direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3⋊S3×Dic5, C30.10D6, C15⋊5(C4×S3), C3⋊1(S3×Dic5), C6.17(S3×D5), C3⋊Dic15⋊3C2, (C3×Dic5)⋊1S3, (C3×C6).21D10, C32⋊5(C2×Dic5), (C3×C30).9C22, (C32×Dic5)⋊4C2, C5⋊4(C4×C3⋊S3), (C5×C3⋊S3)⋊4C4, C2.2(D5×C3⋊S3), (C3×C15)⋊17(C2×C4), C10.2(C2×C3⋊S3), (C2×C3⋊S3).3D5, (C10×C3⋊S3).1C2, SmallGroup(360,66)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — C3⋊S3×Dic5 |
Generators and relations for C3⋊S3×Dic5
G = < a,b,c,d,e | a3=b3=c2=d10=1, e2=d5, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 456 in 96 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C5, S3, C6, C2×C4, C32, C10, C10, Dic3, C12, D6, C15, C3⋊S3, C3×C6, Dic5, Dic5, C2×C10, C4×S3, C5×S3, C30, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×Dic5, C3×C15, C3×Dic5, Dic15, S3×C10, C4×C3⋊S3, C5×C3⋊S3, C3×C30, S3×Dic5, C32×Dic5, C3⋊Dic15, C10×C3⋊S3, C3⋊S3×Dic5
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, D6, C3⋊S3, Dic5, D10, C4×S3, C2×C3⋊S3, C2×Dic5, S3×D5, C4×C3⋊S3, S3×Dic5, D5×C3⋊S3, C3⋊S3×Dic5
►On 180 points
Generators in S180
(1 33 17)(2 34 18)(3 35 19)(4 36 20)(5 37 11)(6 38 12)(7 39 13)(8 40 14)(9 31 15)(10 32 16)(21 82 93)(22 83 94)(23 84 95)(24 85 96)(25 86 97)(26 87 98)(27 88 99)(28 89 100)(29 90 91)(30 81 92)(41 156 171)(42 157 172)(43 158 173)(44 159 174)(45 160 175)(46 151 176)(47 152 177)(48 153 178)(49 154 179)(50 155 180)(51 76 68)(52 77 69)(53 78 70)(54 79 61)(55 80 62)(56 71 63)(57 72 64)(58 73 65)(59 74 66)(60 75 67)(101 126 165)(102 127 166)(103 128 167)(104 129 168)(105 130 169)(106 121 170)(107 122 161)(108 123 162)(109 124 163)(110 125 164)(111 136 150)(112 137 141)(113 138 142)(114 139 143)(115 140 144)(116 131 145)(117 132 146)(118 133 147)(119 134 148)(120 135 149)
(1 61 89)(2 62 90)(3 63 81)(4 64 82)(5 65 83)(6 66 84)(7 67 85)(8 68 86)(9 69 87)(10 70 88)(11 73 22)(12 74 23)(13 75 24)(14 76 25)(15 77 26)(16 78 27)(17 79 28)(18 80 29)(19 71 30)(20 72 21)(31 52 98)(32 53 99)(33 54 100)(34 55 91)(35 56 92)(36 57 93)(37 58 94)(38 59 95)(39 60 96)(40 51 97)(41 161 117)(42 162 118)(43 163 119)(44 164 120)(45 165 111)(46 166 112)(47 167 113)(48 168 114)(49 169 115)(50 170 116)(101 136 160)(102 137 151)(103 138 152)(104 139 153)(105 140 154)(106 131 155)(107 132 156)(108 133 157)(109 134 158)(110 135 159)(121 145 180)(122 146 171)(123 147 172)(124 148 173)(125 149 174)(126 150 175)(127 141 176)(128 142 177)(129 143 178)(130 144 179)
(11 37)(12 38)(13 39)(14 40)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 57)(22 58)(23 59)(24 60)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(41 132)(42 133)(43 134)(44 135)(45 136)(46 137)(47 138)(48 139)(49 140)(50 131)(61 89)(62 90)(63 81)(64 82)(65 83)(66 84)(67 85)(68 86)(69 87)(70 88)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 99)(79 100)(80 91)(101 165)(102 166)(103 167)(104 168)(105 169)(106 170)(107 161)(108 162)(109 163)(110 164)(111 160)(112 151)(113 152)(114 153)(115 154)(116 155)(117 156)(118 157)(119 158)(120 159)(141 176)(142 177)(143 178)(144 179)(145 180)(146 171)(147 172)(148 173)(149 174)(150 175)
(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)
