metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.16D4, C6.90(D4×D5), C10.91(S3×D4), C30.242(C2×D4), C5⋊4(C23.9D6), C23.D5⋊12S3, D30⋊4C4⋊33C2, C23.23(S3×D5), C6.86(C4○D20), C6.D4⋊12D5, C6.Dic10⋊37C2, C30.Q8⋊36C2, (C2×Dic5).64D6, (C22×C6).41D10, (C22×C10).56D6, C10.86(C4○D12), C30.155(C4○D4), C6.59(D4⋊2D5), C3⋊4(D10.12D4), (C2×C30).204C23, C2.41(D10⋊D6), C10.58(D4⋊2S3), (C2×Dic3).126D10, C15⋊24(C22.D4), (C22×C30).66C22, C2.29(Dic3.D10), C2.29(Dic5.D6), (C6×Dic5).119C22, (C22×D15).67C22, (C2×Dic15).140C22, (C10×Dic3).119C22, C22.233(C2×S3×D5), (C2×D30.C2)⋊18C2, (C2×C15⋊7D4).13C2, (C3×C23.D5)⋊14C2, (C5×C6.D4)⋊14C2, (C2×C6).216(C22×D5), (C2×C10).216(C22×S3), SmallGroup(480,638)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.16D4
G = < a,b,c,d | a30=b2=c4=1, d2=a15, bab=a-1, cac-1=dad-1=a19, cbc-1=a3b, dbd-1=a18b, dcd-1=a15c-1 >
Subgroups: 876 in 156 conjugacy classes, 46 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, D15, C30, C30, C22.D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, D30, D30, C2×C30, C2×C30, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C5⋊D4, C23.9D6, D30.C2, C6×Dic5, C10×Dic3, C2×Dic15, C15⋊7D4, C22×D15, C22×C30, D10.12D4, D30⋊4C4, C30.Q8, C6.Dic10, C3×C23.D5, C5×C6.D4, C2×D30.C2, C2×C15⋊7D4, D30.16D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C22.D4, C22×D5, C4○D12, S3×D4, D4⋊2S3, S3×D5, C4○D20, D4×D5, D4⋊2D5, C23.9D6, C2×S3×D5, D10.12D4, Dic5.D6, Dic3.D10, D10⋊D6, D30.16D4
►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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 60)(32 59)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(91 95)(92 94)(96 120)(97 119)(98 118)(99 117)(100 116)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)(121 150)(122 149)(123 148)(124 147)(125 146)(126 145)(127 144)(128 143)(129 142)(130 141)(131 140)(132 139)(133 138)(134 137)(135 136)(151 180)(152 179)(153 178)(154 177)(155 176)(156 175)(157 174)(158 173)(159 172)(160 171)(161 170)(162 169)(163 168)(164 167)(165 166)(181 205)(182 204)(183 203)(184 202)(185 201)(186 200)(187 199)(188 198)(189 197)(190 196)(191 195)(192 194)(206 210)(207 209)(211 215)(212 214)(216 240)(217 239)(218 238)(219 237)(220 236)(221 235)(222 234)(223 233)(224 232)(225 231)(226 230)(227 229)
(1 221 31 201)(2 240 32 190)(3 229 33 209)(4 218 34 198)(5 237 35 187)(6 226 36 206)(7 215 37 195)(8 234 38 184)(9 223 39 203)(10 212 40 192)(11 231 41 181)(12 220 42 200)(13 239 43 189)(14 228 44 208)(15 217 45 197)(16 236 46 186)(17 225 47 205)(18 214 48 194)(19 233 49 183)(20 222 50 202)(21 211 51 191)(22 230 52 210)(23 219 53 199)(24 238 54 188)(25 227 55 207)(26 216 56 196)(27 235 57 185)(28 224 58 204)(29 213 59 193)(30 232 60 182)(61 151 116 121)(62 170 117 140)(63 159 118 129)(64 178 119 148)(65 167 120 137)(66 156 91 126)(67 175 92 145)(68 164 93 134)(69 153 94 123)(70 172 95 142)(71 161 96 131)(72 180 97 150)(73 169 98 139)(74 158 99 128)(75 177 100 147)(76 166 101 136)(77 155 102 125)(78 174 103 144)(79 163 104 133)(80 152 105 122)(81 171 106 141)(82 160 107 130)(83 179 108 149)(84 168 109 138)(85 157 110 127)(86 176 111 146)(87 165 112 135)(88 154 113 124)(89 173 114 143)(90 162 115 132)
