metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊12D4, Dic15⋊15D4, D6⋊C4⋊12D5, C6.64(D4×D5), C5⋊2(Dic3⋊D4), C10.66(S3×D4), C3⋊2(D10⋊D4), C15⋊13(C4⋊D4), (C2×C20).203D6, C30.155(C2×D4), D10⋊C4⋊12S3, C30.93(C4○D4), C6.41(C4○D20), (C2×C12).201D10, Dic15⋊5C4⋊26C2, (C2×Dic5).47D6, (C22×D5).21D6, C10.44(C4○D12), C2.22(C20⋊D6), (C2×C60).169C22, (C2×C30).151C23, (C2×Dic3).47D10, (C22×S3).19D10, C2.17(D10⋊D6), (C6×Dic5).91C22, C2.30(D6.D10), (C10×Dic3).91C22, (C2×Dic15).213C22, (C22×D15).108C22, (C2×C4×D15)⋊11C2, (C5×D6⋊C4)⋊12C2, (C2×C3⋊D20)⋊7C2, (C2×C15⋊D4)⋊7C2, (C2×C5⋊D12)⋊7C2, (C2×C4).182(S3×D5), (D5×C2×C6).35C22, C22.203(C2×S3×D5), (S3×C2×C10).35C22, (C3×D10⋊C4)⋊12C2, (C2×C6).163(C22×D5), (C2×C10).163(C22×S3), SmallGroup(480,537)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊12D4
G = < a,b,c,d | a30=b2=c4=d2=1, bab=cac-1=a-1, dad=a19, cbc-1=a28b, dbd=a3b, dcd=c-1 >
Subgroups: 1196 in 188 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, 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, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, C3×D5, D15, C30, C4⋊D4, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, D30, C2×C30, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, Dic3⋊D4, C15⋊D4, C3⋊D20, C5⋊D12, C6×Dic5, C10×Dic3, C4×D15, C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, C22×D15, D10⋊D4, Dic15⋊5C4, C3×D10⋊C4, C5×D6⋊C4, C2×C15⋊D4, C2×C3⋊D20, C2×C5⋊D12, C2×C4×D15, D30⋊12D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C4⋊D4, C22×D5, C4○D12, S3×D4, S3×D5, C4○D20, D4×D5, Dic3⋊D4, C2×S3×D5, D10⋊D4, D6.D10, C20⋊D6, D10⋊D6, D30⋊12D4
►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)
(1 209)(2 208)(3 207)(4 206)(5 205)(6 204)(7 203)(8 202)(9 201)(10 200)(11 199)(12 198)(13 197)(14 196)(15 195)(16 194)(17 193)(18 192)(19 191)(20 190)(21 189)(22 188)(23 187)(24 186)(25 185)(26 184)(27 183)(28 182)(29 181)(30 210)(31 74)(32 73)(33 72)(34 71)(35 70)(36 69)(37 68)(38 67)(39 66)(40 65)(41 64)(42 63)(43 62)(44 61)(45 90)(46 89)(47 88)(48 87)(49 86)(50 85)(51 84)(52 83)(53 82)(54 81)(55 80)(56 79)(57 78)(58 77)(59 76)(60 75)(91 146)(92 145)(93 144)(94 143)(95 142)(96 141)(97 140)(98 139)(99 138)(100 137)(101 136)(102 135)(103 134)(104 133)(105 132)(106 131)(107 130)(108 129)(109 128)(110 127)(111 126)(112 125)(113 124)(114 123)(115 122)(116 121)(117 150)(118 149)(119 148)(120 147)(151 227)(152 226)(153 225)(154 224)(155 223)(156 222)(157 221)(158 220)(159 219)(160 218)(161 217)(162 216)(163 215)(164 214)(165 213)(166 212)(167 211)(168 240)(169 239)(170 238)(171 237)(172 236)(173 235)(174 234)(175 233)(176 232)(177 231)(178 230)(179 229)(180 228)
