metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.35D4, D6⋊C4⋊9D5, C6.54(D4×D5), D6⋊Dic5⋊4C2, C10.56(S3×D4), D30⋊4C4⋊7C2, C30.107(C2×D4), C10.D4⋊9S3, (C2×C20).181D6, C5⋊2(D6.D4), C6.Dic10⋊3C2, C30.22(C4○D4), C6.22(C4○D20), (C2×C12).179D10, C6.6(D4⋊2D5), (C2×C30).45C23, (C2×Dic3).9D10, (C2×Dic5).11D6, (C22×S3).5D10, C10.25(C4○D12), C3⋊1(D10.12D4), (C2×C60).158C22, C15⋊4(C22.D4), C2.10(D10⋊D6), C2.10(D12⋊D5), C10.25(Q8⋊3S3), (C6×Dic5).27C22, C2.15(D6.D10), (C10×Dic3).26C22, (C22×D15).96C22, (C2×Dic15).184C22, (C2×C4×D15)⋊8C2, (C5×D6⋊C4)⋊9C2, (C2×C4).171(S3×D5), (C2×C5⋊D12).4C2, (S3×C2×C10).5C22, C22.134(C2×S3×D5), (C3×C10.D4)⋊9C2, (C2×C6).57(C22×D5), (C2×C10).57(C22×S3), SmallGroup(480,431)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.35D4
G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, cac-1=a19, dad=a11, cbc-1=a3b, dbd=a25b, dcd=a15c-1 >
Subgroups: 908 in 156 conjugacy classes, 46 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, C20, D10, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C5×S3, D15, C30, C22.D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, D30, D30, C2×C30, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C5⋊D4, D6.D4, C5⋊D12, C6×Dic5, C10×Dic3, C4×D15, C2×Dic15, C2×C60, S3×C2×C10, C22×D15, D10.12D4, D6⋊Dic5, D30⋊4C4, C6.Dic10, C3×C10.D4, C5×D6⋊C4, C2×C5⋊D12, C2×C4×D15, D30.35D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C22.D4, C22×D5, C4○D12, S3×D4, Q8⋊3S3, S3×D5, C4○D20, D4×D5, D4⋊2D5, D6.D4, C2×S3×D5, D10.12D4, D12⋊D5, D6.D10, D10⋊D6, D30.35D4
►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 56)(32 55)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 48)(40 47)(41 46)(42 45)(43 44)(57 60)(58 59)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(84 90)(85 89)(86 88)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(101 102)(113 120)(114 119)(115 118)(116 117)(121 129)(122 128)(123 127)(124 126)(130 150)(131 149)(132 148)(133 147)(134 146)(135 145)(136 144)(137 143)(138 142)(139 141)(151 176)(152 175)(153 174)(154 173)(155 172)(156 171)(157 170)(158 169)(159 168)(160 167)(161 166)(162 165)(163 164)(177 180)(178 179)(182 210)(183 209)(184 208)(185 207)(186 206)(187 205)(188 204)(189 203)(190 202)(191 201)(192 200)(193 199)(194 198)(195 197)(211 235)(212 234)(213 233)(214 232)(215 231)(216 230)(217 229)(218 228)(219 227)(220 226)(221 225)(222 224)(236 240)(237 239)
(1 216 59 148)(2 235 60 137)(3 224 31 126)(4 213 32 145)(5 232 33 134)(6 221 34 123)(7 240 35 142)(8 229 36 131)(9 218 37 150)(10 237 38 139)(11 226 39 128)(12 215 40 147)(13 234 41 136)(14 223 42 125)(15 212 43 144)(16 231 44 133)(17 220 45 122)(18 239 46 141)(19 228 47 130)(20 217 48 149)(21 236 49 138)(22 225 50 127)(23 214 51 146)(24 233 52 135)(25 222 53 124)(26 211 54 143)(27 230 55 132)(28 219 56 121)(29 238 57 140)(30 227 58 129)(61 101 185 178)(62 120 186 167)(63 109 187 156)(64 98 188 175)(65 117 189 164)(66 106 190 153)(67 95 191 172)(68 114 192 161)(69 103 193 180)(70 92 194 169)(71 111 195 158)(72 100 196 177)(73 119 197 166)(74 108 198 155)(75 97 199 174)(76 116 200 163)(77 105 201 152)(78 94 202 171)(79 113 203 160)(80 102 204 179)(81 91 205 168)(82 110 206 157)(83 99 207 176)(84 118 208 165)(85 107 209 154)(86 96 210 173)(87 115 181 162)(88 104 182 151)(89 93 183 170)(90 112 184 159)
