metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.34D4, C6.53(D4×D5), Dic3⋊C4⋊9D5, C10.55(S3×D4), D30⋊4C4⋊6C2, D10⋊C4⋊9S3, (C2×C20).180D6, C30.106(C2×D4), C5⋊1(C23.9D6), C30.Q8⋊4C2, (C22×D5).4D6, C30.21(C4○D4), C6.21(C4○D20), D10⋊Dic3⋊3C2, (C2×C12).178D10, (C2×C30).44C23, (C2×Dic3).8D10, (C2×Dic5).10D6, C10.24(C4○D12), C2.8(D20⋊S3), C10.5(D4⋊2S3), C2.9(D10⋊D6), C3⋊2(D10.13D4), (C2×C60).157C22, C6.24(Q8⋊2D5), C15⋊3(C22.D4), (C6×Dic5).26C22, C2.14(D6.D10), (C10×Dic3).25C22, (C22×D15).95C22, (C2×Dic15).183C22, (C2×C4×D15)⋊7C2, (D5×C2×C6).2C22, (C5×Dic3⋊C4)⋊9C2, (C2×C4).170(S3×D5), (C2×C3⋊D20).3C2, (C3×D10⋊C4)⋊9C2, C22.133(C2×S3×D5), (C2×C6).56(C22×D5), (C2×C10).56(C22×S3), SmallGroup(480,430)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.34D4
G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, cac-1=a11, dad=a19, cbc-1=a25b, 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, C3⋊D20, C6×Dic5, C10×Dic3, C4×D15, C2×Dic15, C2×C60, D5×C2×C6, C22×D15, D10.13D4, D10⋊Dic3, D30⋊4C4, C30.Q8, C3×D10⋊C4, C5×Dic3⋊C4, C2×C3⋊D20, C2×C4×D15, D30.34D4
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, D20⋊S3, D6.D10, D10⋊D6, D30.34D4
►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 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(56 60)(57 59)(61 86)(62 85)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(87 90)(88 89)(91 118)(92 117)(93 116)(94 115)(95 114)(96 113)(97 112)(98 111)(99 110)(100 109)(101 108)(102 107)(103 106)(104 105)(119 120)(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)(151 157)(152 156)(153 155)(158 180)(159 179)(160 178)(161 177)(162 176)(163 175)(164 174)(165 173)(166 172)(167 171)(168 170)(181 202)(182 201)(183 200)(184 199)(185 198)(186 197)(187 196)(188 195)(189 194)(190 193)(191 192)(203 210)(204 209)(205 208)(206 207)(211 229)(212 228)(213 227)(214 226)(215 225)(216 224)(217 223)(218 222)(219 221)(230 240)(231 239)(232 238)(233 237)(234 236)
(1 51 89 213)(2 32 90 224)(3 43 61 235)(4 54 62 216)(5 35 63 227)(6 46 64 238)(7 57 65 219)(8 38 66 230)(9 49 67 211)(10 60 68 222)(11 41 69 233)(12 52 70 214)(13 33 71 225)(14 44 72 236)(15 55 73 217)(16 36 74 228)(17 47 75 239)(18 58 76 220)(19 39 77 231)(20 50 78 212)(21 31 79 223)(22 42 80 234)(23 53 81 215)(24 34 82 226)(25 45 83 237)(26 56 84 218)(27 37 85 229)(28 48 86 240)(29 59 87 221)(30 40 88 232)(91 124 193 173)(92 135 194 154)(93 146 195 165)(94 127 196 176)(95 138 197 157)(96 149 198 168)(97 130 199 179)(98 141 200 160)(99 122 201 171)(100 133 202 152)(101 144 203 163)(102 125 204 174)(103 136 205 155)(104 147 206 166)(105 128 207 177)(106 139 208 158)(107 150 209 169)(108 131 210 180)(109 142 181 161)(110 123 182 172)(111 134 183 153)(112 145 184 164)(113 126 185 175)(114 137 186 156)(115 148 187 167)(116 129 188 178)(117 140 189 159)(118 121 190 170)(119 132 191 151)(120 143 192 162)
(1 143)(2 132)(3 121)(4 140)(5 129)(6 148)(7 137)(8 126)(9 145)(10 134)(11 123)(12 142)(13 131)(14 150)(15 139)(16 128)(17 147)(18 136)(19 125)(20 144)(21 133)(22 122)(23 141)(24 130)(25 149)(26 138)(27 127)(28 146)(29 135)(30 124)(31 115)(32 104)(33 93)(34 112)(35 101)(36 120)(37 109)(38 98)(39 117)(40 106)(41 95)(42 114)(43 103)(44 92)(45 111)(46 100)(47 119)(48 108)(49 97)(50 116)(51 105)(52 94)(53 113)(54 102)(55 91)(56 110)(57 99)(58 118)(59 107)(60 96)(61 170)(62 159)(63 178)(64 167)(65 156)(66 175)(67 164)(68 153)(69 172)(70 161)(71 180)(72 169)(73 158)(74 177)(75 166)(76 155)(77 174)(78 163)(79 152)(80 171)(81 160)(82 179)(83 168)(84 157)(85 176)(86 165)(87 154)(88 173)(89 162)(90 151)(181 229)(182 218)(183 237)(184 226)(185 215)(186 234)(187 223)(188 212)(189 231)(190 220)(191 239)(192 228)(193 217)(194 236)(195 225)(196 214)(197 233)(198 222)(199 211)(200 230)(201 219)(202 238)(203 227)(204 216)(205 235)(206 224)(207 213)(208 232)(209 221)(210 240)
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)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(56,60)(57,59)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(87,90)(88,89)(91,118)(92,117)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105)(119,120)(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)(151,157)(152,156)(153,155)(158,180)(159,179)(160,178)(161,177)(162,176)(163,175)(164,174)(165,173)(166,172)(167,171)(168,170)(181,202)(182,201)(183,200)(184,199)(185,198)(186,197)(187,196)(188,195)(189,194)(190,193)(191,192)(203,210)(204,209)(205,208)(206,207)(211,229)(212,228)(213,227)(214,226)(215,225)(216,224)(217,223)(218,222)(219,221)(230,240)(231,239)(232,238)(233,237)(234,236), (1,51,89,213)(2,32,90,224)(3,43,61,235)(4,54,62,216)(5,35,63,227)(6,46,64,238)(7,57,65,219)(8,38,66,230)(9,49,67,211)(10,60,68,222)(11,41,69,233)(12,52,70,214)(13,33,71,225)(14,44,72,236)(15,55,73,217)(16,36,74,228)(17,47,75,239)(18,58,76,220)(19,39,77,231)(20,50,78,212)(21,31,79,223)(22,42,80,234)(23,53,81,215)(24,34,82,226)(25,45,83,237)(26,56,84,218)(27,37,85,229)(28,48,86,240)(29,59,87,221)(30,40,88,232)(91,124,193,173)(92,135,194,154)(93,146,195,165)(94,127,196,176)(95,138,197,157)(96,149,198,168)(97,130,199,179)(98,141,200,160)(99,122,201,171)(100,133,202,152)(101,144,203,163)(102,125,204,174)(103,136,205,155)(104,147,206,166)(105,128,207,177)(106,139,208,158)(107,150,209,169)(108,131,210,180)(109,142,181,161)(110,123,182,172)(111,134,183,153)(112,145,184,164)(113,126,185,175)(114,137,186,156)(115,148,187,167)(116,129,188,178)(117,140,189,159)(118,121,190,170)(119,132,191,151)(120,143,192,162), (1,143)(2,132)(3,121)(4,140)(5,129)(6,148)(7,137)(8,126)(9,145)(10,134)(11,123)(12,142)(13,131)(14,150)(15,139)(16,128)(17,147)(18,136)(19,125)(20,144)(21,133)(22,122)(23,141)(24,130)(25,149)(26,138)(27,127)(28,146)(29,135)(30,124)(31,115)(32,104)(33,93)(34,112)(35,101)(36,120)(37,109)(38,98)(39,117)(40,106)(41,95)(42,114)(43,103)(44,92)(45,111)(46,100)(47,119)(48,108)(49,97)(50,116)(51,105)(52,94)(53,113)(54,102)(55,91)(56,110)(57,99)(58,118)(59,107)(60,96)(61,170)(62,159)(63,178)(64,167)(65,156)(66,175)(67,164)(68,153)(69,172)(70,161)(71,180)(72,169)(73,158)(74,177)(75,166)(76,155)(77,174)(78,163)(79,152)(80,171)(81,160)(82,179)(83,168)(84,157)(85,176)(86,165)(87,154)(88,173)(89,162)(90,151)(181,229)(182,218)(183,237)(184,226)(185,215)(186,234)(187,223)(188,212)(189,231)(190,220)(191,239)(192,228)(193,217)(194,236)(195,225)(196,214)(197,233)(198,222)(199,211)(200,230)(201,219)(202,238)(203,227)(204,216)(205,235)(206,224)(207,213)(208,232)(209,221)(210,240)>;
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)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(56,60)(57,59)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(87,90)(88,89)(91,118)(92,117)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105)(119,120)(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)(151,157)(152,156)(153,155)(158,180)(159,179)(160,178)(161,177)(162,176)(163,175)(164,174)(165,173)(166,172)(167,171)(168,170)(181,202)(182,201)(183,200)(184,199)(185,198)(186,197)(187,196)(188,195)(189,194)(190,193)(191,192)(203,210)(204,209)(205,208)(206,207)(211,229)(212,228)(213,227)(214,226)(215,225)(216,224)(217,223)(218,222)(219,221)(230,240)(231,239)(232,238)(233,237)(234,236), (1,51,89,213)(2,32,90,224)(3,43,61,235)(4,54,62,216)(5,35,63,227)(6,46,64,238)(7,57,65,219)(8,38,66,230)(9,49,67,211)(10,60,68,222)(11,41,69,233)(12,52,70,214)(13,33,71,225)(14,44,72,236)(15,55,73,217)(16,36,74,228)(17,47,75,239)(18,58,76,220)(19,39,77,231)(20,50,78,212)(21,31,79,223)(22,42,80,234)(23,53,81,215)(24,34,82,226)(25,45,83,237)(26,56,84,218)(27,37,85,229)(28,48,86,240)(29,59,87,221)(30,40,88,232)(91,124,193,173)(92,135,194,154)(93,146,195,165)(94,127,196,176)(95,138,197,157)(96,149,198,168)(97,130,199,179)(98,141,200,160)(99,122,201,171)(100,133,202,152)(101,144,203,163)(102,125,204,174)(103,136,205,155)(104,147,206,166)(105,128,207,177)(106,139,208,158)(107,150,209,169)(108,131,210,180)(109,142,181,161)(110,123,182,172)(111,134,183,153)(112,145,184,164)(113,126,185,175)(114,137,186,156)(115,148,187,167)(116,129,188,178)(117,140,189,159)(118,121,190,170)(119,132,191,151)(120,143,192,162), (1,143)(2,132)(3,121)(4,140)(5,129)(6,148)(7,137)(8,126)(9,145)(10,134)(11,123)(12,142)(13,131)(14,150)(15,139)(16,128)(17,147)(18,136)(19,125)(20,144)(21,133)(22,122)(23,141)(24,130)(25,149)(26,138)(27,127)(28,146)(29,135)(30,124)(31,115)(32,104)(33,93)(34,112)(35,101)(36,120)(37,109)(38,98)(39,117)(40,106)(41,95)(42,114)(43,103)(44,92)(45,111)(46,100)(47,119)(48,108)(49,97)(50,116)(51,105)(52,94)(53,113)(54,102)(55,91)(56,110)(57,99)(58,118)(59,107)(60,96)(61,170)(62,159)(63,178)(64,167)(65,156)(66,175)(67,164)(68,153)(69,172)(70,161)(71,180)(72,169)(73,158)(74,177)(75,166)(76,155)(77,174)(78,163)(79,152)(80,171)(81,160)(82,179)(83,168)(84,157)(85,176)(86,165)(87,154)(88,173)(89,162)(90,151)(181,229)(182,218)(183,237)(184,226)(185,215)(186,234)(187,223)(188,212)(189,231)(190,220)(191,239)(192,228)(193,217)(194,236)(195,225)(196,214)(197,233)(198,222)(199,211)(200,230)(201,219)(202,238)(203,227)(204,216)(205,235)(206,224)(207,213)(208,232)(209,221)(210,240) );
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),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(41,45),(42,44),(56,60),(57,59),(61,86),(62,85),(63,84),(64,83),(65,82),(66,81),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(87,90),(88,89),(91,118),(92,117),(93,116),(94,115),(95,114),(96,113),(97,112),(98,111),(99,110),(100,109),(101,108),(102,107),(103,106),(104,105),(119,120),(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),(151,157),(152,156),(153,155),(158,180),(159,179),(160,178),(161,177),(162,176),(163,175),(164,174),(165,173),(166,172),(167,171),(168,170),(181,202),(182,201),(183,200),(184,199),(185,198),(186,197),(187,196),(188,195),(189,194),(190,193),(191,192),(203,210),(204,209),(205,208),(206,207),(211,229),(212,228),(213,227),(214,226),(215,225),(216,224),(217,223),(218,222),(219,221),(230,240),(231,239),(232,238),(233,237),(234,236)], [(1,51,89,213),(2,32,90,224),(3,43,61,235),(4,54,62,216),(5,35,63,227),(6,46,64,238),(7,57,65,219),(8,38,66,230),(9,49,67,211),(10,60,68,222),(11,41,69,233),(12,52,70,214),(13,33,71,225),(14,44,72,236),(15,55,73,217),(16,36,74,228),(17,47,75,239),(18,58,76,220),(19,39,77,231),(20,50,78,212),(21,31,79,223),(22,42,80,234),(23,53,81,215),(24,34,82,226),(25,45,83,237),(26,56,84,218),(27,37,85,229),(28,48,86,240),(29,59,87,221),(30,40,88,232),(91,124,193,173),(92,135,194,154),(93,146,195,165),(94,127,196,176),(95,138,197,157),(96,149,198,168),(97,130,199,179),(98,141,200,160),(99,122,201,171),(100,133,202,152),(101,144,203,163),(102,125,204,174),(103,136,205,155),(104,147,206,166),(105,128,207,177),(106,139,208,158),(107,150,209,169),(108,131,210,180),(109,142,181,161),(110,123,182,172),(111,134,183,153),(112,145,184,164),(113,126,185,175),(114,137,186,156),(115,148,187,167),(116,129,188,178),(117,140,189,159),(118,121,190,170),(119,132,191,151),(120,143,192,162)], [(1,143),(2,132),(3,121),(4,140),(5,129),(6,148),(7,137),(8,126),(9,145),(10,134),(11,123),(12,142),(13,131),(14,150),(15,139),(16,128),(17,147),(18,136),(19,125),(20,144),(21,133),(22,122),(23,141),(24,130),(25,149),(26,138),(27,127),(28,146),(29,135),(30,124),(31,115),(32,104),(33,93),(34,112),(35,101),(36,120),(37,109),(38,98),(39,117),(40,106),(41,95),(42,114),(43,103),(44,92),(45,111),(46,100),(47,119),(48,108),(49,97),(50,116),(51,105),(52,94),(53,113),(54,102),(55,91),(56,110),(57,99),(58,118),(59,107),(60,96),(61,170),(62,159),(63,178),(64,167),(65,156),(66,175),(67,164),(68,153),(69,172),(70,161),(71,180),(72,169),(73,158),(74,177),(75,166),(76,155),(77,174),(78,163),(79,152),(80,171),(81,160),(82,179),(83,168),(84,157),(85,176),(86,165),(87,154),(88,173),(89,162),(90,151),(181,229),(182,218),(183,237),(184,226),(185,215),(186,234),(187,223),(188,212),(189,231),(190,220),(191,239),(192,228),(193,217),(194,236),(195,225),(196,214),(197,233),(198,222),(199,211),(200,230),(201,219),(202,238),(203,227),(204,216),(205,235),(206,224),(207,213),(208,232),(209,221),(210,240)]])
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 | 2 | 2 | 12 | 12 | 20 | 30 | 30 | 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 | D20⋊S3 | D6.D10 | D10⋊D6 |
| kernel | D30.34D4 | D10⋊Dic3 | D30⋊4C4 | C30.Q8 | C3×D10⋊C4 | C5×Dic3⋊C4 | C2×C3⋊D20 | C2×C4×D15 | D10⋊C4 | D30 | Dic3⋊C4 | 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 D30.34D4 ►in GL6(𝔽61)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 45 | 0 | 0 |
| 0 | 0 | 60 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 38 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 28 | 0 | 0 |
| 0 | 0 | 17 | 47 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 60 |
| 0 | 0 | 0 | 0 | 57 | 19 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 59 | 5 | 0 | 0 |
| 0 | 0 | 36 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 50 |
| 0 | 0 | 0 | 0 | 17 | 26 |
G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,60,0,0,0,0,45,17,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,18,0,0,0,0,17,0,0,0,0,0,0,0,60,38,0,0,0,0,0,1],[60,1,0,0,0,0,0,1,0,0,0,0,0,0,14,17,0,0,0,0,28,47,0,0,0,0,0,0,42,57,0,0,0,0,60,19],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,59,36,0,0,0,0,5,2,0,0,0,0,0,0,35,17,0,0,0,0,50,26] >;
D30.34D4 in GAP, Magma, Sage, TeX
D_{30}._{34}D_4 % in TeX
G:=Group("D30.34D4"); // GroupNames label
G:=SmallGroup(480,430);
// by ID
G=gap.SmallGroup(480,430);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,120,422,219,142,1356,18822]);
// 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^25*b,d*b*d=a^3*b,d*c*d=a^15*c^-1>;
// generators/relations