direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3⋊S3×C2×C10, C30⋊8D6, C62⋊5C10, C6⋊2(S3×C10), (C6×C30)⋊11C2, (C2×C30)⋊11S3, C15⋊9(C22×S3), (C3×C15)⋊11C23, (C3×C30)⋊10C22, C32⋊3(C22×C10), C3⋊2(S3×C2×C10), (C2×C6)⋊5(C5×S3), (C3×C6)⋊3(C2×C10), SmallGroup(360,160)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3⋊S3×C2×C10 |
Generators and relations for C3⋊S3×C2×C10
G = < a,b,c,d,e | a2=b10=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 544 in 192 conjugacy classes, 82 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C5, S3, C6, C23, C32, C10, C10, D6, C2×C6, C15, C3⋊S3, C3×C6, C2×C10, C2×C10, C22×S3, C5×S3, C30, C2×C3⋊S3, C62, C22×C10, C3×C15, S3×C10, C2×C30, C22×C3⋊S3, C5×C3⋊S3, C3×C30, S3×C2×C10, C10×C3⋊S3, C6×C30, C3⋊S3×C2×C10
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C3⋊S3, C2×C10, C22×S3, C5×S3, C2×C3⋊S3, C22×C10, S3×C10, C22×C3⋊S3, C5×C3⋊S3, S3×C2×C10, C10×C3⋊S3, C3⋊S3×C2×C10
►On 180 points
Generators in S180
(1 177)(2 178)(3 179)(4 180)(5 171)(6 172)(7 173)(8 174)(9 175)(10 176)(11 132)(12 133)(13 134)(14 135)(15 136)(16 137)(17 138)(18 139)(19 140)(20 131)(21 151)(22 152)(23 153)(24 154)(25 155)(26 156)(27 157)(28 158)(29 159)(30 160)(31 161)(32 162)(33 163)(34 164)(35 165)(36 166)(37 167)(38 168)(39 169)(40 170)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 100)(52 91)(53 92)(54 93)(55 94)(56 95)(57 96)(58 97)(59 98)(60 99)(61 110)(62 101)(63 102)(64 103)(65 104)(66 105)(67 106)(68 107)(69 108)(70 109)(71 147)(72 148)(73 149)(74 150)(75 141)(76 142)(77 143)(78 144)(79 145)(80 146)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)
(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)
(1 113 148)(2 114 149)(3 115 150)(4 116 141)(5 117 142)(6 118 143)(7 119 144)(8 120 145)(9 111 146)(10 112 147)(11 41 22)(12 42 23)(13 43 24)(14 44 25)(15 45 26)(16 46 27)(17 47 28)(18 48 29)(19 49 30)(20 50 21)(31 62 51)(32 63 52)(33 64 53)(34 65 54)(35 66 55)(36 67 56)(37 68 57)(38 69 58)(39 70 59)(40 61 60)(71 176 82)(72 177 83)(73 178 84)(74 179 85)(75 180 86)(76 171 87)(77 172 88)(78 173 89)(79 174 90)(80 175 81)(91 162 102)(92 163 103)(93 164 104)(94 165 105)(95 166 106)(96 167 107)(97 168 108)(98 169 109)(99 170 110)(100 161 101)(121 157 137)(122 158 138)(123 159 139)(124 160 140)(125 151 131)(126 152 132)(127 153 133)(128 154 134)(129 155 135)(130 156 136)
(1 124 161)(2 125 162)(3 126 163)(4 127 164)(5 128 165)(6 129 166)(7 130 167)(8 121 168)(9 122 169)(10 123 170)(11 53 74)(12 54 75)(13 55 76)(14 56 77)(15 57 78)(16 58 79)(17 59 80)(18 60 71)(19 51 72)(20 52 73)(21 63 84)(22 64 85)(23 65 86)(24 66 87)(25 67 88)(26 68 89)(27 69 90)(28 70 81)(29 61 82)(30 62 83)(31 177 49)(32 178 50)(33 179 41)(34 180 42)(35 171 43)(36 172 44)(37 173 45)(38 174 46)(39 175 47)(40 176 48)(91 149 131)(92 150 132)(93 141 133)(94 142 134)(95 143 135)(96 144 136)(97 145 137)(98 146 138)(99 147 139)(100 148 140)(101 113 160)(102 114 151)(103 115 152)(104 116 153)(105 117 154)(106 118 155)(107 119 156)(108 120 157)(109 111 158)(110 112 159)
(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 61)(19 62)(20 63)(21 52)(22 53)(23 54)(24 55)(25 56)(26 57)(27 58)(28 59)(29 60)(30 51)(31 49)(32 50)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 88)(78 89)(79 90)(80 81)(91 151)(92 152)(93 153)(94 154)(95 155)(96 156)(97 157)(98 158)(99 159)(100 160)(101 140)(102 131)(103 132)(104 133)(105 134)(106 135)(107 136)(108 137)(109 138)(110 139)(111 146)(112 147)(113 148)(114 149)(115 150)(116 141)(117 142)(118 143)(119 144)(120 145)(121 168)(122 169)(123 170)(124 161)(125 162)(126 163)(127 164)(128 165)(129 166)(130 167)
G:=sub<Sym(180)| (1,177)(2,178)(3,179)(4,180)(5,171)(6,172)(7,173)(8,174)(9,175)(10,176)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,131)(21,151)(22,152)(23,153)(24,154)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,161)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,169)(40,170)(41,126)(42,127)(43,128)(44,129)(45,130)(46,121)(47,122)(48,123)(49,124)(50,125)(51,100)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,110)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,147)(72,148)(73,149)(74,150)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120), (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), (1,113,148)(2,114,149)(3,115,150)(4,116,141)(5,117,142)(6,118,143)(7,119,144)(8,120,145)(9,111,146)(10,112,147)(11,41,22)(12,42,23)(13,43,24)(14,44,25)(15,45,26)(16,46,27)(17,47,28)(18,48,29)(19,49,30)(20,50,21)(31,62,51)(32,63,52)(33,64,53)(34,65,54)(35,66,55)(36,67,56)(37,68,57)(38,69,58)(39,70,59)(40,61,60)(71,176,82)(72,177,83)(73,178,84)(74,179,85)(75,180,86)(76,171,87)(77,172,88)(78,173,89)(79,174,90)(80,175,81)(91,162,102)(92,163,103)(93,164,104)(94,165,105)(95,166,106)(96,167,107)(97,168,108)(98,169,109)(99,170,110)(100,161,101)(121,157,137)(122,158,138)(123,159,139)(124,160,140)(125,151,131)(126,152,132)(127,153,133)(128,154,134)(129,155,135)(130,156,136), (1,124,161)(2,125,162)(3,126,163)(4,127,164)(5,128,165)(6,129,166)(7,130,167)(8,121,168)(9,122,169)(10,123,170)(11,53,74)(12,54,75)(13,55,76)(14,56,77)(15,57,78)(16,58,79)(17,59,80)(18,60,71)(19,51,72)(20,52,73)(21,63,84)(22,64,85)(23,65,86)(24,66,87)(25,67,88)(26,68,89)(27,69,90)(28,70,81)(29,61,82)(30,62,83)(31,177,49)(32,178,50)(33,179,41)(34,180,42)(35,171,43)(36,172,44)(37,173,45)(38,174,46)(39,175,47)(40,176,48)(91,149,131)(92,150,132)(93,141,133)(94,142,134)(95,143,135)(96,144,136)(97,145,137)(98,146,138)(99,147,139)(100,148,140)(101,113,160)(102,114,151)(103,115,152)(104,116,153)(105,117,154)(106,118,155)(107,119,156)(108,120,157)(109,111,158)(110,112,159), (11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,61)(19,62)(20,63)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,51)(31,49)(32,50)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,81)(91,151)(92,152)(93,153)(94,154)(95,155)(96,156)(97,157)(98,158)(99,159)(100,160)(101,140)(102,131)(103,132)(104,133)(105,134)(106,135)(107,136)(108,137)(109,138)(110,139)(111,146)(112,147)(113,148)(114,149)(115,150)(116,141)(117,142)(118,143)(119,144)(120,145)(121,168)(122,169)(123,170)(124,161)(125,162)(126,163)(127,164)(128,165)(129,166)(130,167)>;
G:=Group( (1,177)(2,178)(3,179)(4,180)(5,171)(6,172)(7,173)(8,174)(9,175)(10,176)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,131)(21,151)(22,152)(23,153)(24,154)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,161)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,169)(40,170)(41,126)(42,127)(43,128)(44,129)(45,130)(46,121)(47,122)(48,123)(49,124)(50,125)(51,100)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,110)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,147)(72,148)(73,149)(74,150)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120), (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), (1,113,148)(2,114,149)(3,115,150)(4,116,141)(5,117,142)(6,118,143)(7,119,144)(8,120,145)(9,111,146)(10,112,147)(11,41,22)(12,42,23)(13,43,24)(14,44,25)(15,45,26)(16,46,27)(17,47,28)(18,48,29)(19,49,30)(20,50,21)(31,62,51)(32,63,52)(33,64,53)(34,65,54)(35,66,55)(36,67,56)(37,68,57)(38,69,58)(39,70,59)(40,61,60)(71,176,82)(72,177,83)(73,178,84)(74,179,85)(75,180,86)(76,171,87)(77,172,88)(78,173,89)(79,174,90)(80,175,81)(91,162,102)(92,163,103)(93,164,104)(94,165,105)(95,166,106)(96,167,107)(97,168,108)(98,169,109)(99,170,110)(100,161,101)(121,157,137)(122,158,138)(123,159,139)(124,160,140)(125,151,131)(126,152,132)(127,153,133)(128,154,134)(129,155,135)(130,156,136), (1,124,161)(2,125,162)(3,126,163)(4,127,164)(5,128,165)(6,129,166)(7,130,167)(8,121,168)(9,122,169)(10,123,170)(11,53,74)(12,54,75)(13,55,76)(14,56,77)(15,57,78)(16,58,79)(17,59,80)(18,60,71)(19,51,72)(20,52,73)(21,63,84)(22,64,85)(23,65,86)(24,66,87)(25,67,88)(26,68,89)(27,69,90)(28,70,81)(29,61,82)(30,62,83)(31,177,49)(32,178,50)(33,179,41)(34,180,42)(35,171,43)(36,172,44)(37,173,45)(38,174,46)(39,175,47)(40,176,48)(91,149,131)(92,150,132)(93,141,133)(94,142,134)(95,143,135)(96,144,136)(97,145,137)(98,146,138)(99,147,139)(100,148,140)(101,113,160)(102,114,151)(103,115,152)(104,116,153)(105,117,154)(106,118,155)(107,119,156)(108,120,157)(109,111,158)(110,112,159), (11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,61)(19,62)(20,63)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,51)(31,49)(32,50)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,81)(91,151)(92,152)(93,153)(94,154)(95,155)(96,156)(97,157)(98,158)(99,159)(100,160)(101,140)(102,131)(103,132)(104,133)(105,134)(106,135)(107,136)(108,137)(109,138)(110,139)(111,146)(112,147)(113,148)(114,149)(115,150)(116,141)(117,142)(118,143)(119,144)(120,145)(121,168)(122,169)(123,170)(124,161)(125,162)(126,163)(127,164)(128,165)(129,166)(130,167) );
G=PermutationGroup([[(1,177),(2,178),(3,179),(4,180),(5,171),(6,172),(7,173),(8,174),(9,175),(10,176),(11,132),(12,133),(13,134),(14,135),(15,136),(16,137),(17,138),(18,139),(19,140),(20,131),(21,151),(22,152),(23,153),(24,154),(25,155),(26,156),(27,157),(28,158),(29,159),(30,160),(31,161),(32,162),(33,163),(34,164),(35,165),(36,166),(37,167),(38,168),(39,169),(40,170),(41,126),(42,127),(43,128),(44,129),(45,130),(46,121),(47,122),(48,123),(49,124),(50,125),(51,100),(52,91),(53,92),(54,93),(55,94),(56,95),(57,96),(58,97),(59,98),(60,99),(61,110),(62,101),(63,102),(64,103),(65,104),(66,105),(67,106),(68,107),(69,108),(70,109),(71,147),(72,148),(73,149),(74,150),(75,141),(76,142),(77,143),(78,144),(79,145),(80,146),(81,111),(82,112),(83,113),(84,114),(85,115),(86,116),(87,117),(88,118),(89,119),(90,120)], [(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)], [(1,113,148),(2,114,149),(3,115,150),(4,116,141),(5,117,142),(6,118,143),(7,119,144),(8,120,145),(9,111,146),(10,112,147),(11,41,22),(12,42,23),(13,43,24),(14,44,25),(15,45,26),(16,46,27),(17,47,28),(18,48,29),(19,49,30),(20,50,21),(31,62,51),(32,63,52),(33,64,53),(34,65,54),(35,66,55),(36,67,56),(37,68,57),(38,69,58),(39,70,59),(40,61,60),(71,176,82),(72,177,83),(73,178,84),(74,179,85),(75,180,86),(76,171,87),(77,172,88),(78,173,89),(79,174,90),(80,175,81),(91,162,102),(92,163,103),(93,164,104),(94,165,105),(95,166,106),(96,167,107),(97,168,108),(98,169,109),(99,170,110),(100,161,101),(121,157,137),(122,158,138),(123,159,139),(124,160,140),(125,151,131),(126,152,132),(127,153,133),(128,154,134),(129,155,135),(130,156,136)], [(1,124,161),(2,125,162),(3,126,163),(4,127,164),(5,128,165),(6,129,166),(7,130,167),(8,121,168),(9,122,169),(10,123,170),(11,53,74),(12,54,75),(13,55,76),(14,56,77),(15,57,78),(16,58,79),(17,59,80),(18,60,71),(19,51,72),(20,52,73),(21,63,84),(22,64,85),(23,65,86),(24,66,87),(25,67,88),(26,68,89),(27,69,90),(28,70,81),(29,61,82),(30,62,83),(31,177,49),(32,178,50),(33,179,41),(34,180,42),(35,171,43),(36,172,44),(37,173,45),(38,174,46),(39,175,47),(40,176,48),(91,149,131),(92,150,132),(93,141,133),(94,142,134),(95,143,135),(96,144,136),(97,145,137),(98,146,138),(99,147,139),(100,148,140),(101,113,160),(102,114,151),(103,115,152),(104,116,153),(105,117,154),(106,118,155),(107,119,156),(108,120,157),(109,111,158),(110,112,159)], [(11,64),(12,65),(13,66),(14,67),(15,68),(16,69),(17,70),(18,61),(19,62),(20,63),(21,52),(22,53),(23,54),(24,55),(25,56),(26,57),(27,58),(28,59),(29,60),(30,51),(31,49),(32,50),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(71,82),(72,83),(73,84),(74,85),(75,86),(76,87),(77,88),(78,89),(79,90),(80,81),(91,151),(92,152),(93,153),(94,154),(95,155),(96,156),(97,157),(98,158),(99,159),(100,160),(101,140),(102,131),(103,132),(104,133),(105,134),(106,135),(107,136),(108,137),(109,138),(110,139),(111,146),(112,147),(113,148),(114,149),(115,150),(116,141),(117,142),(118,143),(119,144),(120,145),(121,168),(122,169),(123,170),(124,161),(125,162),(126,163),(127,164),(128,165),(129,166),(130,167)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 5A | 5B | 5C | 5D | 6A | ··· | 6L | 10A | ··· | 10L | 10M | ··· | 10AB | 15A | ··· | 15P | 30A | ··· | 30AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | S3 | D6 | C5×S3 | S3×C10 |
| kernel | C3⋊S3×C2×C10 | C10×C3⋊S3 | C6×C30 | C22×C3⋊S3 | C2×C3⋊S3 | C62 | C2×C30 | C30 | C2×C6 | C6 |
| # reps | 1 | 6 | 1 | 4 | 24 | 4 | 4 | 12 | 16 | 48 |
Matrix representation of C3⋊S3×C2×C10 ►in GL4(𝔽31) generated by
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 1 |
| 0 | 0 | 30 | 0 |
| 0 | 30 | 0 | 0 |
| 1 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 30 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 30 | 1 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[15,0,0,0,0,15,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,30,30,0,0,1,0],[0,1,0,0,30,30,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,30,30,0,0,0,0,30,0,0,0,1,1] >;
C3⋊S3×C2×C10 in GAP, Magma, Sage, TeX
C_3\rtimes S_3\times C_2\times C_{10} % in TeX
G:=Group("C3:S3xC2xC10"); // GroupNames label
G:=SmallGroup(360,160);
// by ID
G=gap.SmallGroup(360,160);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-3,-3,2404,8645]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^10=c^3=d^3=e^2=1,a*b=b*a,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=c^-1,e*d*e=d^-1>;
// generators/relations