(1 127 6 122)(2 126 7 121)(3 125 8 130)(4 124 9 129)(5 123 10 128)(11 108 16 103)(12 107 17 102)(13 106 18 101)(14 105 19 110)(15 104 20 109)(21 158 26 153)(22 157 27 152)(23 156 28 151)(24 155 29 160)(25 154 30 159)(31 168 36 163)(32 167 37 162)(33 166 38 161)(34 165 39 170)(35 164 40 169)(41 100 46 95)(42 99 47 94)(43 98 48 93)(44 97 49 92)(45 96 50 91)(51 115 56 120)(52 114 57 119)(53 113 58 118)(54 112 59 117)(55 111 60 116)(61 141 66 146)(62 150 67 145)(63 149 68 144)(64 148 69 143)(65 147 70 142)(71 135 76 140)(72 134 77 139)(73 133 78 138)(74 132 79 137)(75 131 80 136)(81 174 86 179)(82 173 87 178)(83 172 88 177)(84 171 89 176)(85 180 90 175)
G:=sub<Sym(180)| (1,33,17)(2,34,18)(3,35,19)(4,36,20)(5,37,11)(6,38,12)(7,39,13)(8,40,14)(9,31,15)(10,32,16)(21,82,93)(22,83,94)(23,84,95)(24,85,96)(25,86,97)(26,87,98)(27,88,99)(28,89,100)(29,90,91)(30,81,92)(41,156,171)(42,157,172)(43,158,173)(44,159,174)(45,160,175)(46,151,176)(47,152,177)(48,153,178)(49,154,179)(50,155,180)(51,76,68)(52,77,69)(53,78,70)(54,79,61)(55,80,62)(56,71,63)(57,72,64)(58,73,65)(59,74,66)(60,75,67)(101,126,165)(102,127,166)(103,128,167)(104,129,168)(105,130,169)(106,121,170)(107,122,161)(108,123,162)(109,124,163)(110,125,164)(111,136,150)(112,137,141)(113,138,142)(114,139,143)(115,140,144)(116,131,145)(117,132,146)(118,133,147)(119,134,148)(120,135,149), (1,61,89)(2,62,90)(3,63,81)(4,64,82)(5,65,83)(6,66,84)(7,67,85)(8,68,86)(9,69,87)(10,70,88)(11,73,22)(12,74,23)(13,75,24)(14,76,25)(15,77,26)(16,78,27)(17,79,28)(18,80,29)(19,71,30)(20,72,21)(31,52,98)(32,53,99)(33,54,100)(34,55,91)(35,56,92)(36,57,93)(37,58,94)(38,59,95)(39,60,96)(40,51,97)(41,161,117)(42,162,118)(43,163,119)(44,164,120)(45,165,111)(46,166,112)(47,167,113)(48,168,114)(49,169,115)(50,170,116)(101,136,160)(102,137,151)(103,138,152)(104,139,153)(105,140,154)(106,131,155)(107,132,156)(108,133,157)(109,134,158)(110,135,159)(121,145,180)(122,146,171)(123,147,172)(124,148,173)(125,149,174)(126,150,175)(127,141,176)(128,142,177)(129,143,178)(130,144,179), (11,37)(12,38)(13,39)(14,40)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,57)(22,58)(23,59)(24,60)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(41,132)(42,133)(43,134)(44,135)(45,136)(46,137)(47,138)(48,139)(49,140)(50,131)(61,89)(62,90)(63,81)(64,82)(65,83)(66,84)(67,85)(68,86)(69,87)(70,88)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,91)(101,165)(102,166)(103,167)(104,168)(105,169)(106,170)(107,161)(108,162)(109,163)(110,164)(111,160)(112,151)(113,152)(114,153)(115,154)(116,155)(117,156)(118,157)(119,158)(120,159)(141,176)(142,177)(143,178)(144,179)(145,180)(146,171)(147,172)(148,173)(149,174)(150,175), (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), (1,127,6,122)(2,126,7,121)(3,125,8,130)(4,124,9,129)(5,123,10,128)(11,108,16,103)(12,107,17,102)(13,106,18,101)(14,105,19,110)(15,104,20,109)(21,158,26,153)(22,157,27,152)(23,156,28,151)(24,155,29,160)(25,154,30,159)(31,168,36,163)(32,167,37,162)(33,166,38,161)(34,165,39,170)(35,164,40,169)(41,100,46,95)(42,99,47,94)(43,98,48,93)(44,97,49,92)(45,96,50,91)(51,115,56,120)(52,114,57,119)(53,113,58,118)(54,112,59,117)(55,111,60,116)(61,141,66,146)(62,150,67,145)(63,149,68,144)(64,148,69,143)(65,147,70,142)(71,135,76,140)(72,134,77,139)(73,133,78,138)(74,132,79,137)(75,131,80,136)(81,174,86,179)(82,173,87,178)(83,172,88,177)(84,171,89,176)(85,180,90,175)>;
G:=Group( (1,33,17)(2,34,18)(3,35,19)(4,36,20)(5,37,11)(6,38,12)(7,39,13)(8,40,14)(9,31,15)(10,32,16)(21,82,93)(22,83,94)(23,84,95)(24,85,96)(25,86,97)(26,87,98)(27,88,99)(28,89,100)(29,90,91)(30,81,92)(41,156,171)(42,157,172)(43,158,173)(44,159,174)(45,160,175)(46,151,176)(47,152,177)(48,153,178)(49,154,179)(50,155,180)(51,76,68)(52,77,69)(53,78,70)(54,79,61)(55,80,62)(56,71,63)(57,72,64)(58,73,65)(59,74,66)(60,75,67)(101,126,165)(102,127,166)(103,128,167)(104,129,168)(105,130,169)(106,121,170)(107,122,161)(108,123,162)(109,124,163)(110,125,164)(111,136,150)(112,137,141)(113,138,142)(114,139,143)(115,140,144)(116,131,145)(117,132,146)(118,133,147)(119,134,148)(120,135,149), (1,61,89)(2,62,90)(3,63,81)(4,64,82)(5,65,83)(6,66,84)(7,67,85)(8,68,86)(9,69,87)(10,70,88)(11,73,22)(12,74,23)(13,75,24)(14,76,25)(15,77,26)(16,78,27)(17,79,28)(18,80,29)(19,71,30)(20,72,21)(31,52,98)(32,53,99)(33,54,100)(34,55,91)(35,56,92)(36,57,93)(37,58,94)(38,59,95)(39,60,96)(40,51,97)(41,161,117)(42,162,118)(43,163,119)(44,164,120)(45,165,111)(46,166,112)(47,167,113)(48,168,114)(49,169,115)(50,170,116)(101,136,160)(102,137,151)(103,138,152)(104,139,153)(105,140,154)(106,131,155)(107,132,156)(108,133,157)(109,134,158)(110,135,159)(121,145,180)(122,146,171)(123,147,172)(124,148,173)(125,149,174)(126,150,175)(127,141,176)(128,142,177)(129,143,178)(130,144,179), (11,37)(12,38)(13,39)(14,40)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,57)(22,58)(23,59)(24,60)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(41,132)(42,133)(43,134)(44,135)(45,136)(46,137)(47,138)(48,139)(49,140)(50,131)(61,89)(62,90)(63,81)(64,82)(65,83)(66,84)(67,85)(68,86)(69,87)(70,88)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,91)(101,165)(102,166)(103,167)(104,168)(105,169)(106,170)(107,161)(108,162)(109,163)(110,164)(111,160)(112,151)(113,152)(114,153)(115,154)(116,155)(117,156)(118,157)(119,158)(120,159)(141,176)(142,177)(143,178)(144,179)(145,180)(146,171)(147,172)(148,173)(149,174)(150,175), (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), (1,127,6,122)(2,126,7,121)(3,125,8,130)(4,124,9,129)(5,123,10,128)(11,108,16,103)(12,107,17,102)(13,106,18,101)(14,105,19,110)(15,104,20,109)(21,158,26,153)(22,157,27,152)(23,156,28,151)(24,155,29,160)(25,154,30,159)(31,168,36,163)(32,167,37,162)(33,166,38,161)(34,165,39,170)(35,164,40,169)(41,100,46,95)(42,99,47,94)(43,98,48,93)(44,97,49,92)(45,96,50,91)(51,115,56,120)(52,114,57,119)(53,113,58,118)(54,112,59,117)(55,111,60,116)(61,141,66,146)(62,150,67,145)(63,149,68,144)(64,148,69,143)(65,147,70,142)(71,135,76,140)(72,134,77,139)(73,133,78,138)(74,132,79,137)(75,131,80,136)(81,174,86,179)(82,173,87,178)(83,172,88,177)(84,171,89,176)(85,180,90,175) );
G=PermutationGroup([[(1,33,17),(2,34,18),(3,35,19),(4,36,20),(5,37,11),(6,38,12),(7,39,13),(8,40,14),(9,31,15),(10,32,16),(21,82,93),(22,83,94),(23,84,95),(24,85,96),(25,86,97),(26,87,98),(27,88,99),(28,89,100),(29,90,91),(30,81,92),(41,156,171),(42,157,172),(43,158,173),(44,159,174),(45,160,175),(46,151,176),(47,152,177),(48,153,178),(49,154,179),(50,155,180),(51,76,68),(52,77,69),(53,78,70),(54,79,61),(55,80,62),(56,71,63),(57,72,64),(58,73,65),(59,74,66),(60,75,67),(101,126,165),(102,127,166),(103,128,167),(104,129,168),(105,130,169),(106,121,170),(107,122,161),(108,123,162),(109,124,163),(110,125,164),(111,136,150),(112,137,141),(113,138,142),(114,139,143),(115,140,144),(116,131,145),(117,132,146),(118,133,147),(119,134,148),(120,135,149)], [(1,61,89),(2,62,90),(3,63,81),(4,64,82),(5,65,83),(6,66,84),(7,67,85),(8,68,86),(9,69,87),(10,70,88),(11,73,22),(12,74,23),(13,75,24),(14,76,25),(15,77,26),(16,78,27),(17,79,28),(18,80,29),(19,71,30),(20,72,21),(31,52,98),(32,53,99),(33,54,100),(34,55,91),(35,56,92),(36,57,93),(37,58,94),(38,59,95),(39,60,96),(40,51,97),(41,161,117),(42,162,118),(43,163,119),(44,164,120),(45,165,111),(46,166,112),(47,167,113),(48,168,114),(49,169,115),(50,170,116),(101,136,160),(102,137,151),(103,138,152),(104,139,153),(105,140,154),(106,131,155),(107,132,156),(108,133,157),(109,134,158),(110,135,159),(121,145,180),(122,146,171),(123,147,172),(124,148,173),(125,149,174),(126,150,175),(127,141,176),(128,142,177),(129,143,178),(130,144,179)], [(11,37),(12,38),(13,39),(14,40),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(21,57),(22,58),(23,59),(24,60),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(41,132),(42,133),(43,134),(44,135),(45,136),(46,137),(47,138),(48,139),(49,140),(50,131),(61,89),(62,90),(63,81),(64,82),(65,83),(66,84),(67,85),(68,86),(69,87),(70,88),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,99),(79,100),(80,91),(101,165),(102,166),(103,167),(104,168),(105,169),(106,170),(107,161),(108,162),(109,163),(110,164),(111,160),(112,151),(113,152),(114,153),(115,154),(116,155),(117,156),(118,157),(119,158),(120,159),(141,176),(142,177),(143,178),(144,179),(145,180),(146,171),(147,172),(148,173),(149,174),(150,175)], [(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)], [(1,127,6,122),(2,126,7,121),(3,125,8,130),(4,124,9,129),(5,123,10,128),(11,108,16,103),(12,107,17,102),(13,106,18,101),(14,105,19,110),(15,104,20,109),(21,158,26,153),(22,157,27,152),(23,156,28,151),(24,155,29,160),(25,154,30,159),(31,168,36,163),(32,167,37,162),(33,166,38,161),(34,165,39,170),(35,164,40,169),(41,100,46,95),(42,99,47,94),(43,98,48,93),(44,97,49,92),(45,96,50,91),(51,115,56,120),(52,114,57,119),(53,113,58,118),(54,112,59,117),(55,111,60,116),(61,141,66,146),(62,150,67,145),(63,149,68,144),(64,148,69,143),(65,147,70,142),(71,135,76,140),(72,134,77,139),(73,133,78,138),(74,132,79,137),(75,131,80,136),(81,174,86,179),(82,173,87,178),(83,172,88,177),(84,171,89,176),(85,180,90,175)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 10A | 10B | 10C | 10D | 10E | 10F | 12A | ··· | 12H | 15A | ··· | 15H | 30A | ··· | 30H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 2 | 2 | 5 | 5 | 45 | 45 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 10 | ··· | 10 | 4 | ··· | 4 | 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 | D6 | Dic5 | D10 | C4×S3 | S3×D5 | S3×Dic5 |
| kernel | C3⋊S3×Dic5 | C32×Dic5 | C3⋊Dic15 | C10×C3⋊S3 | C5×C3⋊S3 | C3×Dic5 | C2×C3⋊S3 | C30 | C3⋊S3 | C3×C6 | C15 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 4 | 4 | 2 | 8 | 8 | 8 |
Matrix representation of C3⋊S3×Dic5 ►in GL6(𝔽61)
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 58 | 0 | 0 |
| 0 | 0 | 1 | 59 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 36 |
| 0 | 0 | 0 | 0 | 24 | 28 |
G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,58,59,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,60,0,0,0,0,1,0],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,33,24,0,0,0,0,36,28] >;
C3⋊S3×Dic5 in GAP, Magma, Sage, TeX
C_3\rtimes S_3\times {\rm Dic}_5 % in TeX
G:=Group("C3:S3xDic5"); // GroupNames label
G:=SmallGroup(360,66);
// by ID
G=gap.SmallGroup(360,66);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,201,730,10373]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^10=1,e^2=d^5,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations