metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.28D6, C60.42C23, Dic6.14D10, Dic30.16C22, Q8⋊D5⋊4S3, (S3×Q8)⋊2D5, C5⋊2C8.9D6, C5⋊Dic12⋊7C2, C15⋊7Q16⋊8C2, C5⋊7(D4.D6), Q8.11(S3×D5), (C3×Q8).6D10, (C5×Q8).38D6, (S3×C10).37D4, (C4×S3).11D10, C30.204(C2×D4), C30.D4⋊8C2, C10.152(S3×D4), D6.Dic5⋊7C2, D20⋊5S3.1C2, D6.15(C5⋊D4), C3⋊2(C20.C23), C15⋊21(C8.C22), C20.42(C22×S3), (C5×Dic3).17D4, C12.42(C22×D5), (S3×C20).14C22, C15⋊3C8.16C22, (C3×D20).16C22, (Q8×C15).12C22, Dic3.12(C5⋊D4), (C5×Dic6).15C22, (C5×S3×Q8)⋊2C2, C4.42(C2×S3×D5), (C3×Q8⋊D5)⋊6C2, C6.55(C2×C5⋊D4), C2.33(S3×C5⋊D4), (C3×C5⋊2C8).12C22, SmallGroup(480,594)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.28D6
G = < a,b,c,d | a20=b2=1, c6=d2=a10, bab=a-1, cac-1=dad-1=a11, cbc-1=dbd-1=a15b, dcd-1=a10c5 >
Subgroups: 572 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, D5, C10, C10, Dic3, Dic3, C12, C12, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C5×S3, C3×D5, C30, C8.C22, C5⋊2C8, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×Q8, C5×Q8, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, D4⋊2S3, S3×Q8, C5×Dic3, C5×Dic3, Dic15, C60, C60, C6×D5, S3×C10, C4.Dic5, Q8⋊D5, Q8⋊D5, C5⋊Q16, C4○D20, Q8×C10, D4.D6, C3×C5⋊2C8, C15⋊3C8, D5×Dic3, C15⋊D4, C3×D20, C5×Dic6, C5×Dic6, S3×C20, S3×C20, Dic30, Q8×C15, C20.C23, D6.Dic5, C30.D4, C5⋊Dic12, C3×Q8⋊D5, C15⋊7Q16, D20⋊5S3, C5×S3×Q8, D20.28D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8.C22, C5⋊D4, C22×D5, S3×D4, S3×D5, C2×C5⋊D4, D4.D6, C2×S3×D5, C20.C23, S3×C5⋊D4, D20.28D6
►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)
(1 146)(2 145)(3 144)(4 143)(5 142)(6 141)(7 160)(8 159)(9 158)(10 157)(11 156)(12 155)(13 154)(14 153)(15 152)(16 151)(17 150)(18 149)(19 148)(20 147)(21 228)(22 227)(23 226)(24 225)(25 224)(26 223)(27 222)(28 221)(29 240)(30 239)(31 238)(32 237)(33 236)(34 235)(35 234)(36 233)(37 232)(38 231)(39 230)(40 229)(41 72)(42 71)(43 70)(44 69)(45 68)(46 67)(47 66)(48 65)(49 64)(50 63)(51 62)(52 61)(53 80)(54 79)(55 78)(56 77)(57 76)(58 75)(59 74)(60 73)(81 172)(82 171)(83 170)(84 169)(85 168)(86 167)(87 166)(88 165)(89 164)(90 163)(91 162)(92 161)(93 180)(94 179)(95 178)(96 177)(97 176)(98 175)(99 174)(100 173)(101 200)(102 199)(103 198)(104 197)(105 196)(106 195)(107 194)(108 193)(109 192)(110 191)(111 190)(112 189)(113 188)(114 187)(115 186)(116 185)(117 184)(118 183)(119 182)(120 181)(121 212)(122 211)(123 210)(124 209)(125 208)(126 207)(127 206)(128 205)(129 204)(130 203)(131 202)(132 201)(133 220)(134 219)(135 218)(136 217)(137 216)(138 215)(139 214)(140 213)
(1 185 169 215 228 57 11 195 179 205 238 47)(2 196 170 206 229 48 12 186 180 216 239 58)(3 187 171 217 230 59 13 197 161 207 240 49)(4 198 172 208 231 50 14 188 162 218 221 60)(5 189 173 219 232 41 15 199 163 209 222 51)(6 200 174 210 233 52 16 190 164 220 223 42)(7 191 175 201 234 43 17 181 165 211 224 53)(8 182 176 212 235 54 18 192 166 202 225 44)(9 193 177 203 236 45 19 183 167 213 226 55)(10 184 178 214 237 56 20 194 168 204 227 46)(21 61 156 111 94 133 31 71 146 101 84 123)(22 72 157 102 95 124 32 62 147 112 85 134)(23 63 158 113 96 135 33 73 148 103 86 125)(24 74 159 104 97 126 34 64 149 114 87 136)(25 65 160 115 98 137 35 75 150 105 88 127)(26 76 141 106 99 128 36 66 151 116 89 138)(27 67 142 117 100 139 37 77 152 107 90 129)(28 78 143 108 81 130 38 68 153 118 91 140)(29 69 144 119 82 121 39 79 154 109 92 131)(30 80 145 110 83 132 40 70 155 120 93 122)
(1 42 11 52)(2 53 12 43)(3 44 13 54)(4 55 14 45)(5 46 15 56)(6 57 16 47)(7 48 17 58)(8 59 18 49)(9 50 19 60)(10 41 20 51)(21 116 31 106)(22 107 32 117)(23 118 33 108)(24 109 34 119)(25 120 35 110)(26 111 36 101)(27 102 37 112)(28 113 38 103)(29 104 39 114)(30 115 40 105)(61 151 71 141)(62 142 72 152)(63 153 73 143)(64 144 74 154)(65 155 75 145)(66 146 76 156)(67 157 77 147)(68 148 78 158)(69 159 79 149)(70 150 80 160)(81 125 91 135)(82 136 92 126)(83 127 93 137)(84 138 94 128)(85 129 95 139)(86 140 96 130)(87 131 97 121)(88 122 98 132)(89 133 99 123)(90 124 100 134)(161 212 171 202)(162 203 172 213)(163 214 173 204)(164 205 174 215)(165 216 175 206)(166 207 176 217)(167 218 177 208)(168 209 178 219)(169 220 179 210)(170 211 180 201)(181 239 191 229)(182 230 192 240)(183 221 193 231)(184 232 194 222)(185 223 195 233)(186 234 196 224)(187 225 197 235)(188 236 198 226)(189 227 199 237)(190 238 200 228)
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), (1,146)(2,145)(3,144)(4,143)(5,142)(6,141)(7,160)(8,159)(9,158)(10,157)(11,156)(12,155)(13,154)(14,153)(15,152)(16,151)(17,150)(18,149)(19,148)(20,147)(21,228)(22,227)(23,226)(24,225)(25,224)(26,223)(27,222)(28,221)(29,240)(30,239)(31,238)(32,237)(33,236)(34,235)(35,234)(36,233)(37,232)(38,231)(39,230)(40,229)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(81,172)(82,171)(83,170)(84,169)(85,168)(86,167)(87,166)(88,165)(89,164)(90,163)(91,162)(92,161)(93,180)(94,179)(95,178)(96,177)(97,176)(98,175)(99,174)(100,173)(101,200)(102,199)(103,198)(104,197)(105,196)(106,195)(107,194)(108,193)(109,192)(110,191)(111,190)(112,189)(113,188)(114,187)(115,186)(116,185)(117,184)(118,183)(119,182)(120,181)(121,212)(122,211)(123,210)(124,209)(125,208)(126,207)(127,206)(128,205)(129,204)(130,203)(131,202)(132,201)(133,220)(134,219)(135,218)(136,217)(137,216)(138,215)(139,214)(140,213), (1,185,169,215,228,57,11,195,179,205,238,47)(2,196,170,206,229,48,12,186,180,216,239,58)(3,187,171,217,230,59,13,197,161,207,240,49)(4,198,172,208,231,50,14,188,162,218,221,60)(5,189,173,219,232,41,15,199,163,209,222,51)(6,200,174,210,233,52,16,190,164,220,223,42)(7,191,175,201,234,43,17,181,165,211,224,53)(8,182,176,212,235,54,18,192,166,202,225,44)(9,193,177,203,236,45,19,183,167,213,226,55)(10,184,178,214,237,56,20,194,168,204,227,46)(21,61,156,111,94,133,31,71,146,101,84,123)(22,72,157,102,95,124,32,62,147,112,85,134)(23,63,158,113,96,135,33,73,148,103,86,125)(24,74,159,104,97,126,34,64,149,114,87,136)(25,65,160,115,98,137,35,75,150,105,88,127)(26,76,141,106,99,128,36,66,151,116,89,138)(27,67,142,117,100,139,37,77,152,107,90,129)(28,78,143,108,81,130,38,68,153,118,91,140)(29,69,144,119,82,121,39,79,154,109,92,131)(30,80,145,110,83,132,40,70,155,120,93,122), (1,42,11,52)(2,53,12,43)(3,44,13,54)(4,55,14,45)(5,46,15,56)(6,57,16,47)(7,48,17,58)(8,59,18,49)(9,50,19,60)(10,41,20,51)(21,116,31,106)(22,107,32,117)(23,118,33,108)(24,109,34,119)(25,120,35,110)(26,111,36,101)(27,102,37,112)(28,113,38,103)(29,104,39,114)(30,115,40,105)(61,151,71,141)(62,142,72,152)(63,153,73,143)(64,144,74,154)(65,155,75,145)(66,146,76,156)(67,157,77,147)(68,148,78,158)(69,159,79,149)(70,150,80,160)(81,125,91,135)(82,136,92,126)(83,127,93,137)(84,138,94,128)(85,129,95,139)(86,140,96,130)(87,131,97,121)(88,122,98,132)(89,133,99,123)(90,124,100,134)(161,212,171,202)(162,203,172,213)(163,214,173,204)(164,205,174,215)(165,216,175,206)(166,207,176,217)(167,218,177,208)(168,209,178,219)(169,220,179,210)(170,211,180,201)(181,239,191,229)(182,230,192,240)(183,221,193,231)(184,232,194,222)(185,223,195,233)(186,234,196,224)(187,225,197,235)(188,236,198,226)(189,227,199,237)(190,238,200,228)>;
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), (1,146)(2,145)(3,144)(4,143)(5,142)(6,141)(7,160)(8,159)(9,158)(10,157)(11,156)(12,155)(13,154)(14,153)(15,152)(16,151)(17,150)(18,149)(19,148)(20,147)(21,228)(22,227)(23,226)(24,225)(25,224)(26,223)(27,222)(28,221)(29,240)(30,239)(31,238)(32,237)(33,236)(34,235)(35,234)(36,233)(37,232)(38,231)(39,230)(40,229)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(81,172)(82,171)(83,170)(84,169)(85,168)(86,167)(87,166)(88,165)(89,164)(90,163)(91,162)(92,161)(93,180)(94,179)(95,178)(96,177)(97,176)(98,175)(99,174)(100,173)(101,200)(102,199)(103,198)(104,197)(105,196)(106,195)(107,194)(108,193)(109,192)(110,191)(111,190)(112,189)(113,188)(114,187)(115,186)(116,185)(117,184)(118,183)(119,182)(120,181)(121,212)(122,211)(123,210)(124,209)(125,208)(126,207)(127,206)(128,205)(129,204)(130,203)(131,202)(132,201)(133,220)(134,219)(135,218)(136,217)(137,216)(138,215)(139,214)(140,213), (1,185,169,215,228,57,11,195,179,205,238,47)(2,196,170,206,229,48,12,186,180,216,239,58)(3,187,171,217,230,59,13,197,161,207,240,49)(4,198,172,208,231,50,14,188,162,218,221,60)(5,189,173,219,232,41,15,199,163,209,222,51)(6,200,174,210,233,52,16,190,164,220,223,42)(7,191,175,201,234,43,17,181,165,211,224,53)(8,182,176,212,235,54,18,192,166,202,225,44)(9,193,177,203,236,45,19,183,167,213,226,55)(10,184,178,214,237,56,20,194,168,204,227,46)(21,61,156,111,94,133,31,71,146,101,84,123)(22,72,157,102,95,124,32,62,147,112,85,134)(23,63,158,113,96,135,33,73,148,103,86,125)(24,74,159,104,97,126,34,64,149,114,87,136)(25,65,160,115,98,137,35,75,150,105,88,127)(26,76,141,106,99,128,36,66,151,116,89,138)(27,67,142,117,100,139,37,77,152,107,90,129)(28,78,143,108,81,130,38,68,153,118,91,140)(29,69,144,119,82,121,39,79,154,109,92,131)(30,80,145,110,83,132,40,70,155,120,93,122), (1,42,11,52)(2,53,12,43)(3,44,13,54)(4,55,14,45)(5,46,15,56)(6,57,16,47)(7,48,17,58)(8,59,18,49)(9,50,19,60)(10,41,20,51)(21,116,31,106)(22,107,32,117)(23,118,33,108)(24,109,34,119)(25,120,35,110)(26,111,36,101)(27,102,37,112)(28,113,38,103)(29,104,39,114)(30,115,40,105)(61,151,71,141)(62,142,72,152)(63,153,73,143)(64,144,74,154)(65,155,75,145)(66,146,76,156)(67,157,77,147)(68,148,78,158)(69,159,79,149)(70,150,80,160)(81,125,91,135)(82,136,92,126)(83,127,93,137)(84,138,94,128)(85,129,95,139)(86,140,96,130)(87,131,97,121)(88,122,98,132)(89,133,99,123)(90,124,100,134)(161,212,171,202)(162,203,172,213)(163,214,173,204)(164,205,174,215)(165,216,175,206)(166,207,176,217)(167,218,177,208)(168,209,178,219)(169,220,179,210)(170,211,180,201)(181,239,191,229)(182,230,192,240)(183,221,193,231)(184,232,194,222)(185,223,195,233)(186,234,196,224)(187,225,197,235)(188,236,198,226)(189,227,199,237)(190,238,200,228) );
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)], [(1,146),(2,145),(3,144),(4,143),(5,142),(6,141),(7,160),(8,159),(9,158),(10,157),(11,156),(12,155),(13,154),(14,153),(15,152),(16,151),(17,150),(18,149),(19,148),(20,147),(21,228),(22,227),(23,226),(24,225),(25,224),(26,223),(27,222),(28,221),(29,240),(30,239),(31,238),(32,237),(33,236),(34,235),(35,234),(36,233),(37,232),(38,231),(39,230),(40,229),(41,72),(42,71),(43,70),(44,69),(45,68),(46,67),(47,66),(48,65),(49,64),(50,63),(51,62),(52,61),(53,80),(54,79),(55,78),(56,77),(57,76),(58,75),(59,74),(60,73),(81,172),(82,171),(83,170),(84,169),(85,168),(86,167),(87,166),(88,165),(89,164),(90,163),(91,162),(92,161),(93,180),(94,179),(95,178),(96,177),(97,176),(98,175),(99,174),(100,173),(101,200),(102,199),(103,198),(104,197),(105,196),(106,195),(107,194),(108,193),(109,192),(110,191),(111,190),(112,189),(113,188),(114,187),(115,186),(116,185),(117,184),(118,183),(119,182),(120,181),(121,212),(122,211),(123,210),(124,209),(125,208),(126,207),(127,206),(128,205),(129,204),(130,203),(131,202),(132,201),(133,220),(134,219),(135,218),(136,217),(137,216),(138,215),(139,214),(140,213)], [(1,185,169,215,228,57,11,195,179,205,238,47),(2,196,170,206,229,48,12,186,180,216,239,58),(3,187,171,217,230,59,13,197,161,207,240,49),(4,198,172,208,231,50,14,188,162,218,221,60),(5,189,173,219,232,41,15,199,163,209,222,51),(6,200,174,210,233,52,16,190,164,220,223,42),(7,191,175,201,234,43,17,181,165,211,224,53),(8,182,176,212,235,54,18,192,166,202,225,44),(9,193,177,203,236,45,19,183,167,213,226,55),(10,184,178,214,237,56,20,194,168,204,227,46),(21,61,156,111,94,133,31,71,146,101,84,123),(22,72,157,102,95,124,32,62,147,112,85,134),(23,63,158,113,96,135,33,73,148,103,86,125),(24,74,159,104,97,126,34,64,149,114,87,136),(25,65,160,115,98,137,35,75,150,105,88,127),(26,76,141,106,99,128,36,66,151,116,89,138),(27,67,142,117,100,139,37,77,152,107,90,129),(28,78,143,108,81,130,38,68,153,118,91,140),(29,69,144,119,82,121,39,79,154,109,92,131),(30,80,145,110,83,132,40,70,155,120,93,122)], [(1,42,11,52),(2,53,12,43),(3,44,13,54),(4,55,14,45),(5,46,15,56),(6,57,16,47),(7,48,17,58),(8,59,18,49),(9,50,19,60),(10,41,20,51),(21,116,31,106),(22,107,32,117),(23,118,33,108),(24,109,34,119),(25,120,35,110),(26,111,36,101),(27,102,37,112),(28,113,38,103),(29,104,39,114),(30,115,40,105),(61,151,71,141),(62,142,72,152),(63,153,73,143),(64,144,74,154),(65,155,75,145),(66,146,76,156),(67,157,77,147),(68,148,78,158),(69,159,79,149),(70,150,80,160),(81,125,91,135),(82,136,92,126),(83,127,93,137),(84,138,94,128),(85,129,95,139),(86,140,96,130),(87,131,97,121),(88,122,98,132),(89,133,99,123),(90,124,100,134),(161,212,171,202),(162,203,172,213),(163,214,173,204),(164,205,174,215),(165,216,175,206),(166,207,176,217),(167,218,177,208),(168,209,178,219),(169,220,179,210),(170,211,180,201),(181,239,191,229),(182,230,192,240),(183,221,193,231),(184,232,194,222),(185,223,195,233),(186,234,196,224),(187,225,197,235),(188,236,198,226),(189,227,199,237),(190,238,200,228)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 20A | ··· | 20F | 20G | ··· | 20L | 24A | 24B | 30A | 30B | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | 24 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 20 | 2 | 2 | 4 | 6 | 12 | 60 | 2 | 2 | 2 | 40 | 20 | 60 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 8 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 | 20 | 20 | 4 | 4 | 8 | ··· | 8 |
48 irreducible representations
| dim | 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 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8.C22 | S3×D4 | S3×D5 | D4.D6 | C2×S3×D5 | C20.C23 | S3×C5⋊D4 | D20.28D6 |
| kernel | D20.28D6 | D6.Dic5 | C30.D4 | C5⋊Dic12 | C3×Q8⋊D5 | C15⋊7Q16 | D20⋊5S3 | C5×S3×Q8 | Q8⋊D5 | C5×Dic3 | S3×C10 | S3×Q8 | C5⋊2C8 | D20 | C5×Q8 | Dic6 | C4×S3 | C3×Q8 | Dic3 | D6 | C15 | C10 | Q8 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D20.28D6 ►in GL8(𝔽241)
| 51 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 191 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 191 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 123 | 0 | 0 |
| 0 | 0 | 0 | 0 | 192 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 215 | 1 | 49 |
| 0 | 0 | 0 | 0 | 215 | 106 | 118 | 240 |
| 59 | 38 | 0 | 0 | 0 | 0 | 0 | 0 |
| 86 | 182 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 59 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 86 | 182 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 184 | 135 | 123 | 2 |
| 0 | 0 | 0 | 0 | 99 | 191 | 2 | 49 |
| 0 | 0 | 0 | 0 | 79 | 48 | 24 | 177 |
| 0 | 0 | 0 | 0 | 90 | 53 | 87 | 83 |
| 165 | 138 | 165 | 138 | 0 | 0 | 0 | 0 |
| 152 | 76 | 152 | 76 | 0 | 0 | 0 | 0 |
| 76 | 103 | 0 | 0 | 0 | 0 | 0 | 0 |
| 89 | 165 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 43 | 195 | 0 | 0 |
| 0 | 0 | 0 | 0 | 145 | 198 | 0 | 0 |
| 0 | 0 | 0 | 0 | 148 | 83 | 199 | 145 |
| 0 | 0 | 0 | 0 | 29 | 74 | 46 | 42 |
| 76 | 103 | 76 | 103 | 0 | 0 | 0 | 0 |
| 89 | 165 | 89 | 165 | 0 | 0 | 0 | 0 |
| 0 | 0 | 165 | 138 | 0 | 0 | 0 | 0 |
| 0 | 0 | 152 | 76 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 199 | 59 | 0 | 0 |
| 0 | 0 | 0 | 0 | 207 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 150 | 48 | 198 | 207 |
| 0 | 0 | 0 | 0 | 59 | 188 | 182 | 43 |
G:=sub<GL(8,GF(241))| [51,191,0,0,0,0,0,0,240,1,0,0,0,0,0,0,0,0,51,191,0,0,0,0,0,0,240,1,0,0,0,0,0,0,0,0,240,192,64,215,0,0,0,0,123,1,215,106,0,0,0,0,0,0,1,118,0,0,0,0,0,0,49,240],[59,86,0,0,0,0,0,0,38,182,0,0,0,0,0,0,0,0,59,86,0,0,0,0,0,0,38,182,0,0,0,0,0,0,0,0,184,99,79,90,0,0,0,0,135,191,48,53,0,0,0,0,123,2,24,87,0,0,0,0,2,49,177,83],[165,152,76,89,0,0,0,0,138,76,103,165,0,0,0,0,165,152,0,0,0,0,0,0,138,76,0,0,0,0,0,0,0,0,0,0,43,145,148,29,0,0,0,0,195,198,83,74,0,0,0,0,0,0,199,46,0,0,0,0,0,0,145,42],[76,89,0,0,0,0,0,0,103,165,0,0,0,0,0,0,76,89,165,152,0,0,0,0,103,165,138,76,0,0,0,0,0,0,0,0,199,207,150,59,0,0,0,0,59,42,48,188,0,0,0,0,0,0,198,182,0,0,0,0,0,0,207,43] >;
D20.28D6 in GAP, Magma, Sage, TeX
D_{20}._{28}D_6 % in TeX
G:=Group("D20.28D6"); // GroupNames label
G:=SmallGroup(480,594);
// by ID
G=gap.SmallGroup(480,594);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,100,675,185,80,1356,18822]);
// 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=d*a*d^-1=a^11,c*b*c^-1=d*b*d^-1=a^15*b,d*c*d^-1=a^10*c^5>;
// generators/relations