98,216,113,231)(99,215,114,230)(100,214,115,229)(101,213,116,228)(102,212,117,227)(103,211,118,226)(104,240,119,225)(105,239,120,224)(121,205,136,190)(122,204,137,189)(123,203,138,188)(124,202,139,187)(125,201,140,186)(126,200,141,185)(127,199,142,184)(128,198,143,183)(129,197,144,182)(130,196,145,181)(131,195,146,210)(132,194,147,209)(133,193,148,208)(134,192,149,207)(135,191,150,206), (1,129,82,115)(2,130,83,116)(3,131,84,117)(4,132,85,118)(5,133,86,119)(6,134,87,120)(7,135,88,91)(8,136,89,92)(9,137,90,93)(10,138,61,94)(11,139,62,95)(12,140,63,96)(13,141,64,97)(14,142,65,98)(15,143,66,99)(16,144,67,100)(17,145,68,101)(18,146,69,102)(19,147,70,103)(20,148,71,104)(21,149,72,105)(22,150,73,106)(23,121,74,107)(24,122,75,108)(25,123,76,109)(26,124,77,110)(27,125,78,111)(28,126,79,112)(29,127,80,113)(30,128,81,114)(31,222,161,190)(32,223,162,191)(33,224,163,192)(34,225,164,193)(35,226,165,194)(36,227,166,195)(37,228,167,196)(38,229,168,197)(39,230,169,198)(40,231,170,199)(41,232,171,200)(42,233,172,201)(43,234,173,202)(44,235,174,203)(45,236,175,204)(46,237,176,205)(47,238,177,206)(48,239,178,207)(49,240,179,208)(50,211,180,209)(51,212,151,210)(52,213,152,181)(53,214,153,182)(54,215,154,183)(55,216,155,184)(56,217,156,185)(57,218,157,186)(58,219,158,187)(59,220,159,188)(60,221,160,189), (1,129)(2,148)(3,137)(4,126)(5,145)(6,134)(7,123)(8,142)(9,131)(10,150)(11,139)(12,128)(13,147)(14,136)(15,125)(16,144)(17,133)(18,122)(19,141)(20,130)(21,149)(22,138)(23,127)(24,146)(25,135)(26,124)(27,143)(28,132)(29,121)(30,140)(31,216)(32,235)(33,224)(34,213)(35,232)(36,221)(37,240)(38,229)(39,218)(40,237)(41,226)(42,215)(43,234)(44,223)(45,212)(46,231)(47,220)(48,239)(49,228)(50,217)(51,236)(52,225)(53,214)(54,233)(55,222)(56,211)(57,230)(58,219)(59,238)(60,227)(61,106)(62,95)(63,114)(64,103)(65,92)(66,111)(67,100)(68,119)(69,108)(70,97)(71,116)(72,105)(73,94)(74,113)(75,102)(76,91)(77,110)(78,99)(79,118)(80,107)(81,96)(82,115)(83,104)(84,93)(85,112)(86,101)(87,120)(88,109)(89,98)(90,117)(151,204)(152,193)(153,182)(154,201)(155,190)(156,209)(157,198)(158,187)(159,206)(160,195)(161,184)(162,203)(163,192)(164,181)(165,200)(166,189)(167,208)(168,197)(169,186)(170,205)(171,194)(172,183)(173,202)(174,191)(175,210)(176,199)(177,188)(178,207)(179,196)(180,185) );

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)], [(1,168,16,153),(2,167,17,152),(3,166,18,151),(4,165,19,180),(5,164,20,179),(6,163,21,178),(7,162,22,177),(8,161,23,176),(9,160,24,175),(10,159,25,174),(11,158,26,173),(12,157,27,172),(13,156,28,171),(14,155,29,170),(15,154,30,169),(31,74,46,89),(32,73,47,88),(33,72,48,87),(34,71,49,86),(35,70,50,85),(36,69,51,84),(37,68,52,83),(38,67,53,82),(39,66,54,81),(40,65,55,80),(41,64,56,79),(42,63,57,78),(43,62,58,77),(44,61,59,76),(45,90,60,75),(91,223,106,238),(92,222,107,237),(93,221,108,236),(94,220,109,235),(95,219,110,234),(96,218,111,233),(97,217,112,232),(98,216,113,231),(99,215,114,230),(100,214,115,229),(101,213,116,228),(102,212,117,227),(103,211,118,226),(104,240,119,225),(105,239,120,224),(121,205,136,190),(122,204,137,189),(123,203,138,188),(124,202,139,187),(125,201,140,186),(126,200,141,185),(127,199,142,184),(128,198,143,183),(129,197,144,182),(130,196,145,181),(131,195,146,210),(132,194,147,209),(133,193,148,208),(134,192,149,207),(135,191,150,206)], [(1,129,82,115),(2,130,83,116),(3,131,84,117),(4,132,85,118),(5,133,86,119),(6,134,87,120),(7,135,88,91),(8,136,89,92),(9,137,90,93),(10,138,61,94),(11,139,62,95),(12,140,63,96),(13,141,64,97),(14,142,65,98),(15,143,66,99),(16,144,67,100),(17,145,68,101),(18,146,69,102),(19,147,70,103),(20,148,71,104),(21,149,72,105),(22,150,73,106),(23,121,74,107),(24,122,75,108),(25,123,76,109),(26,124,77,110),(27,125,78,111),(28,126,79,112),(29,127,80,113),(30,128,81,114),(31,222,161,190),(32,223,162,191),(33,224,163,192),(34,225,164,193),(35,226,165,194),(36,227,166,195),(37,228,167,196),(38,229,168,197),(39,230,169,198),(40,231,170,199),(41,232,171,200),(42,233,172,201),(43,234,173,202),(44,235,174,203),(45,236,175,204),(46,237,176,205),(47,238,177,206),(48,239,178,207),(49,240,179,208),(50,211,180,209),(51,212,151,210),(52,213,152,181),(53,214,153,182),(54,215,154,183),(55,216,155,184),(56,217,156,185),(57,218,157,186),(58,219,158,187),(59,220,159,188),(60,221,160,189)], [(1,129),(2,148),(3,137),(4,126),(5,145),(6,134),(7,123),(8,142),(9,131),(10,150),(11,139),(12,128),(13,147),(14,136),(15,125),(16,144),(17,133),(18,122),(19,141),(20,130),(21,149),(22,138),(23,127),(24,146),(25,135),(26,124),(27,143),(28,132),(29,121),(30,140),(31,216),(32,235),(33,224),(34,213),(35,232),(36,221),(37,240),(38,229),(39,218),(40,237),(41,226),(42,215),(43,234),(44,223),(45,212),(46,231),(47,220),(48,239),(49,228),(50,217),(51,236),(52,225),(53,214),(54,233),(55,222),(56,211),(57,230),(58,219),(59,238),(60,227),(61,106),(62,95),(63,114),(64,103),(65,92),(66,111),(67,100),(68,119),(69,108),(70,97),(71,116),(72,105),(73,94),(74,113),(75,102),(76,91),(77,110),(78,99),(79,118),(80,107),(81,96),(82,115),(83,104),(84,93),(85,112),(86,101),(87,120),(88,109),(89,98),(90,117),(151,204),(152,193),(153,182),(154,201),(155,190),(156,209),(157,198),(158,187),(159,206),(160,195),(161,184),(162,203),(163,192),(164,181),(165,200),(166,189),(167,208),(168,197),(169,186),(170,205),(171,194),(172,183),(173,202),(174,191),(175,210),(176,199),(177,188),(178,207),(179,196),(180,185)])

66 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A6B6C6D6E6F6G10A···10F12A12B15A15B20A20B20C20D20E···20L30A···30F60A···60H
order12222222344444444444455666666610···10121215152020202020···2030···3060···60
size111110101010222666615151515303022222202020202···24444444412···124···44···4

66 irreducible representations

dim11111112222222222444444444
type+++++++++-+++++-+++-+
imageC1C2C2C2C2C2C4S3D4D5Dic3D6D6C4○D4D10D10C4×D5S3×D4D42S3S3×D5D4×D5Q82D5D5×Dic3C2×S3×D5D20⋊S3C20⋊D6
kernelD208Dic3D10⋊Dic3C5×C4⋊Dic3C4×Dic15C2×D5×Dic3C6×D20C3×D20C2×D20Dic15C4⋊Dic3D20C2×C20C22×D5C30C2×Dic3C2×C12C12C10C10C2×C4C6C6C4C22C2C2
# reps12112181224122428112224244

Matrix representation of D208Dic3 in GL6(𝔽61)

0600000
1440000
001000
000100
0000601
0000600
,
1100000
4500000
0060000
0006000
00003437
00001027
,
6000000
0600000
0014600
00536000
0000600
0000060
,
6000000
4410000
00601500
000100
0000600
0000060

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,44,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[11,4,0,0,0,0,0,50,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,34,10,0,0,0,0,37,27],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,53,0,0,0,0,46,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,44,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,15,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;

D208Dic3 in GAP, Magma, Sage, TeX

D_{20}\rtimes_8{\rm Dic}_3
% in TeX

G:=Group("D20:8Dic3");
// GroupNames label

G:=SmallGroup(480,510);
// by ID

G=gap.SmallGroup(480,510);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,422,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=c^4=d^2=1,b^2=a^15,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^19,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