metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊2D4, Dic3⋊3D20, (C2×D60)⋊2C2, C6.63(D4×D5), C5⋊1(Dic3⋊D4), C3⋊1(C4⋊D20), Dic3⋊C4⋊8D5, (C5×Dic3)⋊2D4, C2.25(S3×D20), C10.65(S3×D4), (C2×C20).29D6, C6.23(C2×D20), D10⋊C4⋊6S3, C15⋊11(C4⋊D4), C30.153(C2×D4), (C2×C12).29D10, D30⋊4C4⋊20C2, C30.91(C4○D4), (C2×C60).14C22, (C22×D5).20D6, C10.17(C4○D12), (C2×C30).149C23, C6.19(Q8⋊2D5), (C2×Dic5).121D6, (C2×Dic3).46D10, C2.16(D10⋊D6), C2.19(C12.28D10), (C6×Dic5).89C22, (C10×Dic3).90C22, (C22×D15).52C22, (C2×C3⋊D20)⋊6C2, (C2×C4).62(S3×D5), (C5×Dic3⋊C4)⋊8C2, (C3×D10⋊C4)⋊6C2, (D5×C2×C6).34C22, C22.201(C2×S3×D5), (C2×D30.C2)⋊11C2, (C2×C6).161(C22×D5), (C2×C10).161(C22×S3), SmallGroup(480,535)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊2D4
G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, cac-1=a11, dad=a19, cbc-1=a10b, dbd=a3b, dcd=c-1 >
Subgroups: 1324 in 188 conjugacy classes, 50 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C3×D5, D15, C30, C4⋊D4, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C5×Dic3, C5×Dic3, C3×Dic5, C60, C6×D5, D30, D30, C2×C30, D10⋊C4, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, Dic3⋊D4, D30.C2, C3⋊D20, C6×Dic5, C10×Dic3, D60, C2×C60, D5×C2×C6, C22×D15, C4⋊D20, D30⋊4C4, C3×D10⋊C4, C5×Dic3⋊C4, C2×D30.C2, C2×C3⋊D20, C2×D60, D30⋊2D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C4⋊D4, D20, C22×D5, C4○D12, S3×D4, S3×D5, C2×D20, D4×D5, Q8⋊2D5, Dic3⋊D4, C2×S3×D5, C4⋊D20, C12.28D10, S3×D20, D10⋊D6, D30⋊2D4
►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 93)(2 92)(3 91)(4 120)(5 119)(6 118)(7 117)(8 116)(9 115)(10 114)(11 113)(12 112)(13 111)(14 110)(15 109)(16 108)(17 107)(18 106)(19 105)(20 104)(21 103)(22 102)(23 101)(24 100)(25 99)(26 98)(27 97)(28 96)(29 95)(30 94)(31 130)(32 129)(33 128)(34 127)(35 126)(36 125)(37 124)(38 123)(39 122)(40 121)(41 150)(42 149)(43 148)(44 147)(45 146)(46 145)(47 144)(48 143)(49 142)(50 141)(51 140)(52 139)(53 138)(54 137)(55 136)(56 135)(57 134)(58 133)(59 132)(60 131)(61 225)(62 224)(63 223)(64 222)(65 221)(66 220)(67 219)(68 218)(69 217)(70 216)(71 215)(72 214)(73 213)(74 212)(75 211)(76 240)(77 239)(78 238)(79 237)(80 236)(81 235)(82 234)(83 233)(84 232)(85 231)(86 230)(87 229)(88 228)(89 227)(90 226)(151 201)(152 200)(153 199)(154 198)(155 197)(156 196)(157 195)(158 194)(159 193)(160 192)(161 191)(162 190)(163 189)(164 188)(165 187)(166 186)(167 185)(168 184)(169 183)(170 182)(171 181)(172 210)(173 209)(174 208)(175 207)(176 206)(177 205)(178 204)(179 203)(180 202)
(1 237 94 80)(2 218 95 61)(3 229 96 72)(4 240 97 83)(5 221 98 64)(6 232 99 75)(7 213 100 86)(8 224 101 67)(9 235 102 78)(10 216 103 89)(11 227 104 70)(12 238 105 81)(13 219 106 62)(14 230 107 73)(15 211 108 84)(16 222 109 65)(17 233 110 76)(18 214 111 87)(19 225 112 68)(20 236 113 79)(21 217 114 90)(22 228 115 71)(23 239 116 82)(24 220 117 63)(25 231 118 74)(26 212 119 85)(27 223 120 66)(28 234 91 77)(29 215 92 88)(30 226 93 69)(31 172 130 200)(32 153 131 181)(33 164 132 192)(34 175 133 203)(35 156 134 184)(36 167 135 195)(37 178 136 206)(38 159 137 187)(39 170 138 198)(40 151 139 209)(41 162 140 190)(42 173 141 201)(43 154 142 182)(44 165 143 193)(45 176 144 204)(46 157 145 185)(47 168 146 196)(48 179 147 207)(49 160 148 188)(50 171 149 199)(51 152 150 210)(52 163 121 191)(53 174 122 202)(54 155 123 183)(55 166 124 194)(56 177 125 205)(57 158 126 186)(58 169 127 197)(59 180 128 208)(60 161 129 189)
