metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.5D4, C4⋊Dic3⋊3D5, C6.38(D4×D5), C30.33(C2×D4), C10.38(S3×D4), (C2×C12).9D10, (C2×C20).221D6, C5⋊2(C23.9D6), Dic15⋊5C4⋊4C2, D30⋊3C4⋊18C2, D10⋊C4⋊15S3, (C22×D5).5D6, C6.64(C4○D20), C10.4(C4○D12), C30.23(C4○D4), D10⋊Dic3⋊4C2, (C2×C30).46C23, C6.5(Q8⋊2D5), (C2×Dic5).94D6, C2.9(C12.28D10), C3⋊3(D10.13D4), C2.16(C20⋊D6), (C2×C60).314C22, (C2×Dic3).87D10, C15⋊5(C22.D4), C10.40(D4⋊2S3), (C6×Dic5).28C22, C2.12(Dic5.D6), (C2×Dic15).49C22, (C10×Dic3).27C22, (C22×D15).22C22, (C2×C4).33(S3×D5), (D5×C2×C6).3C22, (C2×D30.C2)⋊1C2, (C5×C4⋊Dic3)⋊15C2, (C2×C3⋊D20).4C2, C22.135(C2×S3×D5), (C3×D10⋊C4)⋊20C2, (C2×C6).58(C22×D5), (C2×C10).58(C22×S3), SmallGroup(480,432)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.D4
G = < a,b,c,d | a30=b2=c4=d2=1, bab=cac-1=a-1, dad=a19, cbc-1=a13b, dbd=a3b, dcd=a15c-1 >
Subgroups: 940 in 156 conjugacy classes, 46 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C3×D5, D15, C30, C22.D4, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, D30, D30, C2×C30, C10.D4, D10⋊C4, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C23.9D6, D30.C2, C3⋊D20, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, C22×D15, D10.13D4, D10⋊Dic3, Dic15⋊5C4, C3×D10⋊C4, C5×C4⋊Dic3, D30⋊3C4, C2×D30.C2, C2×C3⋊D20, D30.D4
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, Q8⋊2D5, C23.9D6, C2×S3×D5, D10.13D4, C12.28D10, C20⋊D6, Dic5.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 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(52 60)(53 59)(54 58)(55 57)(61 66)(62 65)(63 64)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(91 101)(92 100)(93 99)(94 98)(95 97)(102 120)(103 119)(104 118)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(136 150)(137 149)(138 148)(139 147)(140 146)(141 145)(142 144)(151 165)(152 164)(153 163)(154 162)(155 161)(156 160)(157 159)(166 180)(167 179)(168 178)(169 177)(170 176)(171 175)(172 174)(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)(211 236)(212 235)(213 234)(214 233)(215 232)(216 231)(217 230)(218 229)(219 228)(220 227)(221 226)(222 225)(223 224)(237 240)(238 239)
(1 166 201 34)(2 165 202 33)(3 164 203 32)(4 163 204 31)(5 162 205 60)(6 161 206 59)(7 160 207 58)(8 159 208 57)(9 158 209 56)(10 157 210 55)(11 156 181 54)(12 155 182 53)(13 154 183 52)(14 153 184 51)(15 152 185 50)(16 151 186 49)(17 180 187 48)(18 179 188 47)(19 178 189 46)(20 177 190 45)(21 176 191 44)(22 175 192 43)(23 174 193 42)(24 173 194 41)(25 172 195 40)(26 171 196 39)(27 170 197 38)(28 169 198 37)(29 168 199 36)(30 167 200 35)(61 107 236 124)(62 106 237 123)(63 105 238 122)(64 104 239 121)(65 103 240 150)(66 102 211 149)(67 101 212 148)(68 100 213 147)(69 99 214 146)(70 98 215 145)(71 97 216 144)(72 96 217 143)(73 95 218 142)(74 94 219 141)(75 93 220 140)(76 92 221 139)(77 91 222 138)(78 120 223 137)(79 119 224 136)(80 118 225 135)(81 117 226 134)(82 116 227 133)(83 115 228 132)(84 114 229 131)(85 113 230 130)(86 112 231 129)(87 111 232 128)(88 110 233 127)(89 109 234 126)(90 108 235 125)
(1 119)(2 108)(3 97)(4 116)(5 105)(6 94)(7 113)(8 102)(9 91)(10 110)(11 99)(12 118)(13 107)(14 96)(15 115)(16 104)(17 93)(18 112)(19 101)(20 120)(21 109)(22 98)(23 117)(24 106)(25 95)(26 114)(27 103)(28 92)(29 111)(30 100)(31 212)(32 231)(33 220)(34 239)(35 228)(36 217)(37 236)(38 225)(39 214)(40 233)(41 222)(42 211)(43 230)(44 219)(45 238)(46 227)(47 216)(48 235)(49 224)(50 213)(51 232)(52 221)(53 240)(54 229)(55 218)(56 237)(57 226)(58 215)(59 234)(60 223)(61 169)(62 158)(63 177)(64 166)(65 155)(66 174)(67 163)(68 152)(69 171)(70 160)(71 179)(72 168)(73 157)(74 176)(75 165)(76 154)(77 173)(78 162)(79 151)(80 170)(81 159)(82 178)(83 167)(84 156)(85 175)(86 164)(87 153)(88 172)(89 161)(90 180)(121 186)(122 205)(123 194)(124 183)(125 202)(126 191)(127 210)(128 199)(129 188)(130 207)(131 196)(132 185)(133 204)(134 193)(135 182)(136 201)(137 190)(138 209)(139 198)(140 187)(141 206)(142 195)(143 184)(144 203)(145 192)(146 181)(147 200)(148 189)(149 208)(150 197)
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)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(52,60)(53,59)(54,58)(55,57)(61,66)(62,65)(63,64)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(91,101)(92,100)(93,99)(94,98)(95,97)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,150)(137,149)(138,148)(139,147)(140,146)(141,145)(142,144)(151,165)(152,164)(153,163)(154,162)(155,161)(156,160)(157,159)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(172,174)(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)(211,236)(212,235)(213,234)(214,233)(215,232)(216,231)(217,230)(218,229)(219,228)(220,227)(221,226)(222,225)(223,224)(237,240)(238,239), (1,166,201,34)(2,165,202,33)(3,164,203,32)(4,163,204,31)(5,162,205,60)(6,161,206,59)(7,160,207,58)(8,159,208,57)(9,158,209,56)(10,157,210,55)(11,156,181,54)(12,155,182,53)(13,154,183,52)(14,153,184,51)(15,152,185,50)(16,151,186,49)(17,180,187,48)(18,179,188,47)(19,178,189,46)(20,177,190,45)(21,176,191,44)(22,175,192,43)(23,174,193,42)(24,173,194,41)(25,172,195,40)(26,171,196,39)(27,170,197,38)(28,169,198,37)(29,168,199,36)(30,167,200,35)(61,107,236,124)(62,106,237,123)(63,105,238,122)(64,104,239,121)(65,103,240,150)(66,102,211,149)(67,101,212,148)(68,100,213,147)(69,99,214,146)(70,98,215,145)(71,97,216,144)(72,96,217,143)(73,95,218,142)(74,94,219,141)(75,93,220,140)(76,92,221,139)(77,91,222,138)(78,120,223,137)(79,119,224,136)(80,118,225,135)(81,117,226,134)(82,116,227,133)(83,115,228,132)(84,114,229,131)(85,113,230,130)(86,112,231,129)(87,111,232,128)(88,110,233,127)(89,109,234,126)(90,108,235,125), (1,119)(2,108)(3,97)(4,116)(5,105)(6,94)(7,113)(8,102)(9,91)(10,110)(11,99)(12,118)(13,107)(14,96)(15,115)(16,104)(17,93)(18,112)(19,101)(20,120)(21,109)(22,98)(23,117)(24,106)(25,95)(26,114)(27,103)(28,92)(29,111)(30,100)(31,212)(32,231)(33,220)(34,239)(35,228)(36,217)(37,236)(38,225)(39,214)(40,233)(41,222)(42,211)(43,230)(44,219)(45,238)(46,227)(47,216)(48,235)(49,224)(50,213)(51,232)(52,221)(53,240)(54,229)(55,218)(56,237)(57,226)(58,215)(59,234)(60,223)(61,169)(62,158)(63,177)(64,166)(65,155)(66,174)(67,163)(68,152)(69,171)(70,160)(71,179)(72,168)(73,157)(74,176)(75,165)(76,154)(77,173)(78,162)(79,151)(80,170)(81,159)(82,178)(83,167)(84,156)(85,175)(86,164)(87,153)(88,172)(89,161)(90,180)(121,186)(122,205)(123,194)(124,183)(125,202)(126,191)(127,210)(128,199)(129,188)(130,207)(131,196)(132,185)(133,204)(134,193)(135,182)(136,201)(137,190)(138,209)(139,198)(140,187)(141,206)(142,195)(143,184)(144,203)(145,192)(146,181)(147,200)(148,189)(149,208)(150,197)>;
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)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(52,60)(53,59)(54,58)(55,57)(61,66)(62,65)(63,64)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(91,101)(92,100)(93,99)(94,98)(95,97)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,150)(137,149)(138,148)(139,147)(140,146)(141,145)(142,144)(151,165)(152,164)(153,163)(154,162)(155,161)(156,160)(157,159)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(172,174)(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)(211,236)(212,235)(213,234)(214,233)(215,232)(216,231)(217,230)(218,229)(219,228)(220,227)(221,226)(222,225)(223,224)(237,240)(238,239), (1,166,201,34)(2,165,202,33)(3,164,203,32)(4,163,204,31)(5,162,205,60)(6,161,206,59)(7,160,207,58)(8,159,208,57)(9,158,209,56)(10,157,210,55)(11,156,181,54)(12,155,182,53)(13,154,183,52)(14,153,184,51)(15,152,185,50)(16,151,186,49)(17,180,187,48)(18,179,188,47)(19,178,189,46)(20,177,190,45)(21,176,191,44)(22,175,192,43)(23,174,193,42)(24,173,194,41)(25,172,195,40)(26,171,196,39)(27,170,197,38)(28,169,198,37)(29,168,199,36)(30,167,200,35)(61,107,236,124)(62,106,237,123)(63,105,238,122)(64,104,239,121)(65,103,240,150)(66,102,211,149)(67,101,212,148)(68,100,213,147)(69,99,214,146)(70,98,215,145)(71,97,216,144)(72,96,217,143)(73,95,218,142)(74,94,219,141)(75,93,220,140)(76,92,221,139)(77,91,222,138)(78,120,223,137)(79,119,224,136)(80,118,225,135)(81,117,226,134)(82,116,227,133)(83,115,228,132)(84,114,229,131)(85,113,230,130)(86,112,231,129)(87,111,232,128)(88,110,233,127)(89,109,234,126)(90,108,235,125), (1,119)(2,108)(3,97)(4,116)(5,105)(6,94)(7,113)(8,102)(9,91)(10,110)(11,99)(12,118)(13,107)(14,96)(15,115)(16,104)(17,93)(18,112)(19,101)(20,120)(21,109)(22,98)(23,117)(24,106)(25,95)(26,114)(27,103)(28,92)(29,111)(30,100)(31,212)(32,231)(33,220)(34,239)(35,228)(36,217)(37,236)(38,225)(39,214)(40,233)(41,222)(42,211)(43,230)(44,219)(45,238)(46,227)(47,216)(48,235)(49,224)(50,213)(51,232)(52,221)(53,240)(54,229)(55,218)(56,237)(57,226)(58,215)(59,234)(60,223)(61,169)(62,158)(63,177)(64,166)(65,155)(66,174)(67,163)(68,152)(69,171)(70,160)(71,179)(72,168)(73,157)(74,176)(75,165)(76,154)(77,173)(78,162)(79,151)(80,170)(81,159)(82,178)(83,167)(84,156)(85,175)(86,164)(87,153)(88,172)(89,161)(90,180)(121,186)(122,205)(123,194)(124,183)(125,202)(126,191)(127,210)(128,199)(129,188)(130,207)(131,196)(132,185)(133,204)(134,193)(135,182)(136,201)(137,190)(138,209)(139,198)(140,187)(141,206)(142,195)(143,184)(144,203)(145,192)(146,181)(147,200)(148,189)(149,208)(150,197) );
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),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(52,60),(53,59),(54,58),(55,57),(61,66),(62,65),(63,64),(67,90),(68,89),(69,88),(70,87),(71,86),(72,85),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79),(91,101),(92,100),(93,99),(94,98),(95,97),(102,120),(103,119),(104,118),(105,117),(106,116),(107,115),(108,114),(109,113),(110,112),(121,135),(122,134),(123,133),(124,132),(125,131),(126,130),(127,129),(136,150),(137,149),(138,148),(139,147),(140,146),(141,145),(142,144),(151,165),(152,164),(153,163),(154,162),(155,161),(156,160),(157,159),(166,180),(167,179),(168,178),(169,177),(170,176),(171,175),(172,174),(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),(211,236),(212,235),(213,234),(214,233),(215,232),(216,231),(217,230),(218,229),(219,228),(220,227),(221,226),(222,225),(223,224),(237,240),(238,239)], [(1,166,201,34),(2,165,202,33),(3,164,203,32),(4,163,204,31),(5,162,205,60),(6,161,206,59),(7,160,207,58),(8,159,208,57),(9,158,209,56),(10,157,210,55),(11,156,181,54),(12,155,182,53),(13,154,183,52),(14,153,184,51),(15,152,185,50),(16,151,186,49),(17,180,187,48),(18,179,188,47),(19,178,189,46),(20,177,190,45),(21,176,191,44),(22,175,192,43),(23,174,193,42),(24,173,194,41),(25,172,195,40),(26,171,196,39),(27,170,197,38),(28,169,198,37),(29,168,199,36),(30,167,200,35),(61,107,236,124),(62,106,237,123),(63,105,238,122),(64,104,239,121),(65,103,240,150),(66,102,211,149),(67,101,212,148),(68,100,213,147),(69,99,214,146),(70,98,215,145),(71,97,216,144),(72,96,217,143),(73,95,218,142),(74,94,219,141),(75,93,220,140),(76,92,221,139),(77,91,222,138),(78,120,223,137),(79,119,224,136),(80,118,225,135),(81,117,226,134),(82,116,227,133),(83,115,228,132),(84,114,229,131),(85,113,230,130),(86,112,231,129),(87,111,232,128),(88,110,233,127),(89,109,234,126),(90,108,235,125)], [(1,119),(2,108),(3,97),(4,116),(5,105),(6,94),(7,113),(8,102),(9,91),(10,110),(11,99),(12,118),(13,107),(14,96),(15,115),(16,104),(17,93),(18,112),(19,101),(20,120),(21,109),(22,98),(23,117),(24,106),(25,95),(26,114),(27,103),(28,92),(29,111),(30,100),(31,212),(32,231),(33,220),(34,239),(35,228),(36,217),(37,236),(38,225),(39,214),(40,233),(41,222),(42,211),(43,230),(44,219),(45,238),(46,227),(47,216),(48,235),(49,224),(50,213),(51,232),(52,221),(53,240),(54,229),(55,218),(56,237),(57,226),(58,215),(59,234),(60,223),(61,169),(62,158),(63,177),(64,166),(65,155),(66,174),(67,163),(68,152),(69,171),(70,160),(71,179),(72,168),(73,157),(74,176),(75,165),(76,154),(77,173),(78,162),(79,151),(80,170),(81,159),(82,178),(83,167),(84,156),(85,175),(86,164),(87,153),(88,172),(89,161),(90,180),(121,186),(122,205),(123,194),(124,183),(125,202),(126,191),(127,210),(128,199),(129,188),(130,207),(131,196),(132,185),(133,204),(134,193),(135,182),(136,201),(137,190),(138,209),(139,198),(140,187),(141,206),(142,195),(143,184),(144,203),(145,192),(146,181),(147,200),(148,189),(149,208),(150,197)]])
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 | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| 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 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 20 | 30 | 30 | 2 | 4 | 6 | 6 | 10 | 10 | 12 | 60 | 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 | 1 | 2 | 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 | D6 | C4○D4 | D10 | D10 | C4○D12 | C4○D20 | S3×D4 | D4⋊2S3 | S3×D5 | D4×D5 | Q8⋊2D5 | C2×S3×D5 | C12.28D10 | C20⋊D6 | Dic5.D6 |
| kernel | D30.D4 | D10⋊Dic3 | Dic15⋊5C4 | C3×D10⋊C4 | C5×C4⋊Dic3 | D30⋊3C4 | C2×D30.C2 | C2×C3⋊D20 | D10⋊C4 | D30 | C4⋊Dic3 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C10 | C6 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D30.D4 ►in GL6(𝔽61)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 1 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 36 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 43 |
| 0 | 0 | 0 | 0 | 44 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 43 | 0 | 0 |
| 0 | 0 | 1 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 30 |
| 0 | 0 | 0 | 0 | 55 | 22 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 24 | 0 | 0 |
| 0 | 0 | 11 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 25 |
| 0 | 0 | 0 | 0 | 56 | 59 |
G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,17,16,0,0,0,0,1,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,36,1,0,0,0,0,0,0,0,44,0,0,0,0,43,0],[60,1,0,0,0,0,0,1,0,0,0,0,0,0,18,1,0,0,0,0,43,43,0,0,0,0,0,0,39,55,0,0,0,0,30,22],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,11,0,0,0,0,24,46,0,0,0,0,0,0,2,56,0,0,0,0,25,59] >;
D30.D4 in GAP, Magma, Sage, TeX
D_{30}.D_4 % in TeX
G:=Group("D30.D4"); // GroupNames label
G:=SmallGroup(480,432);
// by ID
G=gap.SmallGroup(480,432);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,64,590,219,100,1356,18822]);
// 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^13*b,d*b*d=a^3*b,d*c*d=a^15*c^-1>;
// generators/relations