(1 133 195 120)(2 132 196 119)(3 131 197 118)(4 130 198 117)(5 129 199 116)(6 128 200 115)(7 127 201 114)(8 126 202 113)(9 125 203 112)(10 124 204 111)(11 123 205 110)(12 122 206 109)(13 121 207 108)(14 150 208 107)(15 149 209 106)(16 148 210 105)(17 147 181 104)(18 146 182 103)(19 145 183 102)(20 144 184 101)(21 143 185 100)(22 142 186 99)(23 141 187 98)(24 140 188 97)(25 139 189 96)(26 138 190 95)(27 137 191 94)(28 136 192 93)(29 135 193 92)(30 134 194 91)(31 172 81 231)(32 171 82 230)(33 170 83 229)(34 169 84 228)(35 168 85 227)(36 167 86 226)(37 166 87 225)(38 165 88 224)(39 164 89 223)(40 163 90 222)(41 162 61 221)(42 161 62 220)(43 160 63 219)(44 159 64 218)(45 158 65 217)(46 157 66 216)(47 156 67 215)(48 155 68 214)(49 154 69 213)(50 153 70 212)(51 152 71 211)(52 151 72 240)(53 180 73 239)(54 179 74 238)(55 178 75 237)(56 177 76 236)(57 176 77 235)(58 175 78 234)(59 174 79 233)(60 173 80 232)
(1 234)(2 223)(3 212)(4 231)(5 220)(6 239)(7 228)(8 217)(9 236)(10 225)(11 214)(12 233)(13 222)(14 211)(15 230)(16 219)(17 238)(18 227)(19 216)(20 235)(21 224)(22 213)(23 232)(24 221)(25 240)(26 229)(27 218)(28 237)(29 226)(30 215)(31 117)(32 106)(33 95)(34 114)(35 103)(36 92)(37 111)(38 100)(39 119)(40 108)(41 97)(42 116)(43 105)(44 94)(45 113)(46 102)(47 91)(48 110)(49 99)(50 118)(51 107)(52 96)(53 115)(54 104)(55 93)(56 112)(57 101)(58 120)(59 109)(60 98)(61 140)(62 129)(63 148)(64 137)(65 126)(66 145)(67 134)(68 123)(69 142)(70 131)(71 150)(72 139)(73 128)(74 147)(75 136)(76 125)(77 144)(78 133)(79 122)(80 141)(81 130)(82 149)(83 138)(84 127)(85 146)(86 135)(87 124)(88 143)(89 132)(90 121)(151 189)(152 208)(153 197)(154 186)(155 205)(156 194)(157 183)(158 202)(159 191)(160 210)(161 199)(162 188)(163 207)(164 196)(165 185)(166 204)(167 193)(168 182)(169 201)(170 190)(171 209)(172 198)(173 187)(174 206)(175 195)(176 184)(177 203)(178 192)(179 181)(180 200)
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), (1,209)(2,208)(3,207)(4,206)(5,205)(6,204)(7,203)(8,202)(9,201)(10,200)(11,199)(12,198)(13,197)(14,196)(15,195)(16,194)(17,193)(18,192)(19,191)(20,190)(21,189)(22,188)(23,187)(24,186)(25,185)(26,184)(27,183)(28,182)(29,181)(30,210)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(91,146)(92,145)(93,144)(94,143)(95,142)(96,141)(97,140)(98,139)(99,138)(100,137)(101,136)(102,135)(103,134)(104,133)(105,132)(106,131)(107,130)(108,129)(109,128)(110,127)(111,126)(112,125)(113,124)(114,123)(115,122)(116,121)(117,150)(118,149)(119,148)(120,147)(151,227)(152,226)(153,225)(154,224)(155,223)(156,222)(157,221)(158,220)(159,219)(160,218)(161,217)(162,216)(163,215)(164,214)(165,213)(166,212)(167,211)(168,240)(169,239)(170,238)(171,237)(172,236)(173,235)(174,234)(175,233)(176,232)(177,231)(178,230)(179,229)(180,228), (1,133,195,120)(2,132,196,119)(3,131,197,118)(4,130,198,117)(5,129,199,116)(6,128,200,115)(7,127,201,114)(8,126,202,113)(9,125,203,112)(10,124,204,111)(11,123,205,110)(12,122,206,109)(13,121,207,108)(14,150,208,107)(15,149,209,106)(16,148,210,105)(17,147,181,104)(18,146,182,103)(19,145,183,102)(20,144,184,101)(21,143,185,100)(22,142,186,99)(23,141,187,98)(24,140,188,97)(25,139,189,96)(26,138,190,95)(27,137,191,94)(28,136,192,93)(29,135,193,92)(30,134,194,91)(31,172,81,231)(32,171,82,230)(33,170,83,229)(34,169,84,228)(35,168,85,227)(36,167,86,226)(37,166,87,225)(38,165,88,224)(39,164,89,223)(40,163,90,222)(41,162,61,221)(42,161,62,220)(43,160,63,219)(44,159,64,218)(45,158,65,217)(46,157,66,216)(47,156,67,215)(48,155,68,214)(49,154,69,213)(50,153,70,212)(51,152,71,211)(52,151,72,240)(53,180,73,239)(54,179,74,238)(55,178,75,237)(56,177,76,236)(57,176,77,235)(58,175,78,234)(59,174,79,233)(60,173,80,232), (1,234)(2,223)(3,212)(4,231)(5,220)(6,239)(7,228)(8,217)(9,236)(10,225)(11,214)(12,233)(13,222)(14,211)(15,230)(16,219)(17,238)(18,227)(19,216)(20,235)(21,224)(22,213)(23,232)(24,221)(25,240)(26,229)(27,218)(28,237)(29,226)(30,215)(31,117)(32,106)(33,95)(34,114)(35,103)(36,92)(37,111)(38,100)(39,119)(40,108)(41,97)(42,116)(43,105)(44,94)(45,113)(46,102)(47,91)(48,110)(49,99)(50,118)(51,107)(52,96)(53,115)(54,104)(55,93)(56,112)(57,101)(58,120)(59,109)(60,98)(61,140)(62,129)(63,148)(64,137)(65,126)(66,145)(67,134)(68,123)(69,142)(70,131)(71,150)(72,139)(73,128)(74,147)(75,136)(76,125)(77,144)(78,133)(79,122)(80,141)(81,130)(82,149)(83,138)(84,127)(85,146)(86,135)(87,124)(88,143)(89,132)(90,121)(151,189)(152,208)(153,197)(154,186)(155,205)(156,194)(157,183)(158,202)(159,191)(160,210)(161,199)(162,188)(163,207)(164,196)(165,185)(166,204)(167,193)(168,182)(169,201)(170,190)(171,209)(172,198)(173,187)(174,206)(175,195)(176,184)(177,203)(178,192)(179,181)(180,200)>;
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), (1,209)(2,208)(3,207)(4,206)(5,205)(6,204)(7,203)(8,202)(9,201)(10,200)(11,199)(12,198)(13,197)(14,196)(15,195)(16,194)(17,193)(18,192)(19,191)(20,190)(21,189)(22,188)(23,187)(24,186)(25,185)(26,184)(27,183)(28,182)(29,181)(30,210)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(91,146)(92,145)(93,144)(94,143)(95,142)(96,141)(97,140)(98,139)(99,138)(100,137)(101,136)(102,135)(103,134)(104,133)(105,132)(106,131)(107,130)(108,129)(109,128)(110,127)(111,126)(112,125)(113,124)(114,123)(115,122)(116,121)(117,150)(118,149)(119,148)(120,147)(151,227)(152,226)(153,225)(154,224)(155,223)(156,222)(157,221)(158,220)(159,219)(160,218)(161,217)(162,216)(163,215)(164,214)(165,213)(166,212)(167,211)(168,240)(169,239)(170,238)(171,237)(172,236)(173,235)(174,234)(175,233)(176,232)(177,231)(178,230)(179,229)(180,228), (1,133,195,120)(2,132,196,119)(3,131,197,118)(4,130,198,117)(5,129,199,116)(6,128,200,115)(7,127,201,114)(8,126,202,113)(9,125,203,112)(10,124,204,111)(11,123,205,110)(12,122,206,109)(13,121,207,108)(14,150,208,107)(15,149,209,106)(16,148,210,105)(17,147,181,104)(18,146,182,103)(19,145,183,102)(20,144,184,101)(21,143,185,100)(22,142,186,99)(23,141,187,98)(24,140,188,97)(25,139,189,96)(26,138,190,95)(27,137,191,94)(28,136,192,93)(29,135,193,92)(30,134,194,91)(31,172,81,231)(32,171,82,230)(33,170,83,229)(34,169,84,228)(35,168,85,227)(36,167,86,226)(37,166,87,225)(38,165,88,224)(39,164,89,223)(40,163,90,222)(41,162,61,221)(42,161,62,220)(43,160,63,219)(44,159,64,218)(45,158,65,217)(46,157,66,216)(47,156,67,215)(48,155,68,214)(49,154,69,213)(50,153,70,212)(51,152,71,211)(52,151,72,240)(53,180,73,239)(54,179,74,238)(55,178,75,237)(56,177,76,236)(57,176,77,235)(58,175,78,234)(59,174,79,233)(60,173,80,232), (1,234)(2,223)(3,212)(4,231)(5,220)(6,239)(7,228)(8,217)(9,236)(10,225)(11,214)(12,233)(13,222)(14,211)(15,230)(16,219)(17,238)(18,227)(19,216)(20,235)(21,224)(22,213)(23,232)(24,221)(25,240)(26,229)(27,218)(28,237)(29,226)(30,215)(31,117)(32,106)(33,95)(34,114)(35,103)(36,92)(37,111)(38,100)(39,119)(40,108)(41,97)(42,116)(43,105)(44,94)(45,113)(46,102)(47,91)(48,110)(49,99)(50,118)(51,107)(52,96)(53,115)(54,104)(55,93)(56,112)(57,101)(58,120)(59,109)(60,98)(61,140)(62,129)(63,148)(64,137)(65,126)(66,145)(67,134)(68,123)(69,142)(70,131)(71,150)(72,139)(73,128)(74,147)(75,136)(76,125)(77,144)(78,133)(79,122)(80,141)(81,130)(82,149)(83,138)(84,127)(85,146)(86,135)(87,124)(88,143)(89,132)(90,121)(151,189)(152,208)(153,197)(154,186)(155,205)(156,194)(157,183)(158,202)(159,191)(160,210)(161,199)(162,188)(163,207)(164,196)(165,185)(166,204)(167,193)(168,182)(169,201)(170,190)(171,209)(172,198)(173,187)(174,206)(175,195)(176,184)(177,203)(178,192)(179,181)(180,200) );
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)], [(1,209),(2,208),(3,207),(4,206),(5,205),(6,204),(7,203),(8,202),(9,201),(10,200),(11,199),(12,198),(13,197),(14,196),(15,195),(16,194),(17,193),(18,192),(19,191),(20,190),(21,189),(22,188),(23,187),(24,186),(25,185),(26,184),(27,183),(28,182),(29,181),(30,210),(31,74),(32,73),(33,72),(34,71),(35,70),(36,69),(37,68),(38,67),(39,66),(40,65),(41,64),(42,63),(43,62),(44,61),(45,90),(46,89),(47,88),(48,87),(49,86),(50,85),(51,84),(52,83),(53,82),(54,81),(55,80),(56,79),(57,78),(58,77),(59,76),(60,75),(91,146),(92,145),(93,144),(94,143),(95,142),(96,141),(97,140),(98,139),(99,138),(100,137),(101,136),(102,135),(103,134),(104,133),(105,132),(106,131),(107,130),(108,129),(109,128),(110,127),(111,126),(112,125),(113,124),(114,123),(115,122),(116,121),(117,150),(118,149),(119,148),(120,147),(151,227),(152,226),(153,225),(154,224),(155,223),(156,222),(157,221),(158,220),(159,219),(160,218),(161,217),(162,216),(163,215),(164,214),(165,213),(166,212),(167,211),(168,240),(169,239),(170,238),(171,237),(172,236),(173,235),(174,234),(175,233),(176,232),(177,231),(178,230),(179,229),(180,228)], [(1,133,195,120),(2,132,196,119),(3,131,197,118),(4,130,198,117),(5,129,199,116),(6,128,200,115),(7,127,201,114),(8,126,202,113),(9,125,203,112),(10,124,204,111),(11,123,205,110),(12,122,206,109),(13,121,207,108),(14,150,208,107),(15,149,209,106),(16,148,210,105),(17,147,181,104),(18,146,182,103),(19,145,183,102),(20,144,184,101),(21,143,185,100),(22,142,186,99),(23,141,187,98),(24,140,188,97),(25,139,189,96),(26,138,190,95),(27,137,191,94),(28,136,192,93),(29,135,193,92),(30,134,194,91),(31,172,81,231),(32,171,82,230),(33,170,83,229),(34,169,84,228),(35,168,85,227),(36,167,86,226),(37,166,87,225),(38,165,88,224),(39,164,89,223),(40,163,90,222),(41,162,61,221),(42,161,62,220),(43,160,63,219),(44,159,64,218),(45,158,65,217),(46,157,66,216),(47,156,67,215),(48,155,68,214),(49,154,69,213),(50,153,70,212),(51,152,71,211),(52,151,72,240),(53,180,73,239),(54,179,74,238),(55,178,75,237),(56,177,76,236),(57,176,77,235),(58