metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊1D4, Dic5⋊3D12, D6⋊C4⋊5D5, (C2×D60)⋊1C2, C6.59(D4×D5), C5⋊1(C12⋊D4), C15⋊4(C4⋊D4), (C3×Dic5)⋊2D4, (C2×C20).19D6, C2.22(D5×D12), C10.61(S3×D4), C3⋊1(D10⋊D4), C6.9(C4○D20), C10.D4⋊6S3, C10.21(C2×D12), C30.133(C2×D4), (C2×C12).17D10, (C2×C60).8C22, D30⋊4C4⋊14C2, C30.63(C4○D4), (C2×Dic5).33D6, (C2×C30).110C23, (C22×S3).12D10, C2.12(D10⋊D6), C2.16(D60⋊C2), C10.14(Q8⋊3S3), (C2×Dic3).103D10, (C6×Dic5).64C22, (C10×Dic3).68C22, (C22×D15).37C22, (C5×D6⋊C4)⋊5C2, (C2×C5⋊D12)⋊2C2, (C2×C4).46(S3×D5), (C2×D30.C2)⋊5C2, C22.176(C2×S3×D5), (C3×C10.D4)⋊6C2, (S3×C2×C10).20C22, (C2×C6).122(C22×D5), (C2×C10).122(C22×S3), SmallGroup(480,496)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊D4
G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, cac-1=a19, dad=a11, cbc-1=a18b, dbd=a25b, dcd=c-1 >
Subgroups: 1292 in 188 conjugacy classes, 50 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C5×S3, D15, C30, C4⋊D4, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, D6⋊C4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C5×Dic3, C3×Dic5, C3×Dic5, C60, S3×C10, D30, D30, C2×C30, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C12⋊D4, D30.C2, C5⋊D12, C6×Dic5, C10×Dic3, D60, C2×C60, S3×C2×C10, C22×D15, D10⋊D4, D30⋊4C4, C3×C10.D4, C5×D6⋊C4, C2×D30.C2, C2×C5⋊D12, C2×D60, D30⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C22×S3, C4⋊D4, C22×D5, C2×D12, S3×D4, Q8⋊3S3, S3×D5, C4○D20, D4×D5, C12⋊D4, C2×S3×D5, D10⋊D4, D60⋊C2, D5×D12, D10⋊D6, D30⋊D4
►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 42)(32 41)(33 40)(34 39)(35 38)(36 37)(43 60)(44 59)(45 58)(46 57)(47 56)(48 55)(49 54)(50 53)(51 52)(61 85)(62 84)(63 83)(64 82)(65 81)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(86 90)(87 89)(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)(121 140)(122 139)(123 138)(124 137)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)(141 150)(142 149)(143 148)(144 147)(145 146)(151 175)(152 174)(153 173)(154 172)(155 171)(156 170)(157 169)(158 168)(159 167)(160 166)(161 165)(162 164)(176 180)(177 179)(181 189)(182 188)(183 187)(184 186)(190 210)(191 209)(192 208)(193 207)(194 206)(195 205)(196 204)(197 203)(198 202)(199 201)(211 212)(213 240)(214 239)(215 238)(216 237)(217 236)(218 235)(219 234)(220 233)(221 232)(222 231)(223 230)(224 229)(225 228)(226 227)
(1 131 52 212)(2 150 53 231)(3 139 54 220)(4 128 55 239)(5 147 56 228)(6 136 57 217)(7 125 58 236)(8 144 59 225)(9 133 60 214)(10 122 31 233)(11 141 32 222)(12 130 33 211)(13 149 34 230)(14 138 35 219)(15 127 36 238)(16 146 37 227)(17 135 38 216)(18 124 39 235)(19 143 40 224)(20 132 41 213)(21 121 42 232)(22 140 43 221)(23 129 44 240)(24 148 45 229)(25 137 46 218)(26 126 47 237)(27 145 48 226)(28 134 49 215)(29 123 50 234)(30 142 51 223)(61 151 96 188)(62 170 97 207)(63 159 98 196)(64 178 99 185)(65 167 100 204)(66 156 101 193)(67 175 102 182)(68 164 103 201)(69 153 104 190)(70 172 105 209)(71 161 106 198)(72 180 107 187)(73 169 108 206)(74 158 109 195)(75 177 110 184)(76 166 111 203)(77 155 112 192)(78 174 113 181)(79 163 114 200)(80 152 115 189)(81 171 116 208)(82 160 117 197)(83 179 118 186)(84 168 119 205)(85 157 120 194)(86 176 91 183)(87 165 92 202)(88 154 93 191)(89 173 94 210)(90 162 95 199)
(1 193)(2 204)(3 185)(4 196)(5 207)(6 188)(7 199)(8 210)(9 191)(10 202)(11 183)(12 194)(13 205)(14 186)(15 197)(16 208)(17 189)(18 200)(19 181)(20 192)(21 203)(22 184)(23 195)(24 206)(25 187)(26 198)(27 209)(28 190)(29 201)(30 182)(31 165)(32 176)(33 157)(34 168)(35 179)(36 160)(37 171)(38 152)(39 163)(40 174)(41 155)(42 166)(43 177)(44 158)(45 169)(46 180)(47 161)(48 172)(49 153)(50 164)(51 175)(52 156)(53 167)(54 178)(55 159)(56 170)(57 151)(58 162)(59 173)(60 154)(61 217)(62 228)(63 239)(64 220)(65 231)(66 212)(67 223)(68 234)(69 215)(70 226)(71 237)(72 218)(73 229)(74 240)(75 221)(76 232)(77 213)(78 224)(79 235)(80 216)(81 227)(82 238)(83 219)(84 230)(85 211)(86 222)(87 233)(88 214)(89 225)(90 236)(91 141)(92 122)(93 133)(94 144)(95 125)(96 136)(97 147)(98 128)(99 139)(100 150)(101 131)(102 142)(103 123)(104 134)(105 145)(106 126)(107 137)(108 148)(109 129)(110 140)(111 121)(112 132)(113 143)(114 124)(115 135)(116 146)(117 127)(118 138)(119 149)(120 130)
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)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(86,90)(87,89)(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)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)(141,150)(142,149)(143,148)(144,147)(145,146)(151,175)(152,174)(153,173)(154,172)(155,171)(156,170)(157,169)(158,168)(159,167)(160,166)(161,165)(162,164)(176,180)(177,179)(181,189)(182,188)(183,187)(184,186)(190,210)(191,209)(192,208)(193,207)(194,206)(195,205)(196,204)(197,203)(198,202)(199,201)(211,212)(213,240)(214,239)(215,238)(216,237)(217,236)(218,235)(219,234)(220,233)(221,232)(222,231)(223,230)(224,229)(225,228)(226,227), (1,131,52,212)(2,150,53,231)(3,139,54,220)(4,128,55,239)(5,147,56,228)(6,136,57,217)(7,125,58,236)(8,144,59,225)(9,133,60,214)(10,122,31,233)(11,141,32,222)(12,130,33,211)(13,149,34,230)(14,138,35,219)(15,127,36,238)(16,146,37,227)(17,135,38,216)(18,124,39,235)(19,143,40,224)(20,132,41,213)(21,121,42,232)(22,140,43,221)(23,129,44,240)(24,148,45,229)(25,137,46,218)(26,126,47,237)(27,145,48,226)(28,134,49,215)(29,123,50,234)(30,142,51,223)(61,151,96,188)(62,170,97,207)(63,159,98,196)(64,178,99,185)(65,167,100,204)(66,156,101,193)(67,175,102,182)(68,164,103,201)(69,153,104,190)(70,172,105,209)(71,161,106,198)(72,180,107,187)(73,169,108,206)(74,158,109,195)(75,177,110,184)(76,166,111,203)(77,155,112,192)(78,174,113,181)(79,163,114,200)(80,152,115,189)(81,171,116,208)(82,160,117,197)(83,179,118,186)(84,168,119,205)(85,157,120,194)(86,176,91,183)(87,165,92,202)(88,154,93,191)(89,173,94,210)(90,162,95,199), (1,193)(2,204)(3,185)(4,196)(5,207)(6,188)(7,199)(8,210)(9,191)(10,202)(11,183)(12,194)(13,205)(14,186)(15,197)(16,208)(17,189)(18,200)(19,181)(20,192)(21,203)(22,184)(23,195)(24,206)(25,187)(26,198)(27,209)(28,190)(29,201)(30,182)(31,165)(32,176)(33,157)(34,168)(35,179)(36,160)(37,171)(38,152)(39,163)(40,174)(41,155)(42,166)(43,177)(44,158)(45,169)(46,180)(47,161)(48,172)(49,153)(50,164)(51,175)(52,156)(53,167)(54,178)(55,159)(56,170)(57,151)(58,162)(59,173)(60,154)(61,217)(62,228)(63,239)(64,220)(65,231)(66,212)(67,223)(68,234)(69,215)(70,226)(71,237)(72,218)(73,229)(74,240)(75,221)(76,232)(77,213)(78,224)(79,235)(80,216)(81,227)(82,238)(83,219)(84,230)(85,211)(86,222)(87,233)(88,214)(89,225)(90,236)(91,141)(92,122)(93,133)(94,144)(95,125)(96,136)(97,147)(98,128)(99,139)(100,150)(101,131)(102,142)(103,123)(104,134)(105,145)(106,126)(107,137)(108,148)(109,129)(110,140)(111,121)(112,132)(113,143)(114,124)(115,135)(116,146)(117,127)(118,138)(119,149)(120,130)>;
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)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(86,90)(87,89)(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)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)(141,150)(142,149)(143,148)(144,147)(145,146)(151,175)(152,174)(153,173)(154,172)(155,171)(156,170)(157,169)(158,168)(159,167)(160,166)(161,165)(162,164)(176,180)(177,179)(181,189)(182,188)(183,187)(184,186)(190,210)(191,209)(192,208)(193,207)(194,206)(195,205)(196,204)(197,203)(198,202)(199,201)(211,212)(213,240)(214,239)(215,238)(216,237)(217,236)(218,235)(219,234)(220,233)(221,232)(222,231)(223,230)(224,229)(225,228)(226,227), (1,131,52,212)(2,150,53,231)(3,139,54,220)(4,128,55,239)(5,147,56,228)(6,136,57,217)(7,125,58,236)(8,144,59,225)(9,133,60,214)(10,122,31,233)(11,141,32,222)(12,130,33,211)(13,149,34,230)(14,138,35,219)(15,127,36,238)(16,146,37,227)(17,135,38,216)(18,124,39,235)(19,143,40,224)(20,132,41,213)(21,121,42,232)(22,140,43,221)(23,129,44,240)(24,148,45,229)(25,137,46,218)(26,126,47,237)(27,145,48,226)(28,134,49,215)(29,123,50,234)(30,142,51,223)(61,151,96,188)(62,170,97,207)(63,159,98,196)(64,178,99,185)(65,167,100,204)(66,156,101,193)(67,175,102,182)(68,164,103,201)(69,153,104,190)(70,172,105,209)(71,161,106,198)(72,180,107,187)(73,169,108,206)(74,158,109,195)(75,177,110,184)(76,166,111,203)(77,155,112,192)(78,174,113,181)(79,163,114,200)(80,152,115,189)(81,171,116,208)(82,160,117,197)(83,179,118,186)(84,168,119,205)(85,157,120,194)(86,176,91,183)(87,165,92,202)(88,154,93,191)(89,173,94,210)(90,162,95,199), (1,193)(2,204)(3,185)(4,196)(5,207)(6,188)(7,199)(8,210)(9,191)(10,202)(11,183)(12,194)(13,205)(14,186)(15,197)(16,208)(17,189)(18,200)(19,181)(20,192)(21,203)(22,184)(23,195)(24,206)(25,187)(26,198)(27,209)(28,190)(29,201)(30,182)(31,165)(32,176)(33,157)(34,168)(35,179)(36,160)(37,171)(38,152)(39,163)(40,174)(41,155)(42,166)(43,177)(44,158)(45,169)(46,180)(47,161)(48,172)(49,153)(50,164)(51,175)(52,156)(53,167)(54,178)(55,159)(56,170)(57,151)(58,162)(59,173)(60,154)(61,217)(62,228)(63,239)(64,220)(65,231)(66,212)(67,223)(68,234)(69,215)(70,226)(71,237)(72,218)(73,229)(74,240)(75,221)(76,232)(77,213)(78,224)(79,235)(80,216)(81,227)(82,238)(83,219)(84,230)(85,211)(86,222)(87,233)(88,214)(89,225)(90,236)(91,141)(92,122)(93,133)(94,144)(95,125)(96,136)(97,147)(98,128)(99,139)(100,150)(101,131)(102,142)(103,123)(104,134)(105,145)(106,126)(107,137)(108,148)(109,129)(110,140)(111,121)(112,132)(113,143)(114,124)(115,135)(116,146)(117,127)(118,138)(119,149)(120,130) );
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),(31,42),(32,41),(33,40),(34,39),(35,38),(36,37),(43,60),(44,59),(45,58),(46,57),(47,56),(48,55),(49,54),(50,53),(51,52),(61,85),(62,84),(63,83),(64,82),(65,81),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(86,90),(87,89),(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),(121,140),(122,139),(123,138),(124,137),(125,136),(126,135),(127,134),(128,133),(129,132),(130,131),(141,150),(142,149),(143,148),(144,147),(145,146),(151,175),(152,174),(153,173),(154,172),(155,171),(156,170),(157,169),(158,168),(159,167),(160,166),(161,165),(162,164),(176,180),(177,179),(181,189),(182,188),(183,187),(184,186),(190,210),(191,209),(192,208),(193,207),(194,206),(195,205),(196,204),(197,203),(198,202),(199,201),(211,212),(213,240),(214,239),(215,238),(216,237),(217,236),(218,235),(219,234),(220,233),(221,232),(222,231),(223,230),(224,229),(225,228),(226,227)], [(1,131,52,212),(2,150,53,231),(3,139,54,220),(4,128,55,239),(5,147,56,228),(6,136,57,217),(7,125,58,236),(8,144,59,225),(9,133,60,214),(10,122,31,233),(11,141,32,222),(12,130,33,211),(13,149,34,230),(14,138,35,219),(15,127,36,238),(16,146,37,227),(17,135,38,216),(18,124,39,235),(19,143,40,224),(20,132,41,213),(21,121,42,232),(22,140,43,221),(23,129,44,240),(24,148,45,229),(25,137,46,218),(26,126,47,237),(27,145,48,226),(28,134,49,215),(29,123,50,234),(30,142,51,223),(61,151,96,188),(62,170,97,207),(63,159,98,196),(64,178,99,185),(65,167,100,204),(66,156,101,193),(67,175,102,182),(68,164,103,201),(69,153,104,190),(70,172,105,209),(71,161,106,198),(72,180,107,187),(73,169,108,206),(74,158,109,195),(75,177,110,184),(76,166,111,203),(77,155,112,192),(78,174,113,181),(79,163,114,200),(80,152,115,189),(81,171,116,208),(82,160,117,197),(83,179,118,186),(84,168,119,205),(85,157,120,194),(86,176,91,183),(87,165,92,202),(88,154,93,191),(89,173,94,210),(90,162,95,199)], [(1,193),(2,204),(3,185),(4,196),(5,207),(6,188),(7,199),(8,210),(9,191),(10,202),(11,183),(12,194),(13,205),(14,186),(15,197),(16,208),(17,189),(18,200),(19,181),(20,192),(21,203),(22,184),(23,195),(24,206),(25,187),(26,198),(27,209),(28,190),(29,201),(30,182),(31,165),(32,176),(33,157),(34,168),(35,179),(36,160),(37,171),(38,152),(39,163),(40,174),(41,155),(42,166),(43,177),(44,158),(45,169),(46,180),(47,161),(48,172),(49,153),(50,164),(51,175),(52,156),(53,167),(54,178),(55,159),(56,170),(57,151),(58,162),(59,173),(60,154),(61,217),(62,228),(63,239),(64,220),(65,231),(66,212),(67,223),(68,234),(69,215),(70,226),(71,237),(72,218),(73,229),(74,240),(75,221),(76,232),(77,213),(78,224),(79,235),(80,216),(81,227),(82,238),(83,219),(84,230),(85,211),(86,222),(87,233),(88,214),(89,225),(90,236),(91,141),(92,122),(93,133),(94,144),(95,125),(96,136),(97,147),(98,128),(99,139),(100,150),(101,131),(102,142),(103,123),(104,134),(105,145),(106,126),(107,137),(108,148),(109,129),(110,140),(111,121),(112,132),(113,143),(114,124),(115,135),(116,146),(117,127),(118,138),(119,149),(120,130)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 12E | 12F | 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 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 30 | 30 | 60 | 2 | 4 | 6 | 6 | 10 | 10 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 20 | 20 | 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 | 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 | C4○D4 | D10 | D10 | D10 | D12 | C4○D20 | S3×D4 | Q8⋊3S3 | S3×D5 | D4×D5 | C2×S3×D5 | D60⋊C2 | D5×D12 | D10⋊D6 |
| kernel | D30⋊D4 | D30⋊4C4 | C3×C10.D4 | C5×D6⋊C4 | C2×D30.C2 | C2×C5⋊D12 | C2×D60 | C10.D4 | C3×Dic5 | D30 | D6⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | Dic5 | C6 | C10 | C10 | C2×C4 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of D30⋊D4 ►in GL4(𝔽61) generated by
| 0 | 60 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 18 | 1 |
| 0 | 0 | 60 | 0 |
| 0 | 60 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 18 | 1 |
| 0 | 0 | 43 | 43 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 15 | 11 |
| 38 | 15 | 0 | 0 |
| 38 | 23 | 0 | 0 |
| 0 | 0 | 31 | 17 |
| 0 | 0 | 44 | 30 |
G:=sub<GL(4,GF(61))| [0,1,0,0,60,1,0,0,0,0,18,60,0,0,1,0],[0,60,0,0,60,0,0,0,0,0,18,43,0,0,1,43],[1,0,0,0,0,1,0,0,0,0,50,15,0,0,0,11],[38,38,0,0,15,23,0,0,0,0,31,44,0,0,17,30] >;
D30⋊D4 in GAP, Magma, Sage, TeX
D_{30}\rtimes D_4 % in TeX
G:=Group("D30:D4"); // GroupNames label
G:=SmallGroup(480,496);
// by ID
G=gap.SmallGroup(480,496);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,254,219,142,1356,18822]);
// 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^19,d*a*d=a^11,c*b*c^-1=a^18*b,d*b*d=a^25*b,d*c*d=c^-1>;
// generators/relations