(1 170)(2 159)(3 178)(4 167)(5 156)(6 175)(7 164)(8 153)(9 172)(10 161)(11 180)(12 169)(13 158)(14 177)(15 166)(16 155)(17 174)(18 163)(19 152)(20 171)(21 160)(22 179)(23 168)(24 157)(25 176)(26 165)(27 154)(28 173)(29 162)(30 151)(31 235)(32 224)(33 213)(34 232)(35 221)(36 240)(37 229)(38 218)(39 237)(40 226)(41 215)(42 234)(43 223)(44 212)(45 231)(46 220)(47 239)(48 228)(49 217)(50 236)(51 225)(52 214)(53 233)(54 222)(55 211)(56 230)(57 219)(58 238)(59 227)(60 216)(61 137)(62 126)(63 145)(64 134)(65 123)(66 142)(67 131)(68 150)(69 139)(70 128)(71 147)(72 136)(73 125)(74 144)(75 133)(76 122)(77 141)(78 130)(79 149)(80 138)(81 127)(82 146)(83 135)(84 124)(85 143)(86 132)(87 121)(88 140)(89 129)(90 148)(91 201)(92 190)(93 209)(94 198)(95 187)(96 206)(97 195)(98 184)(99 203)(100 192)(101 181)(102 200)(103 189)(104 208)(105 197)(106 186)(107 205)(108 194)(109 183)(110 202)(111 191)(112 210)(113 199)(114 188)(115 207)(116 196)(117 185)(118 204)(119 193)(120 182)
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), (1,93)(2,92)(3,91)(4,120)(5,119)(6,118)(7,117)(8,116)(9,115)(10,114)(11,113)(12,112)(13,111)(14,110)(15,109)(16,108)(17,107)(18,106)(19,105)(20,104)(21,103)(22,102)(23,101)(24,100)(25,99)(26,98)(27,97)(28,96)(29,95)(30,94)(31,130)(32,129)(33,128)(34,127)(35,126)(36,125)(37,124)(38,123)(39,122)(40,121)(41,150)(42,149)(43,148)(44,147)(45,146)(46,145)(47,144)(48,143)(49,142)(50,141)(51,140)(52,139)(53,138)(54,137)(55,136)(56,135)(57,134)(58,133)(59,132)(60,131)(61,225)(62,224)(63,223)(64,222)(65,221)(66,220)(67,219)(68,218)(69,217)(70,216)(71,215)(72,214)(73,213)(74,212)(75,211)(76,240)(77,239)(78,238)(79,237)(80,236)(81,235)(82,234)(83,233)(84,232)(85,231)(86,230)(87,229)(88,228)(89,227)(90,226)(151,201)(152,200)(153,199)(154,198)(155,197)(156,196)(157,195)(158,194)(159,193)(160,192)(161,191)(162,190)(163,189)(164,188)(165,187)(166,186)(167,185)(168,184)(169,183)(170,182)(171,181)(172,210)(173,209)(174,208)(175,207)(176,206)(177,205)(178,204)(179,203)(180,202), (1,237,94,80)(2,218,95,61)(3,229,96,72)(4,240,97,83)(5,221,98,64)(6,232,99,75)(7,213,100,86)(8,224,101,67)(9,235,102,78)(10,216,103,89)(11,227,104,70)(12,238,105,81)(13,219,106,62)(14,230,107,73)(15,211,108,84)(16,222,109,65)(17,233,110,76)(18,214,111,87)(19,225,112,68)(20,236,113,79)(21,217,114,90)(22,228,115,71)(23,239,116,82)(24,220,117,63)(25,231,118,74)(26,212,119,85)(27,223,120,66)(28,234,91,77)(29,215,92,88)(30,226,93,69)(31,172,130,200)(32,153,131,181)(33,164,132,192)(34,175,133,203)(35,156,134,184)(36,167,135,195)(37,178,136,206)(38,159,137,187)(39,170,138,198)(40,151,139,209)(41,162,140,190)(42,173,141,201)(43,154,142,182)(44,165,143,193)(45,176,144,204)(46,157,145,185)(47,168,146,196)(48,179,147,207)(49,160,148,188)(50,171,149,199)(51,152,150,210)(52,163,121,191)(53,174,122,202)(54,155,123,183)(55,166,124,194)(56,177,125,205)(57,158,126,186)(58,169,127,197)(59,180,128,208)(60,161,129,189), (1,170)(2,159)(3,178)(4,167)(5,156)(6,175)(7,164)(8,153)(9,172)(10,161)(11,180)(12,169)(13,158)(14,177)(15,166)(16,155)(17,174)(18,163)(19,152)(20,171)(21,160)(22,179)(23,168)(24,157)(25,176)(26,165)(27,154)(28,173)(29,162)(30,151)(31,235)(32,224)(33,213)(34,232)(35,221)(36,240)(37,229)(38,218)(39,237)(40,226)(41,215)(42,234)(43,223)(44,212)(45,231)(46,220)(47,239)(48,228)(49,217)(50,236)(51,225)(52,214)(53,233)(54,222)(55,211)(56,230)(57,219)(58,238)(59,227)(60,216)(61,137)(62,126)(63,145)(64,134)(65,123)(66,142)(67,131)(68,150)(69,139)(70,128)(71,147)(72,136)(73,125)(74,144)(75,133)(76,122)(77,141)(78,130)(79,149)(80,138)(81,127)(82,146)(83,135)(84,124)(85,143)(86,132)(87,121)(88,140)(89,129)(90,148)(91,201)(92,190)(93,209)(94,198)(95,187)(96,206)(97,195)(98,184)(99,203)(100,192)(101,181)(102,200)(103,189)(104,208)(105,197)(106,186)(107,205)(108,194)(109,183)(110,202)(111,191)(112,210)(113,199)(114,188)(115,207)(116,196)(117,185)(118,204)(119,193)(120,182)>;
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), (1,93)(2,92)(3,91)(4,120)(5,119)(6,118)(7,117)(8,116)(9,115)(10,114)(11,113)(12,112)(13,111)(14,110)(15,109)(16,108)(17,107)(18,106)(19,105)(20,104)(21,103)(22,102)(23,101)(24,100)(25,99)(26,98)(27,97)(28,96)(29,95)(30,94)(31,130)(32,129)(33,128)(34,127)(35,126)(36,125)(37,124)(38,123)(39,122)(40,121)(41,150)(42,149)(43,148)(44,147)(45,146)(46,145)(47,144)(48,143)(49,142)(50,141)(51,140)(52,139)(53,138)(54,137)(55,136)(56,135)(57,134)(58,133)(59,132)(60,131)(61,225)(62,224)(63,223)(64,222)(65,221)(66,220)(67,219)(68,218)(69,217)(70,216)(71,215)(72,214)(73,213)(74,212)(75,211)(76,240)(77,239)(78,238)(79,237)(80,236)(81,235)(82,234)(83,233)(84,232)(85,231)(86,230)(87,229)(88,228)(89,227)(90,226)(151,201)(152,200)(153,199)(154,198)(155,197)(156,196)(157,195)(158,194)(159,193)(160,192)(161,191)(162,190)(163,189)(164,188)(165,187)(166,186)(167,185)(168,184)(169,183)(170,182)(171,181)(172,210)(173,209)(174,208)(175,207)(176,206)(177,205)(178,204)(179,203)(180,202), (1,237,94,80)(2,218,95,61)(3,229,96,72)(4,240,97,83)(5,221,98,64)(6,232,99,75)(7,213,100,86)(8,224,101,67)(9,235,102,78)(10,216,103,89)(11,227,104,70)(12,238,105,81)(13,219,106,62)(14,230,107,73)(15,211,108,84)(16,222,109,65)(17,233,110,76)(18,214,111,87)(19,225,112,68)(20,236,113,79)(21,217,114,90)(22,228,115,71)(23,239,116,82)(24,220,117,63)(25,231,118,74)(26,212,119,85)(27,223,120,66)(28,234,91,77)(29,215,92,88)(30,226,93,69)(31,172,130,200)(32,153,131,181)(33,164,132,192)(34,175,133,203)(35,156,134,184)(36,167,135,195)(37,178,136,206)(38,159,137,187)(39,170,138,198)(40,151,139,209)(41,162,140,190)(42,173,141,201)(43,154,142,182)(44,165,143,193)(45,176,144,204)(46,157,145,185)(47,168,146,196)(48,179,147,207)(49,160,148,188)(50,171,149,199)(51,152,150,210)(52,163,121,191)(53,174,122,202)(54,155,123,183)(55,166,124,194)(56,177,125,205)(57,158,126,186)(58,169,127,197)(59,180,128,208)(60,161,129,189), (1,170)(2,159)(3,178)(4,167)(5,156)(6,175)(7,164)(8,153)(9,172)(10,161)(11,180)(12,169)(13,158)(14,177)(15,166)(16,155)(17,174)(18,163)(19,152)(20,171)(21,160)(22,179)(23,168)(24,157)(25,176)(26,165)(27,154)(28,173)(29,162)(30,151)(31,235)(32,224)(33,213)(34,232)(35,221)(36,240)(37,229)(38,218)(39,237)(40,226)(41,215)(42,234)(43,223)(44,212)(45,231)(46,220)(47,239)(48,228)(49,217)(50,236)(51,225)(52,214)(53,233)(54,222)(55,211)(56,230)(57,219)(58,238)(59,227)(60,216)(61,137)(62,126)(63,145)(64,134)(65,123)(66,142)(67,131)(68,150)(69,139)(70,128)(71,147)(72,136)(73,125)(74,144)(75,133)(76,122)(77,141)(78,130)(79,149)(80,138)(81,127)(82,146)(83,135)(84,124)(85,143)(86,132)(87,121)(88,140)(89,129)(90,148)(91,201)(92,190)(93,209)(94,198)(95,187)(96,206)(97,195)(98,184)(99,203)(100,192)(101,181)(102,200)(103,189)(104,208)(105,197)(106,186)(107,205)(108,194)(109,183)(110,202)(111,191)(112,210)(113,199)(114,188)(115,207)(116,196)(117,185)(118,204)(119,193)(120,182) );
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)], [(1,93),(2,92),(3,91),(4,120),(5,119),(6,118),(7,117),(8,116),(9,115),(10,114),(11,113),(12,112),(13,111),(14,110),(15,109),(16,108),(17,107),(18,106),(19,105),(20,104),(21,103),(22,102),(23,101),(24,100),(25,99),(26,98),(27,97),(28,96),(29,95),(30,94),(31,130),(32,129),(33,128),(34,127),(35,126),(36,125),(37,124),(38,123),(39,122),(40,121),(41,150),(42,149),(43,148),(44,147),(45,146),(46,145),(47,144),(48,143),(49,142),(50,141),(51,140),(52,139),(53,138),(54,137),(55,136),(56,135),(57,134),(58,133),(59,132),(60,131),(61,225),(62,224),(63,223),(64,222),(65,221),(66,220),(67,219),(68,218),(69,217),(70,216),(71,215),(72,214),(73,213),(74,212),(75,211),(76,240),(77,239),(78,238),(79,237),(80,236),(81,235),(82,234),(83,233),(84,232),(85,231),(86,230),(87,229),(88,228),(89,227),(90,226),(151,201),(152,200),(153,199),(154,198),(155,197),(156,196),(157,195),(158,194),(159,193),(160,192),(161,191),(162,190),(163,189),(164,188),(165,187),(166,186),(167,185),(168,184),(169,183),(170,182),(171,181),(172,210),(173,209),(174,208),(175,207),(176,206),(177,205),(178,204),(179,203),(180,202)], [(1,237,94,80),(2,218,95,61),(3,229,96,72),(4,240,97,83),(5,221,98,64),(6,232,99,75),(7,213,100,86),(8,224,101,67),(9,235,102,78),(10,216,103,89),(11,227,104,70),(12,238,105,81),(13,219,106,62),(14,230,107,73),(15,211,108,84),(16,222,109,65),(17,233,110,76),(18,214,111,87),(19,225,112,68),(20,236,113,79),(21,217,114,90),(22,228,115,71),(23,239,116,82),(24,220,117,63),(25,231,118,74),(26,212,119,85),(27,223,120,66),(28,234,91,77),(29,215,92,88),(30,226,93,69),(31,172,130,200),(32,153,131,181),(33,164,132,192),(34,175,133,203),(35,156,134,184),(36,167,135,195),(37,178,136,206),(38,159,137,187),(39,170,138,198),(40,151,139,209),(41,162,140,190),(42,173,141,201),(43,154,142,182),(44,165,143,193),(45,176,144,204),(46,157,145,185),(47,168,146,196),(48,179,147,207),(49,160,148,188),(50,171,149,199),(51,152,150,210),(52,163,121,191),(53,174,122,202),(54,155,123,183),(55,166,124,194),(56,177,125,205),(57,158,126,186),(58,169,127,197),(59,180,128,208),(60,161,129,189)], [(1,170),(2,159),(3,178),(4,167),(5,156),(6,175),(7,164),(8,153),(9,172),(10,161),(11,180),(12,169),(13,158),(14,177),(15,166),(16,155),(17,174),(18,163),(19,152),(20,171),(21,160),(22,179),(23,168),(24,157),(25,176),(26,165),(27,154),(28,173),(29,162),(30,151),(31,235),(32,224),(33,213),(34,232),(35,221),(36,240),(37,229),(38,218),(39,237),(40,226),(41,215),(42,234),(43,223),(44,212),(45,231),(46,220),(47,239),(48,228),(49,217),(50,236),(51,225),(52,214),(53,233),(54,222),(55,211),(56,230),(57,219),(58,238),(59,227),(60,216),(61,137),(62,126),(63,145),(64,134),(65,123),(66,142),(67,131),(68,150),(69,139),(70,128),(71,147),(72,136),(73,125),(74,144),(75,133),(76,122),(77,141),(78,130),(79,149),(80,138),(81,127),(82,146),(83,135),(84,124),(85,143),(86,132),(87,121),(88,140),(89,129),(90,148),(91,201),(92,190),(93,209),(94,198),(95,187),(96,206),(97,195),(98,184),(99,203),(100,192),(101,181),(102,200),(103,189),(104,208),(105,197),(106,186),(107,205),(108,194),(109,183),(110,202),(111,191),(112,210),(113,199),(114,188),(115,207),(116,196),(117,185),(118,204),(119,193),(120,182)]])
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 | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 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 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 20 | 30 | 30 | 60 | 2 | 4 | 6 | 6 | 10 | 10 | 12 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D20 | C4○D12 | S3×D4 | S3×D5 | D4×D5 | Q8⋊2D5 | C2×S3×D5 | C12.28D10 | S3×D20 | D10⋊D6 |
| kernel | D30⋊2D4 | D30⋊4C4 | C3×D10⋊C4 | C5×Dic3⋊C4 | C2×D30.C2 | C2×C3⋊D20 | C2×D60 | D10⋊C4 | C5×Dic3 | D30 | Dic3⋊C4 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | Dic3 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 8 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D30⋊2D4 ►in GL4(𝔽61) generated by
| 0 | 60 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 44 |
| 0 | 60 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 17 | 1 |
| 0 | 0 | 17 | 44 |
| 11 | 0 | 0 | 0 |
| 50 | 50 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 18 | 0 | 0 |
| 43 | 52 | 0 | 0 |
| 0 | 0 | 2 | 2 |
| 0 | 0 | 29 | 59 |
G:=sub<GL(4,GF(61))| [0,1,0,0,60,1,0,0,0,0,0,60,0,0,1,44],[0,60,0,0,60,0,0,0,0,0,17,17,0,0,1,44],[11,50,0,0,0,50,0,0,0,0,1,0,0,0,0,1],[9,43,0,0,18,52,0,0,0,0,2,29,0,0,2,59] >;
D30⋊2D4 in GAP, Magma, Sage, TeX
D_{30}\rtimes_2D_4 % in TeX
G:=Group("D30:2D4"); // GroupNames label
G:=SmallGroup(480,535);
// by ID
G=gap.SmallGroup(480,535);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,254,219,142,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=a^11,d*a*d=a^19,c*b*c^-1=a^10*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations