metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.29D6, C30.32C24, C60.56C23, C15⋊52- 1+4, Dic6.30D10, D30.39C23, Dic15.19C23, Dic30.20C22, (S3×Q8)⋊4D5, (Q8×D15)⋊4C2, C5⋊4(Q8○D12), Q8⋊2D5⋊7S3, (D5×Dic6)⋊6C2, (C4×D5).16D6, (C5×Q8).41D6, Q8.26(S3×D5), D20⋊5S3⋊6C2, D20⋊S3⋊6C2, C15⋊Q8.5C22, (C4×S3).16D10, (C3×Q8).24D10, C6.32(C23×D5), D6.D10⋊5C2, C20.56(C22×S3), C10.32(S3×C23), (C6×D5).14C23, D6.28(C22×D5), C12.56(C22×D5), C15⋊D4.3C22, C5⋊D12.5C22, C3⋊D20.3C22, (S3×C10).31C23, (S3×C20).19C22, C3⋊3(Q8.10D10), (C3×D20).19C22, (C4×D15).19C22, (D5×C12).19C22, D10.17(C22×S3), (Q8×C15).19C22, (D5×Dic3).4C22, Dic3.17(C22×D5), Dic5.44(C22×S3), (C5×Dic3).18C23, (C3×Dic5).47C23, (C5×Dic6).20C22, (C5×S3×Q8)⋊4C2, C4.56(C2×S3×D5), C2.35(C22×S3×D5), (C3×Q8⋊2D5)⋊4C2, SmallGroup(480,1104)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.29D6
G = < a,b,c,d | a20=b2=1, c6=d2=a10, bab=a-1, cac-1=a11, ad=da, cbc-1=a10b, bd=db, dcd-1=a10c5 >
Subgroups: 1340 in 292 conjugacy classes, 108 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, D4, Q8, Q8, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, Dic6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C5×S3, C3×D5, D15, C30, 2- 1+4, Dic10, C4×D5, C4×D5, D20, D20, C5⋊D4, C2×C20, C5×Q8, C5×Q8, C2×Dic6, C4○D12, D4⋊2S3, S3×Q8, S3×Q8, C3×C4○D4, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, C4○D20, Q8×D5, Q8⋊2D5, Q8⋊2D5, Q8×C10, Q8○D12, D5×Dic3, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, C3×D20, C5×Dic6, S3×C20, Dic30, C4×D15, Q8×C15, Q8.10D10, D5×Dic6, D20⋊5S3, D20⋊S3, D6.D10, C3×Q8⋊2D5, C5×S3×Q8, Q8×D15, D20.29D6
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, Q8.10D10, C22×S3×D5, D20.29D6
►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)
(1 227)(2 226)(3 225)(4 224)(5 223)(6 222)(7 221)(8 240)(9 239)(10 238)(11 237)(12 236)(13 235)(14 234)(15 233)(16 232)(17 231)(18 230)(19 229)(20 228)(21 88)(22 87)(23 86)(24 85)(25 84)(26 83)(27 82)(28 81)(29 100)(30 99)(31 98)(32 97)(33 96)(34 95)(35 94)(36 93)(37 92)(38 91)(39 90)(40 89)(41 70)(42 69)(43 68)(44 67)(45 66)(46 65)(47 64)(48 63)(49 62)(50 61)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)(101 194)(102 193)(103 192)(104 191)(105 190)(106 189)(107 188)(108 187)(109 186)(110 185)(111 184)(112 183)(113 182)(114 181)(115 200)(116 199)(117 198)(118 197)(119 196)(120 195)(121 155)(122 154)(123 153)(124 152)(125 151)(126 150)(127 149)(128 148)(129 147)(130 146)(131 145)(132 144)(133 143)(134 142)(135 141)(136 160)(137 159)(138 158)(139 157)(140 156)(161 219)(162 218)(163 217)(164 216)(165 215)(166 214)(167 213)(168 212)(169 211)(170 210)(171 209)(172 208)(173 207)(174 206)(175 205)(176 204)(177 203)(178 202)(179 201)(180 220)
(1 127 118 80 34 218 11 137 108 70 24 208)(2 138 119 71 35 209 12 128 109 61 25 219)(3 129 120 62 36 220 13 139 110 72 26 210)(4 140 101 73 37 211 14 130 111 63 27 201)(5 131 102 64 38 202 15 121 112 74 28 212)(6 122 103 75 39 213 16 132 113 65 29 203)(7 133 104 66 40 204 17 123 114 76 30 214)(8 124 105 77 21 215 18 134 115 67 31 205)(9 135 106 68 22 206 19 125 116 78 32 216)(10 126 107 79 23 217 20 136 117 69 33 207)(41 95 172 237 149 187 51 85 162 227 159 197)(42 86 173 228 150 198 52 96 163 238 160 188)(43 97 174 239 151 189 53 87 164 229 141 199)(44 88 175 230 152 200 54 98 165 240 142 190)(45 99 176 221 153 191 55 89 166 231 143 181)(46 90 177 232 154 182 56 100 167 222 144 192)(47 81 178 223 155 193 57 91 168 233 145 183)(48 92 179 234 156 184 58 82 169 224 146 194)(49 83 180 225 157 195 59 93 170 235 147 185)(50 94 161 236 158 186 60 84 171 226 148 196)