(1 121 16 136)(2 140 17 125)(3 129 18 144)(4 148 19 133)(5 137 20 122)(6 126 21 141)(7 145 22 130)(8 134 23 149)(9 123 24 138)(10 142 25 127)(11 131 26 146)(12 150 27 135)(13 139 28 124)(14 128 29 143)(15 147 30 132)(31 151 46 166)(32 170 47 155)(33 159 48 174)(34 178 49 163)(35 167 50 152)(36 156 51 171)(37 175 52 160)(38 164 53 179)(39 153 54 168)(40 172 55 157)(41 161 56 176)(42 180 57 165)(43 169 58 154)(44 158 59 173)(45 177 60 162)(61 201 76 186)(62 190 77 205)(63 209 78 194)(64 198 79 183)(65 187 80 202)(66 206 81 191)(67 195 82 210)(68 184 83 199)(69 203 84 188)(70 192 85 207)(71 181 86 196)(72 200 87 185)(73 189 88 204)(74 208 89 193)(75 197 90 182)(91 226 106 211)(92 215 107 230)(93 234 108 219)(94 223 109 238)(95 212 110 227)(96 231 111 216)(97 220 112 235)(98 239 113 224)(99 228 114 213)(100 217 115 232)(101 236 116 221)(102 225 117 240)(103 214 118 229)(104 233 119 218)(105 222 120 237)
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(91,95)(92,94)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109)(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)(127,144)(128,143)(129,142)(130,141)(131,140)(132,139)(133,138)(134,137)(135,136)(151,180)(152,179)(153,178)(154,177)(155,176)(156,175)(157,174)(158,173)(159,172)(160,171)(161,170)(162,169)(163,168)(164,167)(165,166)(181,205)(182,204)(183,203)(184,202)(185,201)(186,200)(187,199)(188,198)(189,197)(190,196)(191,195)(192,194)(206,210)(207,209)(211,215)(212,214)(216,240)(217,239)(218,238)(219,237)(220,236)(221,235)(222,234)(223,233)(224,232)(225,231)(226,230)(227,229), (1,221,31,201)(2,240,32,190)(3,229,33,209)(4,218,34,198)(5,237,35,187)(6,226,36,206)(7,215,37,195)(8,234,38,184)(9,223,39,203)(10,212,40,192)(11,231,41,181)(12,220,42,200)(13,239,43,189)(14,228,44,208)(15,217,45,197)(16,236,46,186)(17,225,47,205)(18,214,48,194)(19,233,49,183)(20,222,50,202)(21,211,51,191)(22,230,52,210)(23,219,53,199)(24,238,54,188)(25,227,55,207)(26,216,56,196)(27,235,57,185)(28,224,58,204)(29,213,59,193)(30,232,60,182)(61,151,116,121)(62,170,117,140)(63,159,118,129)(64,178,119,148)(65,167,120,137)(66,156,91,126)(67,175,92,145)(68,164,93,134)(69,153,94,123)(70,172,95,142)(71,161,96,131)(72,180,97,150)(73,169,98,139)(74,158,99,128)(75,177,100,147)(76,166,101,136)(77,155,102,125)(78,174,103,144)(79,163,104,133)(80,152,105,122)(81,171,106,141)(82,160,107,130)(83,179,108,149)(84,168,109,138)(85,157,110,127)(86,176,111,146)(87,165,112,135)(88,154,113,124)(89,173,114,143)(90,162,115,132), (1,121,16,136)(2,140,17,125)(3,129,18,144)(4,148,19,133)(5,137,20,122)(6,126,21,141)(7,145,22,130)(8,134,23,149)(9,123,24,138)(10,142,25,127)(11,131,26,146)(12,150,27,135)(13,139,28,124)(14,128,29,143)(15,147,30,132)(31,151,46,166)(32,170,47,155)(33,159,48,174)(34,178,49,163)(35,167,50,152)(36,156,51,171)(37,175,52,160)(38,164,53,179)(39,153,54,168)(40,172,55,157)(41,161,56,176)(42,180,57,165)(43,169,58,154)(44,158,59,173)(45,177,60,162)(61,201,76,186)(62,190,77,205)(63,209,78,194)(64,198,79,183)(65,187,80,202)(66,206,81,191)(67,195,82,210)(68,184,83,199)(69,203,84,188)(70,192,85,207)(71,181,86,196)(72,200,87,185)(73,189,88,204)(74,208,89,193)(75,197,90,182)(91,226,106,211)(92,215,107,230)(93,234,108,219)(94,223,109,238)(95,212,110,227)(96,231,111,216)(97,220,112,235)(98,239,113,224)(99,228,114,213)(100,217,115,232)(101,236,116,221)(102,225,117,240)(103,214,118,229)(104,233,119,218)(105,222,120,237)>;
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(91,95)(92,94)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109)(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)(127,144)(128,143)(129,142)(130,141)(131,140)(132,139)(133,138)(134,137)(135,136)(151,180)(152,179)(153,178)(154,177)(155,176)(156,175)(157,174)(158,173)(159,172)(160,171)(161,170)(162,169)(163,168)(164,167)(165,166)(181,205)(182,204)(183,203)(184,202)(185,201)(186,200)(187,199)(188,198)(189,197)(190,196)(191,195)(192,194)(206,210)(207,209)(211,215)(212,214)(216,240)(217,239)(218,238)(219,237)(220,236)(221,235)(222,234)(223,233)(224,232)(225,231)(226,230)(227,229), (1,221,31,201)(2,240,32,190)(3,229,33,209)(4,218,34,198)(5,237,35,187)(6,226,36,206)(7,215,37,195)(8,234,38,184)(9,223,39,203)(10,212,40,192)(11,231,41,181)(12,220,42,200)(13,239,43,189)(14,228,44,208)(15,217,45,197)(16,236,46,186)(17,225,47,205)(18,214,48,194)(19,233,49,183)(20,222,50,202)(21,211,51,191)(22,230,52,210)(23,219,53,199)(24,238,54,188)(25,227,55,207)(26,216,56,196)(27,235,57,185)(28,224,58,204)(29,213,59,193)(30,232,60,182)(61,151,116,121)(62,170,117,140)(63,159,118,129)(64,178,119,148)(65,167,120,137)(66,156,91,126)(67,175,92,145)(68,164,93,134)(69,153,94,123)(70,172,95,142)(71,161,96,131)(72,180,97,150)(73,169,98,139)(74,158,99,128)(75,177,100,147)(76,166,101,136)(77,155,102,125)(78,174,103,144)(79,163,104,133)(80,152,105,122)(81,171,106,141)(82,160,107,130)(83,179,108,149)(84,168,109,138)(85,157,110,127)(86,176,111,146)(87,165,112,135)(88,154,113,124)(89,173,114,143)(90,162,115,132), (1,121,16,136)(2,140,17,125)(3,129,18,144)(4,148,19,133)(5,137,20,122)(6,126,21,141)(7,145,22,130)(8,134,23,149)(9,123,24,138)(10,142,25,127)(11,131,26,146)(12,150,27,135)(13,139,28,124)(14,128,29,143)(15,147,30,132)(31,151,46,166)(32,170,47,155)(33,159,48,174)(34,178,49,163)(35,167,50,152)(36,156,51,171)(37,175,52,160)(38,164,53,179)(39,153,54,168)(40,172,55,157)(41,161,56,176)(42,180,57,165)(43,169,58,154)(44,158,59,173)(45,177,60,162)(61,201,76,186)(62,190,77,205)(63,209,78,194)(64,198,79,183)(65,187,80,202)(66,206,81,191)(67,195,82,210)(68,184,83,199)(69,203,84,188)(70,192,85,207)(71,181,86,196)(72,200,87,185)(73,189,88,204)(74,208,89,193)(75,197,90,182)(91,226,106,211)(92,215,107,230)(93,234,108,219)(94,223,109,238)(95,212,110,227)(96,231,111,216)(97,220,112,235)(98,239,113,224)(99,228,114,213)(100,217,115,232)(101,236,116,221)(102,225,117,240)(103,214,118,229)(104,233,119,218)(105,222,120,237) );
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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(31,60),(32,59),(33,58),(34,57),(35,56),(36,55),(37,54),(38,53),(39,52),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46),(61,75),(62,74),(63,73),(64,72),(65,71),(66,70),(67,69),(76,90),(77,89),(78,88),(79,87),(80,86),(81,85),(82,84),(91,95),(92,94),(96,120),(97,119),(98,118),(99,117),(100,116),(101,115),(102,114),(103,113),(104,112),(105,111),(106,110),(107,109),(121,150),(122,149),(123,148),(124,147),(125,146),(126,145),(127,144),(128,143),(129,142),(130,141),(131,140),(132,139),(133,138),(134,137),(135,136),(151,180),(152,179),(153,178),(154,177),(155,176),(156,175),(157,174),(158,173),(159,172),(160,171),(161,170),(162,169),(163,168),(164,167),(165,166),(181,205),(182,204),(183,203),(184,202),(185,201),(186,200),(187,199),(188,198),(189,197),(190,196),(191,195),(192,194),(206,210),(207,209),(211,215),(212,214),(216,240),(217,239),(218,238),(219,237),(220,236),(221,235),(222,234),(223,233),(224,232),(225,231),(226,230),(227,229)], [(1,221,31,201),(2,240,32,190),(3,229,33,209),(4,218,34,198),(5,237,35,187),(6,226,36,206),(7,215,37,195),(8,234,38,184),(9,223,39,203),(10,212,40,192),(11,231,41,181),(12,220,42,200),(13,239,43,189),(14,228,44,208),(15,217,45,197),(16,236,46,186),(17,225,47,205),(18,214,48,194),(19,233,49,183),(20,222,50,202),(21,211,51,191),(22,230,52,210),(23,219,53,199),(24,238,54,188),(25,227,55,207),(26,216,56,196),(27,235,57,185),(28,224,58,204),(29,213,59,193),(30,232,60,182),(61,151,116,121),(62,170,117,140),(63,159,118,129),(64,178,119,148),(65,167,120,137),(66,156,91,126),(67,175,92,145),(68,164,93,134),(69,153,94,123),(70,172,95,142),(71,161,96,131),(72,180,97,150),(73,169,98,139),(74,158,99,128),(75,177,100,147),(76,166,101,136),(77,155,102,125),(78,174,103,144),(79,163,104,133),(80,152,105,122),(81,171,106,141),(82,160,107,130),(83,179,108,149),(84,168,109,138),(85,157,110,127),(86,176,111,146),(87,165,112,135),(88,154,113,124),(89,173,114,143),(90,162,115,132)], [(1,121,16,136),(2,140,17,125),(3,129,18,144),(4,148,19,133),(5,137,20,122),(6,126,21,141),(7,145,22,130),(8,134,23,149),(9,123,24,138),(10,142,25,127),(11,131,26,146),(12,150,27,135),(13,139,28,124),(14,128,29,143),(15,147,30,132),(31,151,46,166),(32,170,47,155),(33,159,48,174),(34,178,49,163),(35,167,50,152),(36,156,51,171),(37,175,52,160),(38,164,53,179),(39,153,54,168),(40,172,55,157),(41,161,56,176),(42,180,57,165),(43,169,58,154),(44,158,59,173),(45,177,60,162),(61,201,76,186),(62,190,77,205),(63,209,78,194),(64,198,79,183),(65,187,80,202),(66,206,81,191),(67,195,82,210),(68,184,83,199),(69,203,84,188),(70,192,85,207),(71,181,86,196),(72,200,87,185),(73,189,88,204),(74,208,89,193),(75,197,90,182),(91,226,106,211),(92,215,107,230),(93,234,108,219),(94,223,109,238),(95,212,110,227),(96,231,111,216),(97,220,112,235),(98,239,113,224),(99,228,114,213),(100,217,115,232),(101,236,116,221),(102,225,117,240),(103,214,118,229),(104,233,119,218),(105,222,120,237)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 4 | 30 | 30 | 2 | 6 | 6 | 10 | 10 | 12 | 20 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C4○D12 | C4○D20 | S3×D4 | D4⋊2S3 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | Dic5.D6 | Dic3.D10 | D10⋊D6 |
| kernel | D30.16D4 | D30⋊4C4 | C30.Q8 | C6.Dic10 | C3×C23.D5 | C5×C6.D4 | C2×D30.C2 | C2×C15⋊7D4 | C23.D5 | D30 | C6.D4 | C2×Dic5 | C22×C10 | C30 | C2×Dic3 | C22×C6 | C10 | C6 | C10 | C10 | C23 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D30.16D4 ►in GL4(𝔽61) generated by
| 0 | 1 | 0 | 0 |
| 60 | 18 | 0 | 0 |
| 0 | 0 | 1 | 60 |
| 0 | 0 | 1 | 0 |
| 43 | 1 | 0 | 0 |
| 43 | 18 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 60 | 1 |
| 27 | 27 | 0 | 0 |
| 25 | 34 | 0 | 0 |
| 0 | 0 | 38 | 46 |
| 0 | 0 | 15 | 23 |
| 15 | 46 | 0 | 0 |
| 11 | 46 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
G:=sub<GL(4,GF(61))| [0,60,0,0,1,18,0,0,0,0,1,1,0,0,60,0],[43,43,0,0,1,18,0,0,0,0,60,60,0,0,0,1],[27,25,0,0,27,34,0,0,0,0,38,15,0,0,46,23],[15,11,0,0,46,46,0,0,0,0,11,0,0,0,0,11] >;
D30.16D4 in GAP, Magma, Sage, TeX
D_{30}._{16}D_4 % in TeX
G:=Group("D30.16D4"); // GroupNames label
G:=SmallGroup(480,638);
// by ID
G=gap.SmallGroup(480,638);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,303,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=a^15,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^19,c*b*c^-1=a^3*b,d*b*d^-1=a^18*b,d*c*d^-1=a^15*c^-1>;
// generators/relations