(1 204)(2 185)(3 196)(4 207)(5 188)(6 199)(7 210)(8 191)(9 202)(10 183)(11 194)(12 205)(13 186)(14 197)(15 208)(16 189)(17 200)(18 181)(19 192)(20 203)(21 184)(22 195)(23 206)(24 187)(25 198)(26 209)(27 190)(28 201)(29 182)(30 193)(31 72)(32 83)(33 64)(34 75)(35 86)(36 67)(37 78)(38 89)(39 70)(40 81)(41 62)(42 73)(43 84)(44 65)(45 76)(46 87)(47 68)(48 79)(49 90)(50 71)(51 82)(52 63)(53 74)(54 85)(55 66)(56 77)(57 88)(58 69)(59 80)(60 61)(91 230)(92 211)(93 222)(94 233)(95 214)(96 225)(97 236)(98 217)(99 228)(100 239)(101 220)(102 231)(103 212)(104 223)(105 234)(106 215)(107 226)(108 237)(109 218)(110 229)(111 240)(112 221)(113 232)(114 213)(115 224)(116 235)(117 216)(118 227)(119 238)(120 219)(121 167)(122 178)(123 159)(124 170)(125 151)(126 162)(127 173)(128 154)(129 165)(130 176)(131 157)(132 168)(133 179)(134 160)(135 171)(136 152)(137 163)(138 174)(139 155)(140 166)(141 177)(142 158)(143 169)(144 180)(145 161)(146 172)(147 153)(148 164)(149 175)(150 156)
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)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(57,60)(58,59)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(84,90)(85,89)(86,88)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)(113,120)(114,119)(115,118)(116,117)(121,129)(122,128)(123,127)(124,126)(130,150)(131,149)(132,148)(133,147)(134,146)(135,145)(136,144)(137,143)(138,142)(139,141)(151,176)(152,175)(153,174)(154,173)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,165)(163,164)(177,180)(178,179)(182,210)(183,209)(184,208)(185,207)(186,206)(187,205)(188,204)(189,203)(190,202)(191,201)(192,200)(193,199)(194,198)(195,197)(211,235)(212,234)(213,233)(214,232)(215,231)(216,230)(217,229)(218,228)(219,227)(220,226)(221,225)(222,224)(236,240)(237,239), (1,216,59,148)(2,235,60,137)(3,224,31,126)(4,213,32,145)(5,232,33,134)(6,221,34,123)(7,240,35,142)(8,229,36,131)(9,218,37,150)(10,237,38,139)(11,226,39,128)(12,215,40,147)(13,234,41,136)(14,223,42,125)(15,212,43,144)(16,231,44,133)(17,220,45,122)(18,239,46,141)(19,228,47,130)(20,217,48,149)(21,236,49,138)(22,225,50,127)(23,214,51,146)(24,233,52,135)(25,222,53,124)(26,211,54,143)(27,230,55,132)(28,219,56,121)(29,238,57,140)(30,227,58,129)(61,101,185,178)(62,120,186,167)(63,109,187,156)(64,98,188,175)(65,117,189,164)(66,106,190,153)(67,95,191,172)(68,114,192,161)(69,103,193,180)(70,92,194,169)(71,111,195,158)(72,100,196,177)(73,119,197,166)(74,108,198,155)(75,97,199,174)(76,116,200,163)(77,105,201,152)(78,94,202,171)(79,113,203,160)(80,102,204,179)(81,91,205,168)(82,110,206,157)(83,99,207,176)(84,118,208,165)(85,107,209,154)(86,96,210,173)(87,115,181,162)(88,104,182,151)(89,93,183,170)(90,112,184,159), (1,204)(2,185)(3,196)(4,207)(5,188)(6,199)(7,210)(8,191)(9,202)(10,183)(11,194)(12,205)(13,186)(14,197)(15,208)(16,189)(17,200)(18,181)(19,192)(20,203)(21,184)(22,195)(23,206)(24,187)(25,198)(26,209)(27,190)(28,201)(29,182)(30,193)(31,72)(32,83)(33,64)(34,75)(35,86)(36,67)(37,78)(38,89)(39,70)(40,81)(41,62)(42,73)(43,84)(44,65)(45,76)(46,87)(47,68)(48,79)(49,90)(50,71)(51,82)(52,63)(53,74)(54,85)(55,66)(56,77)(57,88)(58,69)(59,80)(60,61)(91,230)(92,211)(93,222)(94,233)(95,214)(96,225)(97,236)(98,217)(99,228)(100,239)(101,220)(102,231)(103,212)(104,223)(105,234)(106,215)(107,226)(108,237)(109,218)(110,229)(111,240)(112,221)(113,232)(114,213)(115,224)(116,235)(117,216)(118,227)(119,238)(120,219)(121,167)(122,178)(123,159)(124,170)(125,151)(126,162)(127,173)(128,154)(129,165)(130,176)(131,157)(132,168)(133,179)(134,160)(135,171)(136,152)(137,163)(138,174)(139,155)(140,166)(141,177)(142,158)(143,169)(144,180)(145,161)(146,172)(147,153)(148,164)(149,175)(150,156)>;
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)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(57,60)(58,59)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(84,90)(85,89)(86,88)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)(113,120)(114,119)(115,118)(116,117)(121,129)(122,128)(123,127)(124,126)(130,150)(131,149)(132,148)(133,147)(134,146)(135,145)(136,144)(137,143)(138,142)(139,141)(151,176)(152,175)(153,174)(154,173)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,165)(163,164)(177,180)(178,179)(182,210)(183,209)(184,208)(185,207)(186,206)(187,205)(188,204)(189,203)(190,202)(191,201)(192,200)(193,199)(194,198)(195,197)(211,235)(212,234)(213,233)(214,232)(215,231)(216,230)(217,229)(218,228)(219,227)(220,226)(221,225)(222,224)(236,240)(237,239), (1,216,59,148)(2,235,60,137)(3,224,31,126)(4,213,32,145)(5,232,33,134)(6,221,34,123)(7,240,35,142)(8,229,36,131)(9,218,37,150)(10,237,38,139)(11,226,39,128)(12,215,40,147)(13,234,41,136)(14,223,42,125)(15,212,43,144)(16,231,44,133)(17,220,45,122)(18,239,46,141)(19,228,47,130)(20,217,48,149)(21,236,49,138)(22,225,50,127)(23,214,51,146)(24,233,52,135)(25,222,53,124)(26,211,54,143)(27,230,55,132)(28,219,56,121)(29,238,57,140)(30,227,58,129)(61,101,185,178)(62,120,186,167)(63,109,187,156)(64,98,188,175)(65,117,189,164)(66,106,190,153)(67,95,191,172)(68,114,192,161)(69,103,193,180)(70,92,194,169)(71,111,195,158)(72,100,196,177)(73,119,197,166)(74,108,198,155)(75,97,199,174)(76,116,200,163)(77,105,201,152)(78,94,202,171)(79,113,203,160)(80,102,204,179)(81,91,205,168)(82,110,206,157)(83,99,207,176)(84,118,208,165)(85,107,209,154)(86,96,210,173)(87,115,181,162)(88,104,182,151)(89,93,183,170)(90,112,184,159), (1,204)(2,185)(3,196)(4,207)(5,188)(6,199)(7,210)(8,191)(9,202)(10,183)(11,194)(12,205)(13,186)(14,197)(15,208)(16,189)(17,200)(18,181)(19,192)(20,203)(21,184)(22,195)(23,206)(24,187)(25,198)(26,209)(27,190)(28,201)(29,182)(30,193)(31,72)(32,83)(33,64)(34,75)(35,86)(36,67)(37,78)(38,89)(39,70)(40,81)(41,62)(42,73)(43,84)(44,65)(45,76)(46,87)(47,68)(48,79)(49,90)(50,71)(51,82)(52,63)(53,74)(54,85)(55,66)(56,77)(57,88)(58,69)(59,80)(60,61)(91,230)(92,211)(93,222)(94,233)(95,214)(96,225)(97,236)(98,217)(99,228)(100,239)(101,220)(102,231)(103,212)(104,223)(105,234)(106,215)(107,226)(108,237)(109,218)(110,229)(111,240)(112,221)(113,232)(114,213)(115,224)(116,235)(117,216)(118,227)(119,238)(120,219)(121,167)(122,178)(123,159)(124,170)(125,151)(126,162)(127,173)(128,154)(129,165)(130,176)(131,157)(132,168)(133,179)(134,160)(135,171)(136,152)(137,163)(138,174)(139,155)(140,166)(141,177)(142,158)(143,169)(144,180)(145,161)(146,172)(147,153)(148,164)(149,175)(150,156) );
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),(31,56),(32,55),(33,54),(34,53),(35,52),(36,51),(37,50),(38,49),(39,48),(40,47),(41,46),(42,45),(43,44),(57,60),(58,59),(61,83),(62,82),(63,81),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(84,90),(85,89),(86,88),(91,112),(92,111),(93,110),(94,109),(95,108),(96,107),(97,106),(98,105),(99,104),(100,103),(101,102),(113,120),(114,119),(115,118),(116,117),(121,129),(122,128),(123,127),(124,126),(130,150),(131,149),(132,148),(133,147),(134,146),(135,145),(136,144),(137,143),(138,142),(139,141),(151,176),(152,175),(153,174),(154,173),(155,172),(156,171),(157,170),(158,169),(159,168),(160,167),(161,166),(162,165),(163,164),(177,180),(178,179),(182,210),(183,209),(184,208),(185,207),(186,206),(187,205),(188,204),(189,203),(190,202),(191,201),(192,200),(193,199),(194,198),(195,197),(211,235),(212,234),(213,233),(214,232),(215,231),(216,230),(217,229),(218,228),(219,227),(220,226),(221,225),(222,224),(236,240),(237,239)], [(1,216,59,148),(2,235,60,137),(3,224,31,126),(4,213,32,145),(5,232,33,134),(6,221,34,123),(7,240,35,142),(8,229,36,131),(9,218,37,150),(10,237,38,139),(11,226,39,128),(12,215,40,147),(13,234,41,136),(14,223,42,125),(15,212,43,144),(16,231,44,133),(17,220,45,122),(18,239,46,141),(19,228,47,130),(20,217,48,149),(21,236,49,138),(22,225,50,127),(23,214,51,146),(24,233,52,135),(25,222,53,124),(26,211,