(1 178 11 168)(2 179 12 169)(3 180 13 170)(4 161 14 171)(5 162 15 172)(6 163 16 173)(7 164 17 174)(8 165 18 175)(9 166 19 176)(10 167 20 177)(21 152 31 142)(22 153 32 143)(23 154 33 144)(24 155 34 145)(25 156 35 146)(26 157 36 147)(27 158 37 148)(28 159 38 149)(29 160 39 150)(30 141 40 151)(41 102 51 112)(42 103 52 113)(43 104 53 114)(44 105 54 115)(45 106 55 116)(46 107 56 117)(47 108 57 118)(48 109 58 119)(49 110 59 120)(50 111 60 101)(61 184 71 194)(62 185 72 195)(63 186 73 196)(64 187 74 197)(65 188 75 198)(66 189 76 199)(67 190 77 200)(68 191 78 181)(69 192 79 182)(70 193 80 183)(81 137 91 127)(82 138 92 128)(83 139 93 129)(84 140 94 130)(85 121 95 131)(86 122 96 132)(87 123 97 133)(88 124 98 134)(89 125 99 135)(90 126 100 136)(201 236 211 226)(202 237 212 227)(203 238 213 228)(204 239 214 229)(205 240 215 230)(206 221 216 231)(207 222 217 232)(208 223 218 233)(209 224 219 234)(210 225 220 235)
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), (1,227)(2,226)(3,225)(4,224)(5,223)(6,222)(7,221)(8,240)(9,239)(10,238)(11,237)(12,236)(13,235)(14,234)(15,233)(16,232)(17,231)(18,230)(19,229)(20,228)(21,88)(22,87)(23,86)(24,85)(25,84)(26,83)(27,82)(28,81)(29,100)(30,99)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,92)(38,91)(39,90)(40,89)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(101,194)(102,193)(103,192)(104,191)(105,190)(106,189)(107,188)(108,187)(109,186)(110,185)(111,184)(112,183)(113,182)(114,181)(115,200)(116,199)(117,198)(118,197)(119,196)(120,195)(121,155)(122,154)(123,153)(124,152)(125,151)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,160)(137,159)(138,158)(139,157)(140,156)(161,219)(162,218)(163,217)(164,216)(165,215)(166,214)(167,213)(168,212)(169,211)(170,210)(171,209)(172,208)(173,207)(174,206)(175,205)(176,204)(177,203)(178,202)(179,201)(180,220), (1,127,118,80,34,218,11,137,108,70,24,208)(2,138,119,71,35,209,12,128,109,61,25,219)(3,129,120,62,36,220,13,139,110,72,26,210)(4,140,101,73,37,211,14,130,111,63,27,201)(5,131,102,64,38,202,15,121,112,74,28,212)(6,122,103,75,39,213,16,132,113,65,29,203)(7,133,104,66,40,204,17,123,114,76,30,214)(8,124,105,77,21,215,18,134,115,67,31,205)(9,135,106,68,22,206,19,125,116,78,32,216)(10,126,107,79,23,217,20,136,117,69,33,207)(41,95,172,237,149,187,51,85,162,227,159,197)(42,86,173,228,150,198,52,96,163,238,160,188)(43,97,174,239,151,189,53,87,164,229,141,199)(44,88,175,230,152,200,54,98,165,240,142,190)(45,99,176,221,153,191,55,89,166,231,143,181)(46,90,177,232,154,182,56,100,167,222,144,192)(47,81,178,223,155,193,57,91,168,233,145,183)(48,92,179,234,156,184,58,82,169,224,146,194)(49,83,180,225,157,195,59,93,170,235,147,185)(50,94,161,236,158,186,60,84,171,226,148,196), (1,178,11,168)(2,179,12,169)(3,180,13,170)(4,161,14,171)(5,162,15,172)(6,163,16,173)(7,164,17,174)(8,165,18,175)(9,166,19,176)(10,167,20,177)(21,152,31,142)(22,153,32,143)(23,154,33,144)(24,155,34,145)(25,156,35,146)(26,157,36,147)(27,158,37,148)(28,159,38,149)(29,160,39,150)(30,141,40,151)(41,102,51,112)(42,103,52,113)(43,104,53,114)(44,105,54,115)(45,106,55,116)(46,107,56,117)(47,108,57,118)(48,109,58,119)(49,110,59,120)(50,111,60,101)(61,184,71,194)(62,185,72,195)(63,186,73,196)(64,187,74,197)(65,188,75,198)(66,189,76,199)(67,190,77,200)(68,191,78,181)(69,192,79,182)(70,193,80,183)(81,137,91,127)(82,138,92,128)(83,139,93,129)(84,140,94,130)(85,121,95,131)(86,122,96,132)(87,123,97,133)(88,124,98,134)(89,125,99,135)(90,126,100,136)(201,236,211,226)(202,237,212,227)(203,238,213,228)(204,239,214,229)(205,240,215,230)(206,221,216,231)(207,222,217,232)(208,223,218,233)(209,224,219,234)(210,225,220,235)>;
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), (1,227)(2,226)(3,225)(4,224)(5,223)(6,222)(7,221)(8,240)(9,239)(10,238)(11,237)(12,236)(13,235)(14,234)(15,233)(16,232)(17,231)(18,230)(19,229)(20,228)(21,88)(22,87)(23,86)(24,85)(25,84)(26,83)(27,82)(28,81)(29,100)(30,99)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,92)(38,91)(39,90)(40,89)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(101,194)(102,193)(103,192)(104,191)(105,190)(106,189)(107,188)(108,187)(109,186)(110,185)(111,184)(112,183)(113,182)(114,181)(115,200)(116,199)(117,198)(118,197)(119,196)(120,195)(121,155)(122,154)(123,153)(124,152)(125,151)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,160)(137,159)(138,158)(139,157)(140,156)(161,219)(162,218)(163,217)(164,216)(165,215)(166,214)(167,213)(168,212)(169,211)(170,210)(171,209)(172,208)(173,207)(174,206)(175,205)(176,204)(177,203)(178,202)(179,201)(180,220), (1,127,118,80,34,218,11,137,108,70,24,208)(2,138,119,71,35,209,12,128,109,61,25,219)(3,129,120,62,36,220,13,139,110,72,26,210)(4,140,101,73,37,211,14,130,111,63,27,201)(5,131,102,64,38,202,15,121,112,74,28,212)(6,122,103,75,39,213,16,132,113,65,29,203)(7,133,104,66,40,204,17,123,114,76,30,214)(8,124,105,77,21,215,18,134,115,67,31,205)(9,135,106,68,22,206,19,125,116,78,32,216)(10,126,107,79,23,217,20,136,117,69,33,207)(41,95,172,237,149,187,51,85,162,227,159,197)(42,86,173,228,150,198,52,96,163,238,160,188)(43,97,174,239,151,189,53,87,164,229,141,199)(44,88,175,230,152,200,54,98,165,240,142,190)(45,99,176,221,153,191,55,89,166,231,143,181)(46,90,177,232,154,182,56,100,167,222,144,192)(47,81,178,223,155,193,57,91,168,233,145,183)(48,92,179,234,156,184,58,82,169,224,146,194)(49,83,180,225,157,195,59,93,170,235,147,185)(50,94,161,236,158,186,60,84,171,226,148,196), (1,178,11,168)(2,179,12,169)(3,180,13,170)(4,161,14,171)(5,162,15,172)(6,163,16,173)(7,164,17,174)(8,165,18,175)(9,166,19,176)(10,167,20,177)(21,152,31,142)(22,153,32,143)(23,154,33,144)(24,155,34,145)(25,156,35,146)(26,157,36,147)(27,158,37,148)(28,159,38,149)(29,160,39,150)(30,141,40,151)(41,102,51,112)(42,103,52,113)(43,104,53,114)(44,105,54,115)(45,106,55,116)(46,107,56,117)(47,108,57,118)(48,109,58,119)(49,110,59,120)(50,111,60,101)(61,184,71,194)(62,185,72,195)(63,186,73,196)(64,187,74,197)(65,188,75,198)(66,189,76,199)(67,190,77,200)(68,191,78,181)(69,192,79,182)(70,193,80,183)(81,137,91,127)(82,138,92,128)(83,139,93,129)(84,140,94,130)(85,121,95,131)(86,122,96,132)(87,123,97,133)(88,124,98,134)(89,125,99,135)(90,126,100,136)(201,236,211,226)(202,237,212,227)(203,238,213,228)(204,239,214,229)(205,240,215,230)(206,221,216,231)(207,222,217,232)(208,223,218,233)(209,224,219,234)(210,225,220,235) );
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)], [(1,227),(2,226),(3,225),(4,224),(5,223),(6,222),(7,221),(8,240),(9,239),(10,238),(11,237),(12,236),(13,235),(14,234),(15,233),(16,232),(17,231),(18,230),(19,229),(20,228),(21,88),(22,87),(23,86),(24,85),(25,84),(26,83),(27,82),(28,81),(29,100),(30,99),(31,98),(32,97),(33,96),(34,95),(35,94),(36,93),(37,92),(38,91),(39,90),(40,89),(41,70),(42,69),(43,68),(44,67),(45,66),(46,65),(47,64),(48,63),(49,62),(50,61),(51,80),(52,79),(53,78),(54,77),(55,76),(56,75),(57,74),(58,73),(59,72),(60,71),(101,194),(102,193),(103,192),(104,191),(105,190),(106,189),(107,188),(108,187),(109,186),(110,185),(111,184),(112,183),(113,182),(114,181),(115,200),(116,199),(117,198),(118,197),(119,196),(120,195),(121,155),(122,154),(123,153),(124,152),(125,151),(126,150),(127,149),(128,148),(129,147),(130,146),(131,145),(132,144),(133,143),(134,142),(135,141),(136,160),(137,159),(138,158),(139,157),(140,156),(161,219),(162,218),(163,217),(164,216),(165,215),(166,214),(167,213),(168,212),(169,211),(170,210),(171,209),(172,208),(173,207),(174,206),(175,205),(176,204),(177,203),(178,202),(179,201),(180,220)], [(1,127,118,80,34,218,11,137,108,70,24,208),(2,138,119,71,35,209,12,128,109,61,25,219),(3,129,120,62,36,220,13,139,110,72,26,210),(4,140,101,73,37,211,14,130,111,63,27,201),(5,131,102,64,38,202,15,121,112,74,28,212),(6,122,103,75,39,213,16,132,113,65,29,203),(7,133,104,66,40,204,17,123,114,76,30,214),(8,124,105,77,21,215,18,134,115,67,31,205),(9,135,106,68,22,206,19,125,116,78,32,216),(10,126,107,79,23,217,20,136,117,69,33,207),(41,95,172,237,149,187,51,85,162,227,159,197),(42,86,173,228,150,198,52,96,163,238,160,188),(43,97,174,239,151,189,53,87,164,229,141,199),(44,88,175,230,152,200,54,98,165,240,142,190),(45,99,176,221,153,191,55,89,166,231,143,181),(46,90,177,232,154,182,56,100,167,222,144,192),(47,81,178,223,155,193,57,91,168,233,145,183),(48,92,179,234,156,184,58,82,169,224,146,194),(49,83,180,225,157,195,59,93,170,235,147,185),(50,94,161,236,158,186,60,84,171,226,148,196)], [(1,178,11,168),(2,179,12,169),(3,180,13,170),(4,161,14,171),(5,162,15,172),(6,163,16,173),(7,164,17,174),(8,165,18,175),(9,166,19,176),(10,167,20,177),(21,152,31,142),(22,153,32,143),(23,154,33,144),(24,155,34,145),(25,156,35,146),(26,157,36,147),(27,158,37,148),(28,159,38,149),(29,160,39,150),(30,141,40,151),(41,102,51,112),(42,103,52,113),(43,104,53,114),(44,105,54,115),(45,106,55,116),(46,107,56,117),(47,108,57,118),(48,109,58,119),(49,110,59,120),(50,111,60,101),(61,184,71,194),(62,185,72,195),(63,186,73,196),(64,187,74,197),(65,188,75,198),(66,189,76,199),(67,190,77,200),(68,191,78,181),(69,192,79,182),(70,193,80,183),(81,137,91,127),(82,138,92,128),(83,139,93,129),(84,140,94,130),(85,121,95,131),(86,122,96,132),(87,123,97,133),(88,124,98,134),(89,125,99,135),(90,126,100,136),(201,236,211,226),(202,237,212,227),(203,238,213,228),(204,239,214,229),(205,240,215,230),(206,221,216,231),(207,222,217,232),(208,223,218,233),(209,224,219,234),(210,225,220,235)]])
57 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 | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | ··· | 20F | 20G | ··· | 20L | 30A | 30B | 60A | ··· | 60F |
| 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 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 10 | 10 | 10 | 30 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 10 | 30 | 30 | 30 | 2 | 2 | 2 | 20 | 20 | 20 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 10 | 10 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 | 4 | 4 | 8 | ··· | 8 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | 2- 1+4 | S3×D5 | Q8○D12 | C2×S3×D5 | Q8.10D10 | D20.29D6 |
| kernel | D20.29D6 | D5×Dic6 | D20⋊5S3 | D20⋊S3 | D6.D10 | C3×Q8⋊2D5 | C5×S3×Q8 | Q8×D15 | Q8⋊2D5 | S3×Q8 | C4×D5 | D20 | C5×Q8 | Dic6 | C4×S3 | C3×Q8 | C15 | Q8 | C5 | C4 | C3 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 1 | 6 | 6 | 2 | 1 | 2 | 2 | 6 | 4 | 2 |
Matrix representation of D20.29D6 ►in GL8(𝔽61)
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 44 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 30 | 38 | 13 |
| 0 | 0 | 0 | 0 | 53 | 31 | 23 | 25 |
| 0 | 0 | 0 | 0 | 51 | 0 | 0 | 31 |
| 0 | 0 | 0 | 0 | 48 | 51 | 30 | 52 |
| 35 | 2 | 9 | 15 | 0 | 0 | 0 | 0 |
| 48 | 26 | 46 | 52 | 0 | 0 | 0 | 0 |
| 52 | 46 | 26 | 48 | 0 | 0 | 0 | 0 |
| 15 | 9 | 2 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 30 | 13 | 15 |
| 0 | 0 | 0 | 0 | 28 | 14 | 2 | 59 |
| 0 | 0 | 0 | 0 | 48 | 55 | 17 | 31 |
| 0 | 0 | 0 | 0 | 48 | 54 | 30 | 44 |
| 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 0 | 0 | 25 |
| 0 | 0 | 0 | 0 | 0 | 47 | 36 | 36 |
| 0 | 0 | 0 | 0 | 3 | 3 | 14 | 0 |
| 0 | 0 | 0 | 0 | 58 | 0 | 0 | 14 |
| 2 | 35 | 48 | 26 | 0 | 0 | 0 | 0 |
| 26 | 48 | 35 | 2 | 0 | 0 | 0 | 0 |
| 46 | 52 | 59 | 26 | 0 | 0 | 0 | 0 |
| 9 | 15 | 35 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 54 | 39 | 25 | 42 |
| 0 | 0 | 0 | 0 | 14 | 7 | 17 | 44 |
| 0 | 0 | 0 | 0 | 43 | 56 | 15 | 22 |
| 0 | 0 | 0 | 0 | 43 | 48 | 39 | 46 |
G:=sub<GL(8,GF(61))| [0,1,0,0,0,0,0,0,60,44,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,44,0,0,0,0,0,0,0,0,39,53,51,48,0,0,0,0,30,31,0,51,0,0,0,0,38,23,0,30,0,0,0,0,13,25,31,52],[35,48,52,15,0,0,0,0,2,26,46,9,0,0,0,0,9,46,26,2,0,0,0,0,15,52,48,35,0,0,0,0,0,0,0,0,47,28,48,48,0,0,0,0,30,14,55,54,0,0,0,0,13,2,17,30,0,0,0,0,15,59,31,44],[1,0,60,0,0,0,0,0,0,1,0,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,47,0,3,58,0,0,0,0,0,47,3,0,0,0,0,0,0,36,14,0,0,0,0,0,25,36,0,14],[2,26,46,9,0,0,0,0,35,48,52,15,0,0,0,0,48,35,59,35,0,0,0,0,26,2,26,13,0,0,0,0,0,0,0,0,54,14,43,43,0,0,0,0,39,7,56,48,0,0,0,0,25,17,15,39,0,0,0,0,42,44,22,46] >;
D20.29D6 in GAP, Magma, Sage, TeX
D_{20}._{29}D_6 % in TeX
G:=Group("D20.29D6"); // GroupNames label
G:=SmallGroup(480,1104);
// by ID
G=gap.SmallGroup(480,1104);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,219,100,675,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^6=d^2=a^10,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d^-1=a^10*c^5>;
// generators/relations