,175,78,234),(59,174,79,233),(60,173,80,232)], [(1,234),(2,223),(3,212),(4,231),(5,220),(6,239),(7,228),(8,217),(9,236),(10,225),(11,214),(12,233),(13,222),(14,211),(15,230),(16,219),(17,238),(18,227),(19,216),(20,235),(21,224),(22,213),(23,232),(24,221),(25,240),(26,229),(27,218),(28,237),(29,226),(30,215),(31,117),(32,106),(33,95),(34,114),(35,103),(36,92),(37,111),(38,100),(39,119),(40,108),(41,97),(42,116),(43,105),(44,94),(45,113),(46,102),(47,91),(48,110),(49,99),(50,118),(51,107),(52,96),(53,115),(54,104),(55,93),(56,112),(57,101),(58,120),(59,109),(60,98),(61,140),(62,129),(63,148),(64,137),(65,126),(66,145),(67,134),(68,123),(69,142),(70,131),(71,150),(72,139),(73,128),(74,147),(75,136),(76,125),(77,144),(78,133),(79,122),(80,141),(81,130),(82,149),(83,138),(84,127),(85,146),(86,135),(87,124),(88,143),(89,132),(90,121),(151,189),(152,208),(153,197),(154,186),(155,205),(156,194),(157,183),(158,202),(159,191),(160,210),(161,199),(162,188),(163,207),(164,196),(165,185),(166,204),(167,193),(168,182),(169,201),(170,190),(171,209),(172,198),(173,187),(174,206),(175,195),(176,184),(177,203),(178,192),(179,181),(180,200)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 20 | 30 | 30 | 2 | 2 | 2 | 12 | 20 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 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 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C4○D12 | C4○D20 | S3×D4 | S3×D5 | D4×D5 | C2×S3×D5 | D6.D10 | C20⋊D6 | D10⋊D6 |
| kernel | D30⋊12D4 | Dic15⋊5C4 | C3×D10⋊C4 | C5×D6⋊C4 | C2×C15⋊D4 | C2×C3⋊D20 | C2×C5⋊D12 | C2×C4×D15 | D10⋊C4 | Dic15 | D30 | D6⋊C4 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C22×S3 | C10 | C6 | C10 | C2×C4 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of D30⋊12D4 ►in GL6(𝔽61)
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 18 | 60 | 0 | 0 | 0 | 0 |
| 18 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 30 | 60 |
| 43 | 18 | 0 | 0 | 0 | 0 |
| 60 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 11 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 43 | 18 | 0 | 0 | 0 | 0 |
| 60 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 18 | 0 | 0 |
| 0 | 0 | 43 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 55 |
| 0 | 0 | 0 | 0 | 18 | 32 |
G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,43,0,0,0,0,0,0,1,1,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[18,18,0,0,0,0,60,43,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,1,30,0,0,0,0,0,60],[43,60,0,0,0,0,18,18,0,0,0,0,0,0,50,0,0,0,0,0,11,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[43,60,0,0,0,0,18,18,0,0,0,0,0,0,52,43,0,0,0,0,18,9,0,0,0,0,0,0,29,18,0,0,0,0,55,32] >;
D30⋊12D4 in GAP, Magma, Sage, TeX
D_{30}\rtimes_{12}D_4 % in TeX
G:=Group("D30:12D4"); // GroupNames label
G:=SmallGroup(480,537);
// by ID
G=gap.SmallGroup(480,537);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,422,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,d*a*d=a^19,c*b*c^-1=a^28*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations