metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊6D4, C6.84(D4×D5), C5⋊2(D6⋊3D4), (C3×Dic5)⋊3D4, C10.85(S3×D4), C3⋊6(D10⋊D4), C15⋊16(C4⋊D4), C6.D4⋊4D5, C30.213(C2×D4), C23.15(S3×D5), C6.78(C4○D20), Dic5⋊2(C3⋊D4), C30.Q8⋊30C2, (C22×D5).26D6, (C22×C6).23D10, (C22×C10).38D6, C30.135(C4○D4), D10⋊Dic3⋊25C2, (C2×C30).175C23, (C2×Dic3).54D10, (C2×Dic5).126D6, C10.50(D4⋊2S3), C2.37(D10⋊D6), (C22×C30).37C22, C2.23(Dic5.D6), (C6×Dic5).103C22, (C22×D15).59C22, (C10×Dic3).103C22, (C2×Dic15).124C22, (C6×C5⋊D4)⋊1C2, (C2×C5⋊D4)⋊1S3, C2.36(D5×C3⋊D4), (C2×C3⋊D20)⋊11C2, (C2×C15⋊7D4)⋊12C2, C10.57(C2×C3⋊D4), (D5×C2×C6).44C22, C22.219(C2×S3×D5), (C2×D30.C2)⋊13C2, (C5×C6.D4)⋊4C2, (C2×C6).187(C22×D5), (C2×C10).187(C22×S3), SmallGroup(480,609)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊6D4
G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, cac-1=dad=a19, cbc-1=a18b, dbd=a3b, dcd=c-1 >
Subgroups: 1116 in 188 conjugacy classes, 50 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, Dic5, C20, D10, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C3×D5, D15, C30, C30, C4⋊D4, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×D5, C22×C10, C4⋊Dic3, C6.D4, C6.D4, S3×C2×C4, C2×C3⋊D4, C6×D4, C5×Dic3, C3×Dic5, Dic15, C6×D5, D30, D30, C2×C30, C2×C30, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C2×C5⋊D4, D6⋊3D4, D30.C2, C3⋊D20, C6×Dic5, C3×C5⋊D4, C10×Dic3, C2×Dic15, C15⋊7D4, D5×C2×C6, C22×D15, C22×C30, D10⋊D4, D10⋊Dic3, C30.Q8, C5×C6.D4, C2×D30.C2, C2×C3⋊D20, C6×C5⋊D4, C2×C15⋊7D4, D30⋊6D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C4⋊D4, C22×D5, S3×D4, D4⋊2S3, C2×C3⋊D4, S3×D5, C4○D20, D4×D5, D6⋊3D4, C2×S3×D5, D10⋊D4, Dic5.D6, D5×C3⋊D4, D10⋊D6, D30⋊6D4
►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 34)(32 33)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(61 69)(62 68)(63 67)(64 66)(70 90)(71 89)(72 88)(73 87)(74 86)(75 85)(76 84)(77 83)(78 82)(79 81)(91 119)(92 118)(93 117)(94 116)(95 115)(96 114)(97 113)(98 112)(99 111)(100 110)(101 109)(102 108)(103 107)(104 106)(121 149)(122 148)(123 147)(124 146)(125 145)(126 144)(127 143)(128 142)(129 141)(130 140)(131 139)(132 138)(133 137)(134 136)(151 166)(152 165)(153 164)(154 163)(155 162)(156 161)(157 160)(158 159)(167 180)(168 179)(169 178)(170 177)(171 176)(172 175)(173 174)(181 190)(182 189)(183 188)(184 187)(185 186)(191 210)(192 209)(193 208)(194 207)(195 206)(196 205)(197 204)(198 203)(199 202)(200 201)(211 239)(212 238)(213 237)(214 236)(215 235)(216 234)(217 233)(218 232)(219 231)(220 230)(221 229)(222 228)(223 227)(224 226)
(1 33 201 174)(2 52 202 163)(3 41 203 152)(4 60 204 171)(5 49 205 160)(6 38 206 179)(7 57 207 168)(8 46 208 157)(9 35 209 176)(10 54 210 165)(11 43 181 154)(12 32 182 173)(13 51 183 162)(14 40 184 151)(15 59 185 170)(16 48 186 159)(17 37 187 178)(18 56 188 167)(19 45 189 156)(20 34 190 175)(21 53 191 164)(22 42 192 153)(23 31 193 172)(24 50 194 161)(25 39 195 180)(26 58 196 169)(27 47 197 158)(28 36 198 177)(29 55 199 166)(30 44 200 155)(61 230 146 95)(62 219 147 114)(63 238 148 103)(64 227 149 92)(65 216 150 111)(66 235 121 100)(67 224 122 119)(68 213 123 108)(69 232 124 97)(70 221 125 116)(71 240 126 105)(72 229 127 94)(73 218 128 113)(74 237 129 102)(75 226 130 91)(76 215 131 110)(77 234 132 99)(78 223 133 118)(79 212 134 107)(80 231 135 96)(81 220 136 115)(82 239 137 104)(83 228 138 93)(84 217 139 112)(85 236 140 101)(86 225 141 120)(87 214 142 109)(88 233 143 98)(89 222 144 117)(90 211 145 106)
(1 218)(2 237)(3 226)(4 215)(5 234)(6 223)(7 212)(8 231)(9 220)(10 239)(11 228)(12 217)(13 236)(14 225)(15 214)(16 233)(17 222)(18 211)(19 230)(20 219)(21 238)(22 227)(23 216)(24 235)(25 224)(26 213)(27 232)(28 221)(29 240)(30 229)(31 65)(32 84)(33 73)(34 62)(35 81)(36 70)(37 89)(38 78)(39 67)(40 86)(41 75)(42 64)(43 83)(44 72)(45 61)(46 80)(47 69)(48 88)(49 77)(50 66)(51 85)(52 74)(53 63)(54 82)(55 71)(56 90)(57 79)(58 68)(59 87)(60 76)(91 203)(92 192)(93 181)(94 200)(95 189)(96 208)(97 197)(98 186)(99 205)(100 194)(101 183)(102 202)(103 191)(104 210)(105 199)(106 188)(107 207)(108 196)(109 185)(110 204)(111 193)(112 182)(113 201)(114 190)(115 209)(116 198)(117 187)(118 206)(119 195)(120 184)(121 161)(122 180)(123 169)(124 158)(125 177)(126 166)(127 155)(128 174)(129 163)(130 152)(131 171)(132 160)(133 179)(134 168)(135 157)(136 176)(137 165)(138 154)(139 173)(140 162)(141 151)(142 170)(143 159)(144 178)(145 167)(146 156)(147 175)(148 164)(149 153)(150 172)
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)(31,34)(32,33)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(61,69)(62,68)(63,67)(64,66)(70,90)(71,89)(72,88)(73,87)(74,86)(75,85)(76,84)(77,83)(78,82)(79,81)(91,119)(92,118)(93,117)(94,116)(95,115)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)(121,149)(122,148)(123,147)(124,146)(125,145)(126,144)(127,143)(128,142)(129,141)(130,140)(131,139)(132,138)(133,137)(134,136)(151,166)(152,165)(153,164)(154,163)(155,162)(156,161)(157,160)(158,159)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174)(181,190)(182,189)(183,188)(184,187)(185,186)(191,210)(192,209)(193,208)(194,207)(195,206)(196,205)(197,204)(198,203)(199,202)(200,201)(211,239)(212,238)(213,237)(214,236)(215,235)(216,234)(217,233)(218,232)(219,231)(220,230)(221,229)(222,228)(223,227)(224,226), (1,33,201,174)(2,52,202,163)(3,41,203,152)(4,60,204,171)(5,49,205,160)(6,38,206,179)(7,57,207,168)(8,46,208,157)(9,35,209,176)(10,54,210,165)(11,43,181,154)(12,32,182,173)(13,51,183,162)(14,40,184,151)(15,59,185,170)(16,48,186,159)(17,37,187,178)(18,56,188,167)(19,45,189,156)(20,34,190,175)(21,53,191,164)(22,42,192,153)(23,31,193,172)(24,50,194,161)(25,39,195,180)(26,58,196,169)(27,47,197,158)(28,36,198,177)(29,55,199,166)(30,44,200,155)(61,230,146,95)(62,219,147,114)(63,238,148,103)(64,227,149,92)(65,216,150,111)(66,235,121,100)(67,224,122,119)(68,213,123,108)(69,232,124,97)(70,221,125,116)(71,240,126,105)(72,229,127,94)(73,218,128,113)(74,237,129,102)(75,226,130,91)(76,215,131,110)(77,234,132,99)(78,223,133,118)(79,212,134,107)(80,231,135,96)(81,220,136,115)(82,239,137,104)(83,228,138,93)(84,217,139,112)(85,236,140,101)(86,225,141,120)(87,214,142,109)(88,233,143,98)(89,222,144,117)(90,211,145,106), (1,218)(2,237)(3,226)(4,215)(5,234)(6,223)(7,212)(8,231)(9,220)(10,239)(11,228)(12,217)(13,236)(14,225)(15,214)(16,233)(17,222)(18,211)(19,230)(20,219)(21,238)(22,227)(23,216)(24,235)(25,224)(26,213)(27,232)(28,221)(29,240)(30,229)(31,65)(32,84)(33,73)(34,62)(35,81)(36,70)(37,89)(38,78)(39,67)(40,86)(41,75)(42,64)(43,83)(44,72)(45,61)(46,80)(47,69)(48,88)(49,77)(50,66)(51,85)(52,74)(53,63)(54,82)(55,71)(56,90)(57,79)(58,68)(59,87)(60,76)(91,203)(92,192)(93,181)(94,200)(95,189)(96,208)(97,197)(98,186)(99,205)(100,194)(101,183)(102,202)(103,191)(104,210)(105,199)(106,188)(107,207)(108,196)(109,185)(110,204)(111,193)(112,182)(113,201)(114,190)(115,209)(116,198)(117,187)(118,206)(119,195)(120,184)(121,161)(122,180)(123,169)(124,158)(125,177)(126,166)(127,155)(128,174)(129,163)(130,152)(131,171)(132,160)(133,179)(134,168)(135,157)(136,176)(137,165)(138,154)(139,173)(140,162)(141,151)(142,170)(143,159)(144,178)(145,167)(146,156)(147,175)(148,164)(149,153)(150,172)>;
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)(31,34)(32,33)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(61,69)(62,68)(63,67)(64,66)(70,90)(71,89)(72,88)(73,87)(74,86)(75,85)(76,84)(77,83)(78,82)(79,81)(91,119)(92,118)(93,117)(94,116)(95,115)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)(121,149)(122,148)(123,147)(124,146)(125,145)(126,144)(127,143)(128,142)(129,141)(130,140)(131,139)(132,138)(133,137)(134,136)(151,166)(152,165)(153,164)(154,163)(155,162)(156,161)(157,160)(158,159)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174)(181,190)(182,189)(183,188)(184,187)(185,186)(191,210)(192,209)(193,208)(194,207)(195,206)(196,205)(197,204)(198,203)(199,202)(200,201)(211,239)(212,238)(213,237)(214,236)(215,235)(216,234)(217,233)(218,232)(219,231)(220,230)(221,229)(222,228)(223,227)(224,226), (1,33,201,174)(2,52,202,163)(3,41,203,152)(4,60,204,171)(5,49,205,160)(6,38,206,179)(7,57,207,168)(8,46,208,157)(9,35,209,176)(10,54,210,165)(11,43,181,154)(12,32,182,173)(13,51,183,162)(14,40,184,151)(15,59,185,170)(16,48,186,159)(17,37,187,178)(18,56,188,167)(19,45,189,156)(20,34,190,175)(21,53,191,164)(22,42,192,153)(23,31,193,172)(24,50,194,161)(25,39,195,180)(26,58,196,169)(27,47,197,158)(28,36,198,177)(29,55,199,166)(30,44,200,155)(61,230,146,95)(62,219,147,114)(63,238,148,103)(64,227,149,92)(65,216,150,111)(66,235,121,100)(67,224,122,119)(68,213,123,108)(69,232,124,97)(70,221,125,116)(71,240,126,105)(72,229,127,94)(73,218,128,113)(74,237,129,102)(75,226,130,91)(76,215,131,110)(77,234,132,99)(78,223,133,118)(79,212,134,107)(80,231,135,96)(81,220,136,115)(82,239,137,104)(83,228,138,93)(84,217,139,112)(85,236,140,101)(86,225,141,120)(87,214,142,109)(88,233,143,98)(89,222,144,117)(90,211,145,106), (1,218)(2,237)(3,226)(4,215)(5,234)(6,223)(7,212)(8,231)(9,220)(10,239)(11,228)(12,217)(13,236)(14,225)(15,214)(16,233)(17,222)(18,211)(19,230)(20,219)(21,238)(22,227)(23,216)(24,235)(25,224)(26,213)(27,232)(28,221)(29,240)(30,229)(31,65)(32,84)(33,73)(34,62)(35,81)(36,70)(37,89)(38,78)(39,67)(40,86)(41,75)(42,64)(43,83)(44,72)(45,61)(46,80)(47,69)(48,88)(49,77)(50,66)(51,85)(52,74)(53,63)(54,82)(55,71)(56,90)(57,79)(58,68)(59,87)(60,76)(91,203)(92,192)(93,181)(94,200)(95,189)(96,208)(97,197)(98,186)(99,205)(100,194)(101,183)(102,202)(103,191)(104,210)(105,199)(106,188)(107,207)(108,196)(109,185)(110,204)(111,193)(112,182)(113,201)(114,190)(115,209)(116,198)(117,187)(118,206)(119,195)(120,184)(121,161)(122,180)(123,169)(124,158)(125,177)(126,166)(127,155)(128,174)(129,163)(130,152)(131,171)(132,160)(133,179)(134,168)(135,157)(136,176)(137,165)(138,154)(139,173)(140,162)(141,151)(142,170)(143,159)(144,178)(145,167)(146,156)(147,175)(148,164)(149,153)(150,172) );
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),(31,34),(32,33),(35,60),(36,59),(37,58),(38,57),(39,56),(40,55),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,48),(61,69),(62,68),(63,67),(64,66),(70,90),(71,89),(72,88),(73,87),(74,86),(75,85),(76,84),(77,83),(78,82),(79,81),(91,119),(92,118),(93,117),(94,116),(95,115),(96,114),(97,113),(98,112),(99,111),(100,110),(101,109),(102,108),(103,107),(104,106),(121,149),(122,148),(123,147),(124,146),(125,145),(126,144),(127,143),(128,142),(129,141),(130,140),(131,139),(132,138),(133,137),(134,136),(151,166),(152,165),(153,164),(154,163),(155,162),(156,161),(157,160),(158,159),(167,180),(168,179),(169,178),(170,177),(171,176),(172,175),(173,174),(181,190),(182,189),(183,188),(184,187),(185,186),(191,210),(192,209),(193,208),(194,207),(195,206),(196,205),(197,204),(198,203),(199,202),(200,201),(211,239),(212,238),(213,237),(214,236),(215,235),(216,234),(217,233),(218,232),(219,231),(220,230),(221,229),(222,228),(223,227),(224,226)], [(1,33,201,174),(2,52,202,163),(3,41,203,152),(4,60,204,171),(5,49,205,160),(6,38,206,179),(7,57,207,168),(8,46,208,157),(9,35,209,176),(10,54,210,165),(11,43,181,154),(12,32,182,173),(13,51,183,162),(14,40,184,151),(15,59,185,170),(16,48,186,159),(17,37,187,178),(18,56,188,167),(19,45,189,156),(20,34,190,175),(21,53,191,164),(22,42,192,153),(23,31,193,172),(24,50,194,161),(25,39,195,180),(26,58,196,169),(27,47,197,158),(28,36,198,177),(29,55,199,166),(30,44,200,155),(61,230,146,95),(62,219,147,114),(63,238,148,103),(64,227,149,92),(65,216,150,111),(66,235,121,100),(67,224,122,119),(68,213,123,108),(69,232,124,97),(70,221,125,116),(71,240,126,105),(72,229,127,94),(73,218,128,113),(74,237,129,102),(75,226,130,91),(76,215,131,110),(77,234,132,99),(78,223,133,118),(79,212,134,107),(80,231,135,96),(81,220,136,115),(82,239,137,104),(83,228,138,93),(84,217,139,112),(85,236,140,101),(86,225,141,120),(87,214,142,109),(88,233,143,98),(89,222,144,117),(90,211,145,106)], [(1,218),(2,237),(3,226),(4,215),(5,234),(6,223),(7,212),(8,231),(9,220),(10,239),(11,228),(12,217),(13,236),(14,225),(15,214),(16,233),(17,222),(18,211),(19,230),(20,219),(21,238),(22,227),(23,216),(24,235),(25,224),(26,213),(27,232),(28,221),(29,240),(30,229),(31,65),(32,84),(33,73),(34,62),(35,81),(36,70),(37,89),(38,78),(39,67),(40,86),(41,75),(42,64),(43,83),(44,72),(45,61),(46,80),(47,69),(48,88),(49,77),(50,66),(51,85),(52,74),(53,63),(54,82),(55,71),(56,90),(57,79),(58,68),(59,87),(60,76),(91,203),(92,192),(93,181),(94,200),(95,189),(96,208),(97,197),(98,186),(99,205),(100,194),(101,183),(102,202),(103,191),(104,210),(105,199),(106,188),(107,207),(108,196),(109,185),(110,204),(111,193),(112,182),(113,201),(114,190),(115,209),(116,198),(117,187),(118,206),(119,195),(120,184),(121,161),(122,180),(123,169),(124,158),(125,177),(126,166),(127,155),(128,174),(129,163),(130,152),(131,171),(132,160),(133,179),(134,168),(135,157),(136,176),(137,165),(138,154),(139,173),(140,162),(141,151),(142,170),(143,159),(144,178),(145,167),(146,156),(147,175),(148,164),(149,153),(150,172)]])
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 | 6F | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 30 | 30 | 2 | 6 | 6 | 10 | 10 | 12 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 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 | 2 | 2 | 4 | 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 | C3⋊D4 | C4○D20 | S3×D4 | D4⋊2S3 | S3×D5 | D4×D5 | C2×S3×D5 | Dic5.D6 | D5×C3⋊D4 | D10⋊D6 |
| kernel | D30⋊6D4 | D10⋊Dic3 | C30.Q8 | C5×C6.D4 | C2×D30.C2 | C2×C3⋊D20 | C6×C5⋊D4 | C2×C15⋊7D4 | C2×C5⋊D4 | C3×Dic5 | D30 | C6.D4 | C2×Dic5 | C22×D5 | C22×C10 | C30 | C2×Dic3 | C22×C6 | Dic5 | C6 | C10 | C10 | C23 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of D30⋊6D4 ►in GL4(𝔽61) generated by
| 1 | 45 | 0 | 0 |
| 60 | 17 | 0 | 0 |
| 0 | 0 | 2 | 15 |
| 0 | 0 | 12 | 60 |
| 0 | 17 | 0 | 0 |
| 18 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 60 |
| 4 | 4 | 0 | 0 |
| 11 | 57 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 59 | 5 | 0 | 0 |
| 36 | 2 | 0 | 0 |
| 0 | 0 | 34 | 35 |
| 0 | 0 | 28 | 27 |
G:=sub<GL(4,GF(61))| [1,60,0,0,45,17,0,0,0,0,2,12,0,0,15,60],[0,18,0,0,17,0,0,0,0,0,1,12,0,0,0,60],[4,11,0,0,4,57,0,0,0,0,60,0,0,0,0,60],[59,36,0,0,5,2,0,0,0,0,34,28,0,0,35,27] >;
D30⋊6D4 in GAP, Magma, Sage, TeX
D_{30}\rtimes_6D_4 % in TeX
G:=Group("D30:6D4"); // GroupNames label
G:=SmallGroup(480,609);
// by ID
G=gap.SmallGroup(480,609);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,64,254,219,1356,18822]);
// 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=d*a*d=a^19,c*b*c^-1=a^18*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations