metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.39D6, C30.5C24, D12.39D10, C15⋊22- 1+4, C60.111C23, Dic6.42D10, Dic10.42D6, Dic15.4C23, Dic30.56C22, C4○D20⋊6S3, C4○D12⋊5D5, C5⋊2(Q8○D12), C5⋊D4.2D6, (C4×D5).14D6, C3⋊D4.2D10, C15⋊D4.C22, C6.5(C23×D5), C15⋊Q8.2C22, (D5×Dic6)⋊12C2, (C4×S3).13D10, (C2×C20).164D6, C10.5(S3×C23), D20⋊5S3⋊11C2, D12⋊5D5⋊11C2, (C6×D5).4C23, D6.1(C22×D5), (C2×Dic30)⋊20C2, (S3×Dic10)⋊11C2, C30.C23⋊1C2, (C2×C12).162D10, (S3×C10).1C23, (C2×C60).97C22, D10.4(C22×S3), C3⋊2(D4.10D10), (S3×C20).28C22, (C2×C30).224C23, C20.124(C22×S3), (C5×D12).39C22, (D5×C12).29C22, (C3×D20).39C22, C12.123(C22×D5), (C3×Dic5).2C23, (D5×Dic3).2C22, Dic5.2(C22×S3), Dic3.4(C22×D5), (C5×Dic3).4C23, (S3×Dic5).1C22, (C5×Dic6).42C22, (C3×Dic10).41C22, (C2×Dic15).149C22, C4.85(C2×S3×D5), (C5×C4○D12)⋊7C2, (C3×C4○D20)⋊7C2, C2.9(C22×S3×D5), (C2×C4).65(S3×D5), C22.17(C2×S3×D5), (C2×C6).9(C22×D5), (C2×C10).8(C22×S3), (C5×C3⋊D4).2C22, (C3×C5⋊D4).3C22, SmallGroup(480,1077)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.39D6
G = < a,b,c,d | a20=b2=1, c6=d2=a10, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a10c5 >
Subgroups: 1292 in 292 conjugacy classes, 108 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C15, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, Dic6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C5×S3, C3×D5, C30, C30, 2- 1+4, Dic10, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C2×Dic6, C4○D12, C4○D12, D4⋊2S3, S3×Q8, C3×C4○D4, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, C2×C30, C2×Dic10, C4○D20, C4○D20, D4⋊2D5, Q8×D5, C5×C4○D4, Q8○D12, D5×Dic3, S3×Dic5, C15⋊D4, C15⋊Q8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, Dic30, C2×Dic15, C2×C60, D4.10D10, D5×Dic6, D20⋊5S3, S3×Dic10, D12⋊5D5, C30.C23, C3×C4○D20, C5×C4○D12, C2×Dic30, D20.39D6
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22×S3, 2- 1+4, C22×D5, S3×C23, S3×D5, C23×D5, Q8○D12, C2×S3×D5, D4.10D10, C22×S3×D5, D20.39D6
►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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(41 45)(42 44)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(81 91)(82 90)(83 89)(84 88)(85 87)(92 100)(93 99)(94 98)(95 97)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(118 120)(122 140)(123 139)(124 138)(125 137)(126 136)(127 135)(128 134)(129 133)(130 132)(141 149)(142 148)(143 147)(144 146)(150 160)(151 159)(152 158)(153 157)(154 156)(161 167)(162 166)(163 165)(168 180)(169 179)(170 178)(171 177)(172 176)(173 175)(182 200)(183 199)(184 198)(185 197)(186 196)(187 195)(188 194)(189 193)(190 192)(201 213)(202 212)(203 211)(204 210)(205 209)(206 208)(214 220)(215 219)(216 218)(221 229)(222 228)(223 227)(224 226)(230 240)(231 239)(232 238)(233 237)(234 236)
(1 210 89 56 158 30 11 220 99 46 148 40)(2 211 90 57 159 31 12 201 100 47 149 21)(3 212 91 58 160 32 13 202 81 48 150 22)(4 213 92 59 141 33 14 203 82 49 151 23)(5 214 93 60 142 34 15 204 83 50 152 24)(6 215 94 41 143 35 16 205 84 51 153 25)(7 216 95 42 144 36 17 206 85 52 154 26)(8 217 96 43 145 37 18 207 86 53 155 27)(9 218 97 44 146 38 19 208 87 54 156 28)(10 219 98 45 147 39 20 209 88 55 157 29)(61 110 122 236 165 182 71 120 132 226 175 192)(62 111 123 237 166 183 72 101 133 227 176 193)(63 112 124 238 167 184 73 102 134 228 177 194)(64 113 125 239 168 185 74 103 135 229 178 195)(65 114 126 240 169 186 75 104 136 230 179 196)(66 115 127 221 170 187 76 105 137 231 180 197)(67 116 128 222 171 188 77 106 138 232 161 198)(68 117 129 223 172 189 78 107 139 233 162 199)(69 118 130 224 173 190 79 108 140 234 163 200)(70 119 131 225 174 191 80 109 121 235 164 181)
(1 73 11 63)(2 74 12 64)(3 75 13 65)(4 76 14 66)(5 77 15 67)(6 78 16 68)(7 79 17 69)(8 80 18 70)(9 61 19 71)(10 62 20 72)(21 103 31 113)(22 104 32 114)(23 105 33 115)(24 106 34 116)(25 107 35 117)(26 108 36 118)(27 109 37 119)(28 110 38 120)(29 111 39 101)(30 112 40 102)(41 223 51 233)(42 224 52 234)(43 225 53 235)(44 226 54 236)(45 227 55 237)(46 228 56 238)(47 229 57 239)(48 230 58 240)(49 231 59 221)(50 232 60 222)(81 179 91 169)(82 180 92 170)(83 161 93 171)(84 162 94 172)(85 163 95 173)(86 164 96 174)(87 165 97 175)(88 166 98 176)(89 167 99 177)(90 168 100 178)(121 145 131 155)(122 146 132 156)(123 147 133 157)(124 148 134 158)(125 149 135 159)(126 150 136 160)(127 151 137 141)(128 152 138 142)(129 153 139 143)(130 154 140 144)(181 217 191 207)(182 218 192 208)(183 219 193 209)(184 220 194 210)(185 201 195 211)(186 202 196 212)(187 203 197 213)(188 204 198 214)(189 205 199 215)(190 206 200 216)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(41,45)(42,44)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(81,91)(82,90)(83,89)(84,88)(85,87)(92,100)(93,99)(94,98)(95,97)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(118,120)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)(141,149)(142,148)(143,147)(144,146)(150,160)(151,159)(152,158)(153,157)(154,156)(161,167)(162,166)(163,165)(168,180)(169,179)(170,178)(171,177)(172,176)(173,175)(182,200)(183,199)(184,198)(185,197)(186,196)(187,195)(188,194)(189,193)(190,192)(201,213)(202,212)(203,211)(204,210)(205,209)(206,208)(214,220)(215,219)(216,218)(221,229)(222,228)(223,227)(224,226)(230,240)(231,239)(232,238)(233,237)(234,236), (1,210,89,56,158,30,11,220,99,46,148,40)(2,211,90,57,159,31,12,201,100,47,149,21)(3,212,91,58,160,32,13,202,81,48,150,22)(4,213,92,59,141,33,14,203,82,49,151,23)(5,214,93,60,142,34,15,204,83,50,152,24)(6,215,94,41,143,35,16,205,84,51,153,25)(7,216,95,42,144,36,17,206,85,52,154,26)(8,217,96,43,145,37,18,207,86,53,155,27)(9,218,97,44,146,38,19,208,87,54,156,28)(10,219,98,45,147,39,20,209,88,55,157,29)(61,110,122,236,165,182,71,120,132,226,175,192)(62,111,123,237,166,183,72,101,133,227,176,193)(63,112,124,238,167,184,73,102,134,228,177,194)(64,113,125,239,168,185,74,103,135,229,178,195)(65,114,126,240,169,186,75,104,136,230,179,196)(66,115,127,221,170,187,76,105,137,231,180,197)(67,116,128,222,171,188,77,106,138,232,161,198)(68,117,129,223,172,189,78,107,139,233,162,199)(69,118,130,224,173,190,79,108,140,234,163,200)(70,119,131,225,174,191,80,109,121,235,164,181), (1,73,11,63)(2,74,12,64)(3,75,13,65)(4,76,14,66)(5,77,15,67)(6,78,16,68)(7,79,17,69)(8,80,18,70)(9,61,19,71)(10,62,20,72)(21,103,31,113)(22,104,32,114)(23,105,33,115)(24,106,34,116)(25,107,35,117)(26,108,36,118)(27,109,37,119)(28,110,38,120)(29,111,39,101)(30,112,40,102)(41,223,51,233)(42,224,52,234)(43,225,53,235)(44,226,54,236)(45,227,55,237)(46,228,56,238)(47,229,57,239)(48,230,58,240)(49,231,59,221)(50,232,60,222)(81,179,91,169)(82,180,92,170)(83,161,93,171)(84,162,94,172)(85,163,95,173)(86,164,96,174)(87,165,97,175)(88,166,98,176)(89,167,99,177)(90,168,100,178)(121,145,131,155)(122,146,132,156)(123,147,133,157)(124,148,134,158)(125,149,135,159)(126,150,136,160)(127,151,137,141)(128,152,138,142)(129,153,139,143)(130,154,140,144)(181,217,191,207)(182,218,192,208)(183,219,193,209)(184,220,194,210)(185,201,195,211)(186,202,196,212)(187,203,197,213)(188,204,198,214)(189,205,199,215)(190,206,200,216)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(41,45)(42,44)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(81,91)(82,90)(83,89)(84,88)(85,87)(92,100)(93,99)(94,98)(95,97)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(118,120)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)(141,149)(142,148)(143,147)(144,146)(150,160)(151,159)(152,158)(153,157)(154,156)(161,167)(162,166)(163,165)(168,180)(169,179)(170,178)(171,177)(172,176)(173,175)(182,200)(183,199)(184,198)(185,197)(186,196)(187,195)(188,194)(189,193)(190,192)(201,213)(202,212)(203,211)(204,210)(205,209)(206,208)(214,220)(215,219)(216,218)(221,229)(222,228)(223,227)(224,226)(230,240)(231,239)(232,238)(233,237)(234,236), (1,210,89,56,158,30,11,220,99,46,148,40)(2,211,90,57,159,31,12,201,100,47,149,21)(3,212,91,58,160,32,13,202,81,48,150,22)(4,213,92,59,141,33,14,203,82,49,151,23)(5,214,93,60,142,34,15,204,83,50,152,24)(6,215,94,41,143,35,16,205,84,51,153,25)(7,216,95,42,144,36,17,206,85,52,154,26)(8,217,96,43,145,37,18,207,86,53,155,27)(9,218,97,44,146,38,19,208,87,54,156,28)(10,219,98,45,147,39,20,209,88,55,157,29)(61,110,122,236,165,182,71,120,132,226,175,192)(62,111,123,237,166,183,72,101,133,227,176,193)(63,112,124,238,167,184,73,102,134,228,177,194)(64,113,125,239,168,185,74,103,135,229,178,195)(65,114,126,240,169,186,75,104,136,230,179,196)(66,115,127,221,170,187,76,105,137,231,180,197)(67,116,128,222,171,188,77,106,138,232,161,198)(68,117,129,223,172,189,78,107,139,233,162,199)(69,118,130,224,173,190,79,108,140,234,163,200)(70,119,131,225,174,191,80,109,121,235,164,181), (1,73,11,63)(2,74,12,64)(3,75,13,65)(4,76,14,66)(5,77,15,67)(6,78,16,68)(7,79,17,69)(8,80,18,70)(9,61,19,71)(10,62,20,72)(21,103,31,113)(22,104,32,114)(23,105,33,115)(24,106,34,116)(25,107,35,117)(26,108,36,118)(27,109,37,119)(28,110,38,120)(29,111,39,101)(30,112,40,102)(41,223,51,233)(42,224,52,234)(43,225,53,235)(44,226,54,236)(45,227,55,237)(46,228,56,238)(47,229,57,239)(48,230,58,240)(49,231,59,221)(50,232,60,222)(81,179,91,169)(82,180,92,170)(83,161,93,171)(84,162,94,172)(85,163,95,173)(86,164,96,174)(87,165,97,175)(88,166,98,176)(89,167,99,177)(90,168,100,178)(121,145,131,155)(122,146,132,156)(123,147,133,157)(124,148,134,158)(125,149,135,159)(126,150,136,160)(127,151,137,141)(128,152,138,142)(129,153,139,143)(130,154,140,144)(181,217,191,207)(182,218,192,208)(183,219,193,209)(184,220,194,210)(185,201,195,211)(186,202,196,212)(187,203,197,213)(188,204,198,214)(189,205,199,215)(190,206,200,216) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(34,40),(35,39),(36,38),(41,45),(42,44),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(81,91),(82,90),(83,89),(84,88),(85,87),(92,100),(93,99),(94,98),(95,97),(101,117),(102,116),(103,115),(104,114),(105,113),(106,112),(107,111),(108,110),(118,120),(122,140),(123,139),(124,138),(125,137),(126,136),(127,135),(128,134),(129,133),(130,132),(141,149),(142,148),(143,147),(144,146),(150,160),(151,159),(152,158),(153,157),(154,156),(161,167),(162,166),(163,165),(168,180),(169,179),(170,178),(171,177),(172,176),(173,175),(182,200),(183,199),(184,198),(185,197),(186,196),(187,195),(188,194),(189,193),(190,192),(201,213),(202,212),(203,211),(204,210),(205,209),(206,208),(214,220),(215,219),(216,218),(221,229),(222,228),(223,227),(224,226),(230,240),(231,239),(232,238),(233,237),(234,236)], [(1,210,89,56,158,30,11,220,99,46,148,40),(2,211,90,57,159,31,12,201,100,47,149,21),(3,212,91,58,160,32,13,202,81,48,150,22),(4,213,92,59,141,33,14,203,82,49,151,23),(5,214,93,60,142,34,15,204,83,50,152,24),(6,215,94,41,143,35,16,205,84,51,153,25),(7,216,95,42,144,36,17,206,85,52,154,26),(8,217,96,43,145,37,18,207,86,53,155,27),(9,218,97,44,146,38,19,208,87,54,156,28),(10,219,98,45,147,39,20,209,88,55,157,29),(61,110,122,236,165,182,71,120,132,226,175,192),(62,111,123,237,166,183,72,101,133,227,176,193),(63,112,124,238,167,184,73,102,134,228,177,194),(64,113,125,239,168,185,74,103,135,229,178,195),(65,114,126,240,169,186,75,104,136,230,179,196),(66,115,127,221,170,187,76,105,137,231,180,197),(67,116,128,222,171,188,77,106,138,232,161,198),(68,117,129,223,172,189,78,107,139,233,162,199),(69,118,130,224,173,190,79,108,140,234,163,200),(70,119,131,225,174,191,80,109,121,235,164,181)], [(1,73,11,63),(2,74,12,64),(3,75,13,65),(4,76,14,66),(5,77,15,67),(6,78,16,68),(7,79,17,69),(8,80,18,70),(9,61,19,71),(10,62,20,72),(21,103,31,113),(22,104,32,114),(23,105,33,115),(24,106,34,116),(25,107,35,117),(26,108,36,118),(27,109,37,119),(28,110,38,120),(29,111,39,101),(30,112,40,102),(41,223,51,233),(42,224,52,234),(43,225,53,235),(44,226,54,236),(45,227,55,237),(46,228,56,238),(47,229,57,239),(48,230,58,240),(49,231,59,221),(50,232,60,222),(81,179,91,169),(82,180,92,170),(83,161,93,171),(84,162,94,172),(85,163,95,173),(86,164,96,174),(87,165,97,175),(88,166,98,176),(89,167,99,177),(90,168,100,178),(121,145,131,155),(122,146,132,156),(123,147,133,157),(124,148,134,158),(125,149,135,159),(126,150,136,160),(127,151,137,141),(128,152,138,142),(129,153,139,143),(130,154,140,144),(181,217,191,207),(182,218,192,208),(183,219,193,209),(184,220,194,210),(185,201,195,211),(186,202,196,212),(187,203,197,213),(188,204,198,214),(189,205,199,215),(190,206,200,216)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 6D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 6 | 6 | 10 | 10 | 2 | 2 | 2 | 6 | 6 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D6 | D6 | D10 | D10 | D10 | D10 | D10 | 2- 1+4 | S3×D5 | Q8○D12 | C2×S3×D5 | C2×S3×D5 | D4.10D10 | D20.39D6 |
| kernel | D20.39D6 | D5×Dic6 | D20⋊5S3 | S3×Dic10 | D12⋊5D5 | C30.C23 | C3×C4○D20 | C5×C4○D12 | C2×Dic30 | C4○D20 | C4○D12 | Dic10 | C4×D5 | D20 | C5⋊D4 | C2×C20 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | C15 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 8 |
Matrix representation of D20.39D6 ►in GL4(𝔽61) generated by
| 7 | 32 | 0 | 0 |
| 29 | 2 | 0 | 0 |
| 0 | 0 | 7 | 32 |
| 0 | 0 | 29 | 2 |
| 0 | 60 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 60 | 0 |
| 23 | 0 | 38 | 0 |
| 0 | 23 | 0 | 38 |
| 23 | 0 | 46 | 0 |
| 0 | 23 | 0 | 46 |
| 2 | 0 | 23 | 0 |
| 0 | 2 | 0 | 23 |
| 21 | 0 | 59 | 0 |
| 0 | 21 | 0 | 59 |
G:=sub<GL(4,GF(61))| [7,29,0,0,32,2,0,0,0,0,7,29,0,0,32,2],[0,60,0,0,60,0,0,0,0,0,0,60,0,0,60,0],[23,0,23,0,0,23,0,23,38,0,46,0,0,38,0,46],[2,0,21,0,0,2,0,21,23,0,59,0,0,23,0,59] >;
D20.39D6 in GAP, Magma, Sage, TeX
D_{20}._{39}D_6 % in TeX
G:=Group("D20.39D6"); // GroupNames label
G:=SmallGroup(480,1077);
// by ID
G=gap.SmallGroup(480,1077);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,219,100,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^6=d^2=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^10*c^5>;
// generators/relations