54,143),(27,230,55,132),(28,219,56,121),(29,238,57,140),(30,227,58,129),(61,101,185,178),(62,120,186,167),(63,109,187,156),(64,98,188,175),(65,117,189,164),(66,106,190,153),(67,95,191,172),(68,114,192,161),(69,103,193,180),(70,92,194,169),(71,111,195,158),(72,100,196,177),(73,119,197,166),(74,108,198,155),(75,97,199,174),(76,116,200,163),(77,105,201,152),(78,94,202,171),(79,113,203,160),(80,102,204,179),(81,91,205,168),(82,110,206,157),(83,99,207,176),(84,118,208,165),(85,107,209,154),(86,96,210,173),(87,115,181,162),(88,104,182,151),(89,93,183,170),(90,112,184,159)], [(1,204),(2,185),(3,196),(4,207),(5,188),(6,199),(7,210),(8,191),(9,202),(10,183),(11,194),(12,205),(13,186),(14,197),(15,208),(16,189),(17,200),(18,181),(19,192),(20,203),(21,184),(22,195),(23,206),(24,187),(25,198),(26,209),(27,190),(28,201),(29,182),(30,193),(31,72),(32,83),(33,64),(34,75),(35,86),(36,67),(37,78),(38,89),(39,70),(40,81),(41,62),(42,73),(43,84),(44,65),(45,76),(46,87),(47,68),(48,79),(49,90),(50,71),(51,82),(52,63),(53,74),(54,85),(55,66),(56,77),(57,88),(58,69),(59,80),(60,61),(91,230),(92,211),(93,222),(94,233),(95,214),(96,225),(97,236),(98,217),(99,228),(100,239),(101,220),(102,231),(103,212),(104,223),(105,234),(106,215),(107,226),(108,237),(109,218),(110,229),(111,240),(112,221),(113,232),(114,213),(115,224),(116,235),(117,216),(118,227),(119,238),(120,219),(121,167),(122,178),(123,159),(124,170),(125,151),(126,162),(127,173),(128,154),(129,165),(130,176),(131,157),(132,168),(133,179),(134,160),(135,171),(136,152),(137,163),(138,174),(139,155),(140,166),(141,177),(142,158),(143,169),(144,180),(145,161),(146,172),(147,153),(148,164),(149,175),(150,156)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 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 | 3 | 4 | 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 | 2 | 2 | 2 | 12 | 20 | 20 | 30 | 30 | 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 | 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 | C4○D4 | D10 | D10 | D10 | C4○D12 | C4○D20 | S3×D4 | Q8⋊3S3 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | D12⋊D5 | D6.D10 | D10⋊D6 |
| kernel | D30.35D4 | D6⋊Dic5 | D30⋊4C4 | C6.Dic10 | C3×C10.D4 | C5×D6⋊C4 | C2×C5⋊D12 | C2×C4×D15 | C10.D4 | D30 | D6⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | C10 | C6 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D30.35D4 ►in GL8(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 1 |
| 27 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 17 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 41 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 56 | 20 |
| 8 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 59 | 53 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 37 | 45 |
| 0 | 0 | 0 | 0 | 0 | 0 | 55 | 24 |
G:=sub<GL(8,GF(61))| [60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,17,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[60,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,44,44,0,0,0,0,0,0,60,17,0,0,0,0,0,0,0,0,60,3,0,0,0,0,0,0,0,1],[27,17,0,0,0,0,0,0,11,34,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,17,60,0,0,0,0,0,0,44,44,0,0,0,0,0,0,0,0,41,56,0,0,0,0,0,0,7,20],[8,59,0,0,0,0,0,0,1,53,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,37,55,0,0,0,0,0,0,45,24] >;
D30.35D4 in GAP, Magma, Sage, TeX
D_{30}._{35}D_4 % in TeX
G:=Group("D30.35D4"); // GroupNames label
G:=SmallGroup(480,431);
// by ID
G=gap.SmallGroup(480,431);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,422,219,142,1356,18822]);
// 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^19,d*a*d=a^11,c*b*c^-1=a^3*b,d*b*d=a^25*b,d*c*d=a^15*c^-1>;
// generators/relations