direct product, metabelian, supersoluble, monomial, A-group
Aliases: D5×C3⋊Dic3, C30.9D6, (C6×D5).5S3, C3⋊3(D5×Dic3), C6.16(S3×D5), C32⋊10(C4×D5), C3⋊Dic15⋊2C2, (C3×D5)⋊1Dic3, C15⋊4(C2×Dic3), (C3×C6).20D10, (C32×D5)⋊4C4, D10.2(C3⋊S3), (C3×C30).8C22, C2.1(D5×C3⋊S3), C5⋊2(C2×C3⋊Dic3), (D5×C3×C6).2C2, (C3×C15)⋊16(C2×C4), C10.1(C2×C3⋊S3), (C5×C3⋊Dic3)⋊1C2, SmallGroup(360,65)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — D5×C3⋊Dic3 |
Generators and relations for D5×C3⋊Dic3
G = < a,b,c,d,e | a5=b2=c3=d6=1, e2=d3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 440 in 96 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C2×C4, C32, D5, C10, Dic3, C2×C6, C15, C3×C6, C3×C6, Dic5, C20, D10, C2×Dic3, C3×D5, C30, C3⋊Dic3, C3⋊Dic3, C62, C4×D5, C3×C15, C5×Dic3, Dic15, C6×D5, C2×C3⋊Dic3, C32×D5, C3×C30, D5×Dic3, C5×C3⋊Dic3, C3⋊Dic15, D5×C3×C6, D5×C3⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, Dic3, D6, C3⋊S3, D10, C2×Dic3, C3⋊Dic3, C2×C3⋊S3, C4×D5, S3×D5, C2×C3⋊Dic3, D5×Dic3, D5×C3⋊S3, D5×C3⋊Dic3
►On 180 points
Generators in S180
(1 29 68 92 85)(2 30 69 93 86)(3 25 70 94 87)(4 26 71 95 88)(5 27 72 96 89)(6 28 67 91 90)(7 156 23 109 40)(8 151 24 110 41)(9 152 19 111 42)(10 153 20 112 37)(11 154 21 113 38)(12 155 22 114 39)(13 162 99 177 117)(14 157 100 178 118)(15 158 101 179 119)(16 159 102 180 120)(17 160 97 175 115)(18 161 98 176 116)(31 163 78 105 174)(32 164 73 106 169)(33 165 74 107 170)(34 166 75 108 171)(35 167 76 103 172)(36 168 77 104 173)(43 136 147 62 127)(44 137 148 63 128)(45 138 149 64 129)(46 133 150 65 130)(47 134 145 66 131)(48 135 146 61 132)(49 141 59 81 122)(50 142 60 82 123)(51 143 55 83 124)(52 144 56 84 125)(53 139 57 79 126)(54 140 58 80 121)
(1 88)(2 89)(3 90)(4 85)(5 86)(6 87)(7 153)(8 154)(9 155)(10 156)(11 151)(12 152)(13 180)(14 175)(15 176)(16 177)(17 178)(18 179)(19 39)(20 40)(21 41)(22 42)(23 37)(24 38)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 166)(32 167)(33 168)(34 163)(35 164)(36 165)(43 46)(44 47)(45 48)(49 125)(50 126)(51 121)(52 122)(53 123)(54 124)(55 58)(56 59)(57 60)(61 149)(62 150)(63 145)(64 146)(65 147)(66 148)(67 70)(68 71)(69 72)(73 172)(74 173)(75 174)(76 169)(77 170)(78 171)(79 142)(80 143)(81 144)(82 139)(83 140)(84 141)(97 157)(98 158)(99 159)(100 160)(101 161)(102 162)(103 106)(104 107)(105 108)(109 112)(110 113)(111 114)(115 118)(116 119)(117 120)(127 133)(128 134)(129 135)(130 136)(131 137)(132 138)
(1 51 154)(2 52 155)(3 53 156)(4 54 151)(5 49 152)(6 50 153)(7 87 126)(8 88 121)(9 89 122)(10 90 123)(11 85 124)(12 86 125)(13 136 169)(14 137 170)(15 138 171)(16 133 172)(17 134 173)(18 135 174)(19 27 141)(20 28 142)(21 29 143)(22 30 144)(23 25 139)(24 26 140)(31 161 146)(32 162 147)(33 157 148)(34 158 149)(35 159 150)(36 160 145)(37 91 82)(38 92 83)(39 93 84)(40 94 79)(41 95 80)(42 96 81)(43 106 117)(44 107 118)(45 108 119)(46 103 120)(47 104 115)(48 105 116)(55 113 68)(56 114 69)(57 109 70)(58 110 71)(59 111 72)(60 112 67)(61 163 98)(62 164 99)(63 165 100)(64 166 101)(65 167 102)(66 168 97)(73 177 127)(74 178 128)(75 179 129)(76 180 130)(77 175 131)(78 176 132)
(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)
(1 100 4 97)(2 99 5 102)(3 98 6 101)(7 146 10 149)(8 145 11 148)(9 150 12 147)(13 96 16 93)(14 95 17 92)(15 94 18 91)(19 130 22 127)(20 129 23 132)(21 128 24 131)(25 176 28 179)(26 175 29 178)(27 180 30 177)(31 123 34 126)(32 122 35 125)(33 121 36 124)(37 138 40 135)(38 137 41 134)(39 136 42 133)(43 111 46 114)(44 110 47 113)(45 109 48 112)(49 167 52 164)(50 166 53 163)(51 165 54 168)(55 107 58 104)(56 106 59 103)(57 105 60 108)(61 153 64 156)(62 152 65 155)(63 151 66 154)(67 119 70 116)(68 118 71 115)(69 117 72 120)(73 141 76 144)(74 140 77 143)(75 139 78 142)(79 174 82 171)(80 173 83 170)(81 172 84 169)(85 157 88 160)(86 162 89 159)(87 161 90 158)
G:=sub<Sym(180)| (1,29,68,92,85)(2,30,69,93,86)(3,25,70,94,87)(4,26,71,95,88)(5,27,72,96,89)(6,28,67,91,90)(7,156,23,109,40)(8,151,24,110,41)(9,152,19,111,42)(10,153,20,112,37)(11,154,21,113,38)(12,155,22,114,39)(13,162,99,177,117)(14,157,100,178,118)(15,158,101,179,119)(16,159,102,180,120)(17,160,97,175,115)(18,161,98,176,116)(31,163,78,105,174)(32,164,73,106,169)(33,165,74,107,170)(34,166,75,108,171)(35,167,76,103,172)(36,168,77,104,173)(43,136,147,62,127)(44,137,148,63,128)(45,138,149,64,129)(46,133,150,65,130)(47,134,145,66,131)(48,135,146,61,132)(49,141,59,81,122)(50,142,60,82,123)(51,143,55,83,124)(52,144,56,84,125)(53,139,57,79,126)(54,140,58,80,121), (1,88)(2,89)(3,90)(4,85)(5,86)(6,87)(7,153)(8,154)(9,155)(10,156)(11,151)(12,152)(13,180)(14,175)(15,176)(16,177)(17,178)(18,179)(19,39)(20,40)(21,41)(22,42)(23,37)(24,38)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,166)(32,167)(33,168)(34,163)(35,164)(36,165)(43,46)(44,47)(45,48)(49,125)(50,126)(51,121)(52,122)(53,123)(54,124)(55,58)(56,59)(57,60)(61,149)(62,150)(63,145)(64,146)(65,147)(66,148)(67,70)(68,71)(69,72)(73,172)(74,173)(75,174)(76,169)(77,170)(78,171)(79,142)(80,143)(81,144)(82,139)(83,140)(84,141)(97,157)(98,158)(99,159)(100,160)(101,161)(102,162)(103,106)(104,107)(105,108)(109,112)(110,113)(111,114)(115,118)(116,119)(117,120)(127,133)(128,134)(129,135)(130,136)(131,137)(132,138), (1,51,154)(2,52,155)(3,53,156)(4,54,151)(5,49,152)(6,50,153)(7,87,126)(8,88,121)(9,89,122)(10,90,123)(11,85,124)(12,86,125)(13,136,169)(14,137,170)(15,138,171)(16,133,172)(17,134,173)(18,135,174)(19,27,141)(20,28,142)(21,29,143)(22,30,144)(23,25,139)(24,26,140)(31,161,146)(32,162,147)(33,157,148)(34,158,149)(35,159,150)(36,160,145)(37,91,82)(38,92,83)(39,93,84)(40,94,79)(41,95,80)(42,96,81)(43,106,117)(44,107,118)(45,108,119)(46,103,120)(47,104,115)(48,105,116)(55,113,68)(56,114,69)(57,109,70)(58,110,71)(59,111,72)(60,112,67)(61,163,98)(62,164,99)(63,165,100)(64,166,101)(65,167,102)(66,168,97)(73,177,127)(74,178,128)(75,179,129)(76,180,130)(77,175,131)(78,176,132), (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), (1,100,4,97)(2,99,5,102)(3,98,6,101)(7,146,10,149)(8,145,11,148)(9,150,12,147)(13,96,16,93)(14,95,17,92)(15,94,18,91)(19,130,22,127)(20,129,23,132)(21,128,24,131)(25,176,28,179)(26,175,29,178)(27,180,30,177)(31,123,34,126)(32,122,35,125)(33,121,36,124)(37,138,40,135)(38,137,41,134)(39,136,42,133)(43,111,46,114)(44,110,47,113)(45,109,48,112)(49,167,52,164)(50,166,53,163)(51,165,54,168)(55,107,58,104)(56,106,59,103)(57,105,60,108)(61,153,64,156)(62,152,65,155)(63,151,66,154)(67,119,70,116)(68,118,71,115)(69,117,72,120)(73,141,76,144)(74,140,77,143)(75,139,78,142)(79,174,82,171)(80,173,83,170)(81,172,84,169)(85,157,88,160)(86,162,89,159)(87,161,90,158)>;
G:=Group( (1,29,68,92,85)(2,30,69,93,86)(3,25,70,94,87)(4,26,71,95,88)(5,27,72,96,89)(6,28,67,91,90)(7,156,23,109,40)(8,151,24,110,41)(9,152,19,111,42)(10,153,20,112,37)(11,154,21,113,38)(12,155,22,114,39)(13,162,99,177,117)(14,157,100,178,118)(15,158,101,179,119)(16,159,102,180,120)(17,160,97,175,115)(18,161,98,176,116)(31,163,78,105,174)(32,164,73,106,169)(33,165,74,107,170)(34,166,75,108,171)(35,167,76,103,172)(36,168,77,104,173)(43,136,147,62,127)(44,137,148,63,128)(45,138,149,64,129)(46,133,150,65,130)(47,134,145,66,131)(48,135,146,61,132)(49,141,59,81,122)(50,142,60,82,123)(51,143,55,83,124)(52,144,56,84,125)(53,139,57,79,126)(54,140,58,80,121), (1,88)(2,89)(3,90)(4,85)(5,86)(6,87)(7,153)(8,154)(9,155)(10,156)(11,151)(12,152)(13,180)(14,175)(15,176)(16,177)(17,178)(18,179)(19,39)(20,40)(21,41)(22,42)(23,37)(24,38)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,166)(32,167)(33,168)(34,163)(35,164)(36,165)(43,46)(44,47)(45,48)(49,125)(50,126)(51,121)(52,122)(53,123)(54,124)(55,58)(56,59)(57,60)(61,149)(62,150)(63,145)(64,146)(65,147)(66,148)(67,70)(68,71)(69,72)(73,172)(74,173)(75,174)(76,169)(77,170)(78,171)(79,142)(80,143)(81,144)(82,139)(83,140)(84,141)(97,157)(98,158)(99,159)(100,160)(101,161)(102,162)(103,106)(104,107)(105,108)(109,112)(110,113)(111,114)(115,118)(116,119)(117,120)(127,133)(128,134)(129,135)(130,136)(131,137)(132,138), (1,51,154)(2,52,155)(3,53,156)(4,54,151)(5,49,152)(6,50,153)(7,87,126)(8,88,121)(9,89,122)(10,90,123)(11,85,124)(12,86,125)(13,136,169)(14,137,170)(15,138,171)(16,133,172)(17,134,173)(18,135,174)(19,27,141)(20,28,142)(21,29,143)(22,30,144)(23,25,139)(24,26,140)(31,161,146)(32,162,147)(33,157,148)(34,158,149)(35,159,150)(36,160,145)(37,91,82)(38,92,83)(39,93,84)(40,94,79)(41,95,80)(42,96,81)(43,106,117)(44,107,118)(45,108,119)(46,103,120)(47,104,115)(48,105,116)(55,113,68)(56,114,69)(57,109,70)(58,110,71)(59,111,72)(60,112,67)(61,163,98)(62,164,99)(63,165,100)(64,166,101)(65,167,102)(66,168,97)(73,177,127)(74,178,128)(75,179,129)(76,180,130)(77,175,131)(78,176,132), (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), (1,100,4,97)(2,99,5,102)(3,98,6,101)(7,146,10,149)(8,145,11,148)(9,150,12,147)(13,96,16,93)(14,95,17,92)(15,94,18,91)(19,130,22,127)(20,129,23,132)(21,128,24,131)(25,176,28,179)(26,175,29,178)(27,180,30,177)(31,123,34,126)(32,122,35,125)(33,121,36,124)(37,138,40,135)(38,137,41,134)(39,136,42,133)(43,111,46,114)(44,110,47,113)(45,109,48,112)(49,167,52,164)(50,166,53,163)(51,165,54,168)(55,107,58,104)(56,106,59,103)(57,105,60,108)(61,153,64,156)(62,152,65,155)(63,151,66,154)(67,119,70,116)(68,118,71,115)(69,117,72,120)(73,141,76,144)(74,140,77,143)(75,139,78,142)(79,174,82,171)(80,173,83,170)(81,172,84,169)(85,157,88,160)(86,162,89,159)(87,161,90,158) );
G=PermutationGroup([[(1,29,68,92,85),(2,30,69,93,86),(3,25,70,94,87),(4,26,71,95,88),(5,27,72,96,89),(6,28,67,91,90),(7,156,23,109,40),(8,151,24,110,41),(9,152,19,111,42),(10,153,20,112,37),(11,154,21,113,38),(12,155,22,114,39),(13,162,99,177,117),(14,157,100,178,118),(15,158,101,179,119),(16,159,102,180,120),(17,160,97,175,115),(18,161,98,176,116),(31,163,78,105,174),(32,164,73,106,169),(33,165,74,107,170),(34,166,75,108,171),(35,167,76,103,172),(36,168,77,104,173),(43,136,147,62,127),(44,137,148,63,128),(45,138,149,64,129),(46,133,150,65,130),(47,134,145,66,131),(48,135,146,61,132),(49,141,59,81,122),(50,142,60,82,123),(51,143,55,83,124),(52,144,56,84,125),(53,139,57,79,126),(54,140,58,80,121)], [(1,88),(2,89),(3,90),(4,85),(5,86),(6,87),(7,153),(8,154),(9,155),(10,156),(11,151),(12,152),(13,180),(14,175),(15,176),(16,177),(17,178),(18,179),(19,39),(20,40),(21,41),(22,42),(23,37),(24,38),(25,91),(26,92),(27,93),(28,94),(29,95),(30,96),(31,166),(32,167),(33,168),(34,163),(35,164),(36,165),(43,46),(44,47),(45,48),(49,125),(50,126),(51,121),(52,122),(53,123),(54,124),(55,58),(56,59),(57,60),(61,149),(62,150),(63,145),(64,146),(65,147),(66,148),(67,70),(68,71),(69,72),(73,172),(74,173),(75,174),(76,169),(77,170),(78,171),(79,142),(80,143),(81,144),(82,139),(83,140),(84,141),(97,157),(98,158),(99,159),(100,160),(101,161),(102,162),(103,106),(104,107),(105,108),(109,112),(110,113),(111,114),(115,118),(116,119),(117,120),(127,133),(128,134),(129,135),(130,136),(131,137),(132,138)], [(1,51,154),(2,52,155),(3,53,156),(4,54,151),(5,49,152),(6,50,153),(7,87,126),(8,88,121),(9,89,122),(10,90,123),(11,85,124),(12,86,125),(13,136,169),(14,137,170),(15,138,171),(16,133,172),(17,134,173),(18,135,174),(19,27,141),(20,28,142),(21,29,143),(22,30,144),(23,25,139),(24,26,140),(31,161,146),(32,162,147),(33,157,148),(34,158,149),(35,159,150),(36,160,145),(37,91,82),(38,92,83),(39,93,84),(40,94,79),(41,95,80),(42,96,81),(43,106,117),(44,107,118),(45,108,119),(46,103,120),(47,104,115),(48,105,116),(55,113,68),(56,114,69),(57,109,70),(58,110,71),(59,111,72),(60,112,67),(61,163,98),(62,164,99),(63,165,100),(64,166,101),(65,167,102),(66,168,97),(73,177,127),(74,178,128),(75,179,129),(76,180,130),(77,175,131),(78,176,132)], [(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)], [(1,100,4,97),(2,99,5,102),(3,98,6,101),(7,146,10,149),(8,145,11,148),(9,150,12,147),(13,96,16,93),(14,95,17,92),(15,94,18,91),(19,130,22,127),(20,129,23,132),(21,128,24,131),(25,176,28,179),(26,175,29,178),(27,180,30,177),(31,123,34,126),(32,122,35,125),(33,121,36,124),(37,138,40,135),(38,137,41,134),(39,136,42,133),(43,111,46,114),(44,110,47,113),(45,109,48,112),(49,167,52,164),(50,166,53,163),(51,165,54,168),(55,107,58,104),(56,106,59,103),(57,105,60,108),(61,153,64,156),(62,152,65,155),(63,151,66,154),(67,119,70,116),(68,118,71,115),(69,117,72,120),(73,141,76,144),(74,140,77,143),(75,139,78,142),(79,174,82,171),(80,173,83,170),(81,172,84,169),(85,157,88,160),(86,162,89,159),(87,161,90,158)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | ··· | 6L | 10A | 10B | 15A | ··· | 15H | 20A | 20B | 20C | 20D | 30A | ··· | 30H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 10 | 10 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 2 | 2 | 9 | 9 | 45 | 45 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 4 | ··· | 4 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D5 | Dic3 | D6 | D10 | C4×D5 | S3×D5 | D5×Dic3 |
| kernel | D5×C3⋊Dic3 | C5×C3⋊Dic3 | C3⋊Dic15 | D5×C3×C6 | C32×D5 | C6×D5 | C3⋊Dic3 | C3×D5 | C30 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 8 | 4 | 2 | 4 | 8 | 8 |
Matrix representation of D5×C3⋊Dic3 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 18 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 8 | 25 | 0 | 0 | 0 | 0 |
| 17 | 53 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 30 |
| 0 | 0 | 0 | 0 | 49 | 42 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,43,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,18,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[8,17,0,0,0,0,25,53,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,19,49,0,0,0,0,30,42] >;
D5×C3⋊Dic3 in GAP, Magma, Sage, TeX
D_5\times C_3\rtimes {\rm Dic}_3 % in TeX
G:=Group("D5xC3:Dic3"); // GroupNames label
G:=SmallGroup(360,65);
// by ID
G=gap.SmallGroup(360,65);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-5,31,201,730,10373]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^3=d^6=1,e^2=d^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations