metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.9D30, C60.207D4, Q8.14D30, C60.82C23, Dic30.43C22, C4○D4.4D15, (C5×D4).33D6, (C2×C4).22D30, (C2×C30).10D4, (C5×Q8).52D6, D4.D15⋊14C2, C15⋊7Q16⋊14C2, (C3×D4).33D10, C30.393(C2×D4), (C2×C20).160D6, C5⋊5(Q8.14D6), (C3×Q8).35D10, C60.7C4⋊22C2, (C2×Dic30)⋊14C2, (C2×C12).158D10, C3⋊5(D4.9D10), C4.25(C15⋊7D4), C15⋊37(C8.C22), (C2×C60).85C22, C4.19(C22×D15), C12.104(C5⋊D4), C20.104(C3⋊D4), C20.120(C22×S3), C15⋊3C8.22C22, (D4×C15).38C22, C12.120(C22×D5), C22.6(C15⋊7D4), (Q8×C15).40C22, (C3×C4○D4).3D5, (C5×C4○D4).7S3, (C15×C4○D4).3C2, C6.120(C2×C5⋊D4), C2.25(C2×C15⋊7D4), C10.120(C2×C3⋊D4), (C2×C6).22(C5⋊D4), (C2×C10).21(C3⋊D4), SmallGroup(480,916)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.9D30
G = < a,b,c,d | a4=b2=c30=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >
Subgroups: 548 in 120 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C30, C30, C8.C22, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C4.Dic3, D4.S3, C3⋊Q16, C2×Dic6, C3×C4○D4, Dic15, C60, C60, C2×C30, C2×C30, C4.Dic5, D4.D5, C5⋊Q16, C2×Dic10, C5×C4○D4, Q8.14D6, C15⋊3C8, Dic30, Dic30, C2×Dic15, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, D4.9D10, C60.7C4, D4.D15, C15⋊7Q16, C2×Dic30, C15×C4○D4, D4.9D30
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, D15, C8.C22, C5⋊D4, C22×D5, C2×C3⋊D4, D30, C2×C5⋊D4, Q8.14D6, C15⋊7D4, C22×D15, D4.9D10, C2×C15⋊7D4, D4.9D30
►On 240 points
Generators in S240
(1 118 94 60)(2 119 95 46)(3 120 96 47)(4 106 97 48)(5 107 98 49)(6 108 99 50)(7 109 100 51)(8 110 101 52)(9 111 102 53)(10 112 103 54)(11 113 104 55)(12 114 105 56)(13 115 91 57)(14 116 92 58)(15 117 93 59)(16 82 40 74)(17 83 41 75)(18 84 42 61)(19 85 43 62)(20 86 44 63)(21 87 45 64)(22 88 31 65)(23 89 32 66)(24 90 33 67)(25 76 34 68)(26 77 35 69)(27 78 36 70)(28 79 37 71)(29 80 38 72)(30 81 39 73)(121 202 136 187)(122 203 137 188)(123 204 138 189)(124 205 139 190)(125 206 140 191)(126 207 141 192)(127 208 142 193)(128 209 143 194)(129 210 144 195)(130 181 145 196)(131 182 146 197)(132 183 147 198)(133 184 148 199)(134 185 149 200)(135 186 150 201)(151 233 166 218)(152 234 167 219)(153 235 168 220)(154 236 169 221)(155 237 170 222)(156 238 171 223)(157 239 172 224)(158 240 173 225)(159 211 174 226)(160 212 175 227)(161 213 176 228)(162 214 177 229)(163 215 178 230)(164 216 179 231)(165 217 180 232)
(1 128)(2 144)(3 130)(4 146)(5 132)(6 148)(7 134)(8 150)(9 136)(10 122)(11 138)(12 124)(13 140)(14 126)(15 142)(16 240)(17 226)(18 212)(19 228)(20 214)(21 230)(22 216)(23 232)(24 218)(25 234)(26 220)(27 236)(28 222)(29 238)(30 224)(31 231)(32 217)(33 233)(34 219)(35 235)(36 221)(37 237)(38 223)(39 239)(40 225)(41 211)(42 227)(43 213)(44 229)(45 215)(46 195)(47 181)(48 197)(49 183)(50 199)(51 185)(52 201)(53 187)(54 203)(55 189)(56 205)(57 191)(58 207)(59 193)(60 209)(61 175)(62 161)(63 177)(64 163)(65 179)(66 165)(67 151)(68 167)(69 153)(70 169)(71 155)(72 171)(73 157)(74 173)(75 159)(76 152)(77 168)(78 154)(79 170)(80 156)(81 172)(82 158)(83 174)(84 160)(85 176)(86 162)(87 178)(88 164)(89 180)(90 166)(91 125)(92 141)(93 127)(94 143)(95 129)(96 145)(97 131)(98 147)(99 133)(100 149)(101 135)(102 121)(103 137)(104 123)(105 139)(106 182)(107 198)(108 184)(109 200)(110 186)(111 202)(112 188)(113 204)(114 190)(115 206)(116 192)(117 208)(118 194)(119 210)(120 196)
(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 73 94 81)(2 72 95 80)(3 71 96 79)(4 70 97 78)(5 69 98 77)(6 68 99 76)(7 67 100 90)(8 66 101 89)(9 65 102 88)(10 64 103 87)(11 63 104 86)(12 62 105 85)(13 61 91 84)(14 75 92 83)(15 74 93 82)(16 117 40 59)(17 116 41 58)(18 115 42 57)(19 114 43 56)(20 113 44 55)(21 112 45 54)(22 111 31 53)(23 110 32 52)(24 109 33 51)(25 108 34 50)(26 107 35 49)(27 106 36 48)(28 120 37 47)(29 119 38 46)(30 118 39 60)(121 216 136 231)(122 215 137 230)(123 214 138 229)(124 213 139 228)(125 212 140 227)(126 211 141 226)(127 240 142 225)(128 239 143 224)(129 238 144 223)(130 237 145 222)(131 236 146 221)(132 235 147 220)(133 234 148 219)(134 233 149 218)(135 232 150 217)(151 200 166 185)(152 199 167 184)(153 198 168 183)(154 197 169 182)(155 196 170 181)(156 195 171 210)(157 194 172 209)(158 193 173 208)(159 192 174 207)(160 191 175 206)(161 190 176 205)(162 189 177 204)(163 188 178 203)(164 187 179 202)(165 186 180 201)
G:=sub<Sym(240)| (1,118,94,60)(2,119,95,46)(3,120,96,47)(4,106,97,48)(5,107,98,49)(6,108,99,50)(7,109,100,51)(8,110,101,52)(9,111,102,53)(10,112,103,54)(11,113,104,55)(12,114,105,56)(13,115,91,57)(14,116,92,58)(15,117,93,59)(16,82,40,74)(17,83,41,75)(18,84,42,61)(19,85,43,62)(20,86,44,63)(21,87,45,64)(22,88,31,65)(23,89,32,66)(24,90,33,67)(25,76,34,68)(26,77,35,69)(27,78,36,70)(28,79,37,71)(29,80,38,72)(30,81,39,73)(121,202,136,187)(122,203,137,188)(123,204,138,189)(124,205,139,190)(125,206,140,191)(126,207,141,192)(127,208,142,193)(128,209,143,194)(129,210,144,195)(130,181,145,196)(131,182,146,197)(132,183,147,198)(133,184,148,199)(134,185,149,200)(135,186,150,201)(151,233,166,218)(152,234,167,219)(153,235,168,220)(154,236,169,221)(155,237,170,222)(156,238,171,223)(157,239,172,224)(158,240,173,225)(159,211,174,226)(160,212,175,227)(161,213,176,228)(162,214,177,229)(163,215,178,230)(164,216,179,231)(165,217,180,232), (1,128)(2,144)(3,130)(4,146)(5,132)(6,148)(7,134)(8,150)(9,136)(10,122)(11,138)(12,124)(13,140)(14,126)(15,142)(16,240)(17,226)(18,212)(19,228)(20,214)(21,230)(22,216)(23,232)(24,218)(25,234)(26,220)(27,236)(28,222)(29,238)(30,224)(31,231)(32,217)(33,233)(34,219)(35,235)(36,221)(37,237)(38,223)(39,239)(40,225)(41,211)(42,227)(43,213)(44,229)(45,215)(46,195)(47,181)(48,197)(49,183)(50,199)(51,185)(52,201)(53,187)(54,203)(55,189)(56,205)(57,191)(58,207)(59,193)(60,209)(61,175)(62,161)(63,177)(64,163)(65,179)(66,165)(67,151)(68,167)(69,153)(70,169)(71,155)(72,171)(73,157)(74,173)(75,159)(76,152)(77,168)(78,154)(79,170)(80,156)(81,172)(82,158)(83,174)(84,160)(85,176)(86,162)(87,178)(88,164)(89,180)(90,166)(91,125)(92,141)(93,127)(94,143)(95,129)(96,145)(97,131)(98,147)(99,133)(100,149)(101,135)(102,121)(103,137)(104,123)(105,139)(106,182)(107,198)(108,184)(109,200)(110,186)(111,202)(112,188)(113,204)(114,190)(115,206)(116,192)(117,208)(118,194)(119,210)(120,196), (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), (1,73,94,81)(2,72,95,80)(3,71,96,79)(4,70,97,78)(5,69,98,77)(6,68,99,76)(7,67,100,90)(8,66,101,89)(9,65,102,88)(10,64,103,87)(11,63,104,86)(12,62,105,85)(13,61,91,84)(14,75,92,83)(15,74,93,82)(16,117,40,59)(17,116,41,58)(18,115,42,57)(19,114,43,56)(20,113,44,55)(21,112,45,54)(22,111,31,53)(23,110,32,52)(24,109,33,51)(25,108,34,50)(26,107,35,49)(27,106,36,48)(28,120,37,47)(29,119,38,46)(30,118,39,60)(121,216,136,231)(122,215,137,230)(123,214,138,229)(124,213,139,228)(125,212,140,227)(126,211,141,226)(127,240,142,225)(128,239,143,224)(129,238,144,223)(130,237,145,222)(131,236,146,221)(132,235,147,220)(133,234,148,219)(134,233,149,218)(135,232,150,217)(151,200,166,185)(152,199,167,184)(153,198,168,183)(154,197,169,182)(155,196,170,181)(156,195,171,210)(157,194,172,209)(158,193,173,208)(159,192,174,207)(160,191,175,206)(161,190,176,205)(162,189,177,204)(163,188,178,203)(164,187,179,202)(165,186,180,201)>;
G:=Group( (1,118,94,60)(2,119,95,46)(3,120,96,47)(4,106,97,48)(5,107,98,49)(6,108,99,50)(7,109,100,51)(8,110,101,52)(9,111,102,53)(10,112,103,54)(11,113,104,55)(12,114,105,56)(13,115,91,57)(14,116,92,58)(15,117,93,59)(16,82,40,74)(17,83,41,75)(18,84,42,61)(19,85,43,62)(20,86,44,63)(21,87,45,64)(22,88,31,65)(23,89,32,66)(24,90,33,67)(25,76,34,68)(26,77,35,69)(27,78,36,70)(28,79,37,71)(29,80,38,72)(30,81,39,73)(121,202,136,187)(122,203,137,188)(123,204,138,189)(124,205,139,190)(125,206,140,191)(126,207,141,192)(127,208,142,193)(128,209,143,194)(129,210,144,195)(130,181,145,196)(131,182,146,197)(132,183,147,198)(133,184,148,199)(134,185,149,200)(135,186,150,201)(151,233,166,218)(152,234,167,219)(153,235,168,220)(154,236,169,221)(155,237,170,222)(156,238,171,223)(157,239,172,224)(158,240,173,225)(159,211,174,226)(160,212,175,227)(161,213,176,228)(162,214,177,229)(163,215,178,230)(164,216,179,231)(165,217,180,232), (1,128)(2,144)(3,130)(4,146)(5,132)(6,148)(7,134)(8,150)(9,136)(10,122)(11,138)(12,124)(13,140)(14,126)(15,142)(16,240)(17,226)(18,212)(19,228)(20,214)(21,230)(22,216)(23,232)(24,218)(25,234)(26,220)(27,236)(28,222)(29,238)(30,224)(31,231)(32,217)(33,233)(34,219)(35,235)(36,221)(37,237)(38,223)(39,239)(40,225)(41,211)(42,227)(43,213)(44,229)(45,215)(46,195)(47,181)(48,197)(49,183)(50,199)(51,185)(52,201)(53,187)(54,203)(55,189)(56,205)(57,191)(58,207)(59,193)(60,209)(61,175)(62,161)(63,177)(64,163)(65,179)(66,165)(67,151)(68,167)(69,153)(70,169)(71,155)(72,171)(73,157)(74,173)(75,159)(76,152)(77,168)(78,154)(79,170)(80,156)(81,172)(82,158)(83,174)(84,160)(85,176)(86,162)(87,178)(88,164)(89,180)(90,166)(91,125)(92,141)(93,127)(94,143)(95,129)(96,145)(97,131)(98,147)(99,133)(100,149)(101,135)(102,121)(103,137)(104,123)(105,139)(106,182)(107,198)(108,184)(109,200)(110,186)(111,202)(112,188)(113,204)(114,190)(115,206)(116,192)(117,208)(118,194)(119,210)(120,196), (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), (1,73,94,81)(2,72,95,80)(3,71,96,79)(4,70,97,78)(5,69,98,77)(6,68,99,76)(7,67,100,90)(8,66,101,89)(9,65,102,88)(10,64,103,87)(11,63,104,86)(12,62,105,85)(13,61,91,84)(14,75,92,83)(15,74,93,82)(16,117,40,59)(17,116,41,58)(18,115,42,57)(19,114,43,56)(20,113,44,55)(21,112,45,54)(22,111,31,53)(23,110,32,52)(24,109,33,51)(25,108,34,50)(26,107,35,49)(27,106,36,48)(28,120,37,47)(29,119,38,46)(30,118,39,60)(121,216,136,231)(122,215,137,230)(123,214,138,229)(124,213,139,228)(125,212,140,227)(126,211,141,226)(127,240,142,225)(128,239,143,224)(129,238,144,223)(130,237,145,222)(131,236,146,221)(132,235,147,220)(133,234,148,219)(134,233,149,218)(135,232,150,217)(151,200,166,185)(152,199,167,184)(153,198,168,183)(154,197,169,182)(155,196,170,181)(156,195,171,210)(157,194,172,209)(158,193,173,208)(159,192,174,207)(160,191,175,206)(161,190,176,205)(162,189,177,204)(163,188,178,203)(164,187,179,202)(165,186,180,201) );
G=PermutationGroup([[(1,118,94,60),(2,119,95,46),(3,120,96,47),(4,106,97,48),(5,107,98,49),(6,108,99,50),(7,109,100,51),(8,110,101,52),(9,111,102,53),(10,112,103,54),(11,113,104,55),(12,114,105,56),(13,115,91,57),(14,116,92,58),(15,117,93,59),(16,82,40,74),(17,83,41,75),(18,84,42,61),(19,85,43,62),(20,86,44,63),(21,87,45,64),(22,88,31,65),(23,89,32,66),(24,90,33,67),(25,76,34,68),(26,77,35,69),(27,78,36,70),(28,79,37,71),(29,80,38,72),(30,81,39,73),(121,202,136,187),(122,203,137,188),(123,204,138,189),(124,205,139,190),(125,206,140,191),(126,207,141,192),(127,208,142,193),(128,209,143,194),(129,210,144,195),(130,181,145,196),(131,182,146,197),(132,183,147,198),(133,184,148,199),(134,185,149,200),(135,186,150,201),(151,233,166,218),(152,234,167,219),(153,235,168,220),(154,236,169,221),(155,237,170,222),(156,238,171,223),(157,239,172,224),(158,240,173,225),(159,211,174,226),(160,212,175,227),(161,213,176,228),(162,214,177,229),(163,215,178,230),(164,216,179,231),(165,217,180,232)], [(1,128),(2,144),(3,130),(4,146),(5,132),(6,148),(7,134),(8,150),(9,136),(10,122),(11,138),(12,124),(13,140),(14,126),(15,142),(16,240),(17,226),(18,212),(19,228),(20,214),(21,230),(22,216),(23,232),(24,218),(25,234),(26,220),(27,236),(28,222),(29,238),(30,224),(31,231),(32,217),(33,233),(34,219),(35,235),(36,221),(37,237),(38,223),(39,239),(40,225),(41,211),(42,227),(43,213),(44,229),(45,215),(46,195),(47,181),(48,197),(49,183),(50,199),(51,185),(52,201),(53,187),(54,203),(55,189),(56,205),(57,191),(58,207),(59,193),(60,209),(61,175),(62,161),(63,177),(64,163),(65,179),(66,165),(67,151),(68,167),(69,153),(70,169),(71,155),(72,171),(73,157),(74,173),(75,159),(76,152),(77,168),(78,154),(79,170),(80,156),(81,172),(82,158),(83,174),(84,160),(85,176),(86,162),(87,178),(88,164),(89,180),(90,166),(91,125),(92,141),(93,127),(94,143),(95,129),(96,145),(97,131),(98,147),(99,133),(100,149),(101,135),(102,121),(103,137),(104,123),(105,139),(106,182),(107,198),(108,184),(109,200),(110,186),(111,202),(112,188),(113,204),(114,190),(115,206),(116,192),(117,208),(118,194),(119,210),(120,196)], [(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)], [(1,73,94,81),(2,72,95,80),(3,71,96,79),(4,70,97,78),(5,69,98,77),(6,68,99,76),(7,67,100,90),(8,66,101,89),(9,65,102,88),(10,64,103,87),(11,63,104,86),(12,62,105,85),(13,61,91,84),(14,75,92,83),(15,74,93,82),(16,117,40,59),(17,116,41,58),(18,115,42,57),(19,114,43,56),(20,113,44,55),(21,112,45,54),(22,111,31,53),(23,110,32,52),(24,109,33,51),(25,108,34,50),(26,107,35,49),(27,106,36,48),(28,120,37,47),(29,119,38,46),(30,118,39,60),(121,216,136,231),(122,215,137,230),(123,214,138,229),(124,213,139,228),(125,212,140,227),(126,211,141,226),(127,240,142,225),(128,239,143,224),(129,238,144,223),(130,237,145,222),(131,236,146,221),(132,235,147,220),(133,234,148,219),(134,233,149,218),(135,232,150,217),(151,200,166,185),(152,199,167,184),(153,198,168,183),(154,197,169,182),(155,196,170,181),(156,195,171,210),(157,194,172,209),(158,193,173,208),(159,192,174,207),(160,191,175,206),(161,190,176,205),(162,189,177,204),(163,188,178,203),(164,187,179,202),(165,186,180,201)]])
81 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 60 | 60 | 2 | 2 | 2 | 4 | 4 | 4 | 60 | 60 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C3⋊D4 | C3⋊D4 | D15 | C5⋊D4 | C5⋊D4 | D30 | D30 | D30 | C15⋊7D4 | C15⋊7D4 | C8.C22 | Q8.14D6 | D4.9D10 | D4.9D30 |
| kernel | D4.9D30 | C60.7C4 | D4.D15 | C15⋊7Q16 | C2×Dic30 | C15×C4○D4 | C5×C4○D4 | C60 | C2×C30 | C3×C4○D4 | C2×C20 | C5×D4 | C5×Q8 | C2×C12 | C3×D4 | C3×Q8 | C20 | C2×C10 | C4○D4 | C12 | C2×C6 | C2×C4 | D4 | Q8 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of D4.9D30 ►in GL4(𝔽241) generated by
| 44 | 156 | 0 | 0 |
| 88 | 197 | 0 | 0 |
| 114 | 219 | 200 | 85 |
| 135 | 92 | 156 | 41 |
| 230 | 119 | 128 | 46 |
| 56 | 39 | 174 | 174 |
| 149 | 125 | 7 | 92 |
| 119 | 177 | 43 | 206 |
| 161 | 64 | 0 | 0 |
| 67 | 94 | 0 | 0 |
| 167 | 97 | 211 | 177 |
| 236 | 95 | 64 | 16 |
| 94 | 75 | 0 | 0 |
| 226 | 147 | 0 | 0 |
| 194 | 108 | 169 | 166 |
| 106 | 61 | 37 | 72 |
G:=sub<GL(4,GF(241))| [44,88,114,135,156,197,219,92,0,0,200,156,0,0,85,41],[230,56,149,119,119,39,125,177,128,174,7,43,46,174,92,206],[161,67,167,236,64,94,97,95,0,0,211,64,0,0,177,16],[94,226,194,106,75,147,108,61,0,0,169,37,0,0,166,72] >;
D4.9D30 in GAP, Magma, Sage, TeX
D_4._9D_{30} % in TeX
G:=Group("D4.9D30"); // GroupNames label
G:=SmallGroup(480,916);
// by ID
G=gap.SmallGroup(480,916);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,219,675,185,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^30=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations