metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊9D4, Dic15⋊11D4, C23.10D30, (C2×D60)⋊6C2, C2.9(D4×D15), (C2×C4).7D30, C5⋊3(Dic3⋊D4), C22⋊C4⋊4D15, (C2×C20).33D6, C6.100(D4×D5), C3⋊3(D10⋊D4), C15⋊25(C4⋊D4), C10.102(S3×D4), C30.308(C2×D4), C30.4Q8⋊6C2, D30⋊3C4⋊11C2, C6.97(C4○D20), (C2×C12).205D10, (C22×C6).59D10, (C22×C10).74D6, C30.170(C4○D4), C10.97(C4○D12), (C2×C30).281C23, (C2×C60).174C22, (C22×C30).15C22, C2.11(D60⋊11C2), (C2×Dic15).7C22, (C22×D15).4C22, C22.43(C22×D15), (C2×C4×D15)⋊17C2, (C2×C15⋊7D4)⋊2C2, (C5×C22⋊C4)⋊6S3, (C3×C22⋊C4)⋊6D5, (C15×C22⋊C4)⋊8C2, (C2×C6).277(C22×D5), (C2×C10).276(C22×S3), SmallGroup(480,849)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊9D4
G = < a,b,c,d | a30=b2=c4=d2=1, bab=cac-1=dad=a-1, cbc-1=a28b, dbd=a13b, dcd=c-1 >
Subgroups: 1316 in 188 conjugacy classes, 51 normal (47 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, D15, C30, C30, C4⋊D4, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, Dic15, Dic15, C60, D30, D30, C2×C30, C2×C30, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, Dic3⋊D4, C4×D15, D60, C2×Dic15, C15⋊7D4, C2×C60, C22×D15, C22×C30, D10⋊D4, C30.4Q8, D30⋊3C4, C15×C22⋊C4, C2×C4×D15, C2×D60, C2×C15⋊7D4, D30⋊9D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, D15, C4⋊D4, C22×D5, C4○D12, S3×D4, D30, C4○D20, D4×D5, Dic3⋊D4, C22×D15, D10⋊D4, D60⋊11C2, D4×D15, D30⋊9D4
►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)
(1 122)(2 121)(3 150)(4 149)(5 148)(6 147)(7 146)(8 145)(9 144)(10 143)(11 142)(12 141)(13 140)(14 139)(15 138)(16 137)(17 136)(18 135)(19 134)(20 133)(21 132)(22 131)(23 130)(24 129)(25 128)(26 127)(27 126)(28 125)(29 124)(30 123)(31 178)(32 177)(33 176)(34 175)(35 174)(36 173)(37 172)(38 171)(39 170)(40 169)(41 168)(42 167)(43 166)(44 165)(45 164)(46 163)(47 162)(48 161)(49 160)(50 159)(51 158)(52 157)(53 156)(54 155)(55 154)(56 153)(57 152)(58 151)(59 180)(60 179)(61 192)(62 191)(63 190)(64 189)(65 188)(66 187)(67 186)(68 185)(69 184)(70 183)(71 182)(72 181)(73 210)(74 209)(75 208)(76 207)(77 206)(78 205)(79 204)(80 203)(81 202)(82 201)(83 200)(84 199)(85 198)(86 197)(87 196)(88 195)(89 194)(90 193)(91 227)(92 226)(93 225)(94 224)(95 223)(96 222)(97 221)(98 220)(99 219)(100 218)(101 217)(102 216)(103 215)(104 214)(105 213)(106 212)(107 211)(108 240)(109 239)(110 238)(111 237)(112 236)(113 235)(114 234)(115 233)(116 232)(117 231)(118 230)(119 229)(120 228)
(1 202 138 67)(2 201 139 66)(3 200 140 65)(4 199 141 64)(5 198 142 63)(6 197 143 62)(7 196 144 61)(8 195 145 90)(9 194 146 89)(10 193 147 88)(11 192 148 87)(12 191 149 86)(13 190 150 85)(14 189 121 84)(15 188 122 83)(16 187 123 82)(17 186 124 81)(18 185 125 80)(19 184 126 79)(20 183 127 78)(21 182 128 77)(22 181 129 76)(23 210 130 75)(24 209 131 74)(25 208 132 73)(26 207 133 72)(27 206 134 71)(28 205 135 70)(29 204 136 69)(30 203 137 68)(31 211 179 108)(32 240 180 107)(33 239 151 106)(34 238 152 105)(35 237 153 104)(36 236 154 103)(37 235 155 102)(38 234 156 101)(39 233 157 100)(40 232 158 99)(41 231 159 98)(42 230 160 97)(43 229 161 96)(44 228 162 95)(45 227 163 94)(46 226 164 93)(47 225 165 92)(48 224 166 91)(49 223 167 120)(50 222 168 119)(51 221 169 118)(52 220 170 117)(53 219 171 116)(54 218 172 115)(55 217 173 114)(56 216 174 113)(57 215 175 112)(58 214 176 111)(59 213 177 110)(60 212 178 109)
(1 31)(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 45)(18 44)(19 43)(20 42)(21 41)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(61 217)(62 216)(63 215)(64 214)(65 213)(66 212)(67 211)(68 240)(69 239)(70 238)(71 237)(72 236)(73 235)(74 234)(75 233)(76 232)(77 231)(78 230)(79 229)(80 228)(81 227)(82 226)(83 225)(84 224)(85 223)(86 222)(87 221)(88 220)(89 219)(90 218)(91 189)(92 188)(93 187)(94 186)(95 185)(96 184)(97 183)(98 182)(99 181)(100 210)(101 209)(102 208)(103 207)(104 206)(105 205)(106 204)(107 203)(108 202)(109 201)(110 200)(111 199)(112 198)(113 197)(114 196)(115 195)(116 194)(117 193)(118 192)(119 191)(120 190)(121 166)(122 165)(123 164)(124 163)(125 162)(126 161)(127 160)(128 159)(129 158)(130 157)(131 156)(132 155)(133 154)(134 153)(135 152)(136 151)(137 180)(138 179)(139 178)(140 177)(141 176)(142 175)(143 174)(144 173)(145 172)(146 171)(147 170)(148 169)(149 168)(150 167)
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), (1,122)(2,121)(3,150)(4,149)(5,148)(6,147)(7,146)(8,145)(9,144)(10,143)(11,142)(12,141)(13,140)(14,139)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,132)(22,131)(23,130)(24,129)(25,128)(26,127)(27,126)(28,125)(29,124)(30,123)(31,178)(32,177)(33,176)(34,175)(35,174)(36,173)(37,172)(38,171)(39,170)(40,169)(41,168)(42,167)(43,166)(44,165)(45,164)(46,163)(47,162)(48,161)(49,160)(50,159)(51,158)(52,157)(53,156)(54,155)(55,154)(56,153)(57,152)(58,151)(59,180)(60,179)(61,192)(62,191)(63,190)(64,189)(65,188)(66,187)(67,186)(68,185)(69,184)(70,183)(71,182)(72,181)(73,210)(74,209)(75,208)(76,207)(77,206)(78,205)(79,204)(80,203)(81,202)(82,201)(83,200)(84,199)(85,198)(86,197)(87,196)(88,195)(89,194)(90,193)(91,227)(92,226)(93,225)(94,224)(95,223)(96,222)(97,221)(98,220)(99,219)(100,218)(101,217)(102,216)(103,215)(104,214)(105,213)(106,212)(107,211)(108,240)(109,239)(110,238)(111,237)(112,236)(113,235)(114,234)(115,233)(116,232)(117,231)(118,230)(119,229)(120,228), (1,202,138,67)(2,201,139,66)(3,200,140,65)(4,199,141,64)(5,198,142,63)(6,197,143,62)(7,196,144,61)(8,195,145,90)(9,194,146,89)(10,193,147,88)(11,192,148,87)(12,191,149,86)(13,190,150,85)(14,189,121,84)(15,188,122,83)(16,187,123,82)(17,186,124,81)(18,185,125,80)(19,184,126,79)(20,183,127,78)(21,182,128,77)(22,181,129,76)(23,210,130,75)(24,209,131,74)(25,208,132,73)(26,207,133,72)(27,206,134,71)(28,205,135,70)(29,204,136,69)(30,203,137,68)(31,211,179,108)(32,240,180,107)(33,239,151,106)(34,238,152,105)(35,237,153,104)(36,236,154,103)(37,235,155,102)(38,234,156,101)(39,233,157,100)(40,232,158,99)(41,231,159,98)(42,230,160,97)(43,229,161,96)(44,228,162,95)(45,227,163,94)(46,226,164,93)(47,225,165,92)(48,224,166,91)(49,223,167,120)(50,222,168,119)(51,221,169,118)(52,220,170,117)(53,219,171,116)(54,218,172,115)(55,217,173,114)(56,216,174,113)(57,215,175,112)(58,214,176,111)(59,213,177,110)(60,212,178,109), (1,31)(2,60)(3,59)(4,58)(5,57)(6,56)(7,55)(8,54)(9,53)(10,52)(11,51)(12,50)(13,49)(14,48)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,41)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(61,217)(62,216)(63,215)(64,214)(65,213)(66,212)(67,211)(68,240)(69,239)(70,238)(71,237)(72,236)(73,235)(74,234)(75,233)(76,232)(77,231)(78,230)(79,229)(80,228)(81,227)(82,226)(83,225)(84,224)(85,223)(86,222)(87,221)(88,220)(89,219)(90,218)(91,189)(92,188)(93,187)(94,186)(95,185)(96,184)(97,183)(98,182)(99,181)(100,210)(101,209)(102,208)(103,207)(104,206)(105,205)(106,204)(107,203)(108,202)(109,201)(110,200)(111,199)(112,198)(113,197)(114,196)(115,195)(116,194)(117,193)(118,192)(119,191)(120,190)(121,166)(122,165)(123,164)(124,163)(125,162)(126,161)(127,160)(128,159)(129,158)(130,157)(131,156)(132,155)(133,154)(134,153)(135,152)(136,151)(137,180)(138,179)(139,178)(140,177)(141,176)(142,175)(143,174)(144,173)(145,172)(146,171)(147,170)(148,169)(149,168)(150,167)>;
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), (1,122)(2,121)(3,150)(4,149)(5,148)(6,147)(7,146)(8,145)(9,144)(10,143)(11,142)(12,141)(13,140)(14,139)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,132)(22,131)(23,130)(24,129)(25,128)(26,127)(27,126)(28,125)(29,124)(30,123)(31,178)(32,177)(33,176)(34,175)(35,174)(36,173)(37,172)(38,171)(39,170)(40,169)(41,168)(42,167)(43,166)(44,165)(45,164)(46,163)(47,162)(48,161)(49,160)(50,159)(51,158)(52,157)(53,156)(54,155)(55,154)(56,153)(57,152)(58,151)(59,180)(60,179)(61,192)(62,191)(63,190)(64,189)(65,188)(66,187)(67,186)(68,185)(69,184)(70,183)(71,182)(72,181)(73,210)(74,209)(75,208)(76,207)(77,206)(78,205)(79,204)(80,203)(81,202)(82,201)(83,200)(84,199)(85,198)(86,197)(87,196)(88,195)(89,194)(90,193)(91,227)(92,226)(93,225)(94,224)(95,223)(96,222)(97,221)(98,220)(99,219)(100,218)(101,217)(102,216)(103,215)(104,214)(105,213)(106,212)(107,211)(108,240)(109,239)(110,238)(111,237)(112,236)(113,235)(114,234)(115,233)(116,232)(117,231)(118,230)(119,229)(120,228), (1,202,138,67)(2,201,139,66)(3,200,140,65)(4,199,141,64)(5,198,142,63)(6,197,143,62)(7,196,144,61)(8,195,145,90)(9,194,146,89)(10,193,147,88)(11,192,148,87)(12,191,149,86)(13,190,150,85)(14,189,121,84)(15,188,122,83)(16,187,123,82)(17,186,124,81)(18,185,125,80)(19,184,126,79)(20,183,127,78)(21,182,128,77)(22,181,129,76)(23,210,130,75)(24,209,131,74)(25,208,132,73)(26,207,133,72)(27,206,134,71)(28,205,135,70)(29,204,136,69)(30,203,137,68)(31,211,179,108)(32,240,180,107)(33,239,151,106)(34,238,152,105)(35,237,153,104)(36,236,154,103)(37,235,155,102)(38,234,156,101)(39,233,157,100)(40,232,158,99)(41,231,159,98)(42,230,160,97)(43,229,161,96)(44,228,162,95)(45,227,163,94)(46,226,164,93)(47,225,165,92)(48,224,166,91)(49,223,167,120)(50,222,168,119)(51,221,169,118)(52,220,170,117)(53,219,171,116)(54,218,172,115)(55,217,173,114)(56,216,174,113)(57,215,175,112)(58,214,176,111)(59,213,177,110)(60,212,178,109), (1,31)(2,60)(3,59)(4,58)(5,57)(6,56)(7,55)(8,54)(9,53)(10,52)(11,51)(12,50)(13,49)(14,48)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,41)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(61,217)(62,216)(63,215)(64,214)(65,213)(66,212)(67,211)(68,240)(69,239)(70,238)(71,237)(72,236)(73,235)(74,234)(75,233)(76,232)(77,231)(78,230)(79,229)(80,228)(81,227)(82,226)(83,225)(84,224)(85,223)(86,222)(87,221)(88,220)(89,219)(90,218)(91,189)(92,188)(93,187)(94,186)(95,185)(96,184)(97,183)(98,182)(99,181)(100,210)(101,209)(102,208)(103,207)(104,206)(105,205)(106,204)(107,203)(108,202)(109,201)(110,200)(111,199)(112,198)(113,197)(114,196)(115,195)(116,194)(117,193)(118,192)(119,191)(120,190)(121,166)(122,165)(123,164)(124,163)(125,162)(126,161)(127,160)(128,159)(129,158)(130,157)(131,156)(132,155)(133,154)(134,153)(135,152)(136,151)(137,180)(138,179)(139,178)(140,177)(141,176)(142,175)(143,174)(144,173)(145,172)(146,171)(147,170)(148,169)(149,168)(150,167) );
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)], [(1,122),(2,121),(3,150),(4,149),(5,148),(6,147),(7,146),(8,145),(9,144),(10,143),(11,142),(12,141),(13,140),(14,139),(15,138),(16,137),(17,136),(18,135),(19,134),(20,133),(21,132),(22,131),(23,130),(24,129),(25,128),(26,127),(27,126),(28,125),(29,124),(30,123),(31,178),(32,177),(33,176),(34,175),(35,174),(36,173),(37,172),(38,171),(39,170),(40,169),(41,168),(42,167),(43,166),(44,165),(45,164),(46,163),(47,162),(48,161),(49,160),(50,159),(51,158),(52,157),(53,156),(54,155),(55,154),(56,153),(57,152),(58,151),(59,180),(60,179),(61,192),(62,191),(63,190),(64,189),(65,188),(66,187),(67,186),(68,185),(69,184),(70,183),(71,182),(72,181),(73,210),(74,209),(75,208),(76,207),(77,206),(78,205),(79,204),(80,203),(81,202),(82,201),(83,200),(84,199),(85,198),(86,197),(87,196),(88,195),(89,194),(90,193),(91,227),(92,226),(93,225),(94,224),(95,223),(96,222),(97,221),(98,220),(99,219),(100,218),(101,217),(102,216),(103,215),(104,214),(105,213),(106,212),(107,211),(108,240),(109,239),(110,238),(111,237),(112,236),(113,235),(114,234),(115,233),(116,232),(117,231),(118,230),(119,229),(120,228)], [(1,202,138,67),(2,201,139,66),(3,200,140,65),(4,199,141,64),(5,198,142,63),(6,197,143,62),(7,196,144,61),(8,195,145,90),(9,194,146,89),(10,193,147,88),(11,192,148,87),(12,191,149,86),(13,190,150,85),(14,189,121,84),(15,188,122,83),(16,187,123,82),(17,186,124,81),(18,185,125,80),(19,184,126,79),(20,183,127,78),(21,182,128,77),(22,181,129,76),(23,210,130,75),(24,209,131,74),(25,208,132,73),(26,207,133,72),(27,206,134,71),(28,205,135,70),(29,204,136,69),(30,203,137,68),(31,211,179,108),(32,240,180,107),(33,239,151,106),(34,238,152,105),(35,237,153,104),(36,236,154,103),(37,235,155,102),(38,234,156,101),(39,233,157,100),(40,232,158,99),(41,231,159,98),(42,230,160,97),(43,229,161,96),(44,228,162,95),(45,227,163,94),(46,226,164,93),(47,225,165,92),(48,224,166,91),(49,223,167,120),(50,222,168,119),(51,221,169,118),(52,220,170,117),(53,219,171,116),(54,218,172,115),(55,217,173,114),(56,216,174,113),(57,215,175,112),(58,214,176,111),(59,213,177,110),(60,212,178,109)], [(1,31),(2,60),(3,59),(4,58),(5,57),(6,56),(7,55),(8,54),(9,53),(10,52),(11,51),(12,50),(13,49),(14,48),(15,47),(16,46),(17,45),(18,44),(19,43),(20,42),(21,41),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(61,217),(62,216),(63,215),(64,214),(65,213),(66,212),(67,211),(68,240),(69,239),(70,238),(71,237),(72,236),(73,235),(74,234),(75,233),(76,232),(77,231),(78,230),(79,229),(80,228),(81,227),(82,226),(83,225),(84,224),(85,223),(86,222),(87,221),(88,220),(89,219),(90,218),(91,189),(92,188),(93,187),(94,186),(95,185),(96,184),(97,183),(98,182),(99,181),(100,210),(101,209),(102,208),(103,207),(104,206),(105,205),(106,204),(107,203),(108,202),(109,201),(110,200),(111,199),(112,198),(113,197),(114,196),(115,195),(116,194),(117,193),(118,192),(119,191),(120,190),(121,166),(122,165),(123,164),(124,163),(125,162),(126,161),(127,160),(128,159),(129,158),(130,157),(131,156),(132,155),(133,154),(134,153),(135,152),(136,151),(137,180),(138,179),(139,178),(140,177),(141,176),(142,175),(143,174),(144,173),(145,172),(146,171),(147,170),(148,169),(149,168),(150,167)]])
84 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 | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 30 | 30 | 60 | 2 | 2 | 2 | 4 | 30 | 30 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D15 | C4○D12 | D30 | D30 | C4○D20 | D60⋊11C2 | S3×D4 | D4×D5 | D4×D15 |
| kernel | D30⋊9D4 | C30.4Q8 | D30⋊3C4 | C15×C22⋊C4 | C2×C4×D15 | C2×D60 | C2×C15⋊7D4 | C5×C22⋊C4 | Dic15 | D30 | C3×C22⋊C4 | C2×C20 | C22×C10 | C30 | C2×C12 | C22×C6 | C22⋊C4 | C10 | C2×C4 | C23 | C6 | C2 | C10 | C6 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 8 | 16 | 2 | 4 | 8 |
Matrix representation of D30⋊9D4 ►in GL6(𝔽61)
| 45 | 53 | 0 | 0 | 0 | 0 |
| 8 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 18 | 60 | 0 | 0 | 0 | 0 |
| 18 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 46 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 56 | 38 | 0 | 0 | 0 | 0 |
| 9 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 43 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 56 | 38 | 0 | 0 | 0 | 0 |
| 9 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 27 | 0 | 0 |
| 0 | 0 | 43 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(61))| [45,8,0,0,0,0,53,23,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[18,18,0,0,0,0,60,43,0,0,0,0,0,0,60,46,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60],[56,9,0,0,0,0,38,5,0,0,0,0,0,0,11,43,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[56,9,0,0,0,0,38,5,0,0,0,0,0,0,11,43,0,0,0,0,27,50,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
D30⋊9D4 in GAP, Magma, Sage, TeX
D_{30}\rtimes_9D_4 % in TeX
G:=Group("D30:9D4"); // GroupNames label
G:=SmallGroup(480,849);
// by ID
G=gap.SmallGroup(480,849);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,590,219,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^28*b,d*b*d=a^13*b,d*c*d=c^-1>;
// generators/relations