direct product, metabelian, supersoluble, monomial, A-group
Aliases: Dic3×C3⋊Dic3, C33⋊5C42, C62.107D6, C3⋊1Dic32, C33⋊5C4⋊3C4, C6.8(S3×Dic3), C32⋊6(C4×Dic3), (C3×Dic3)⋊2Dic3, (C6×Dic3).17S3, (C32×Dic3)⋊6C4, C6.9(C6.D6), (C3×C62).6C22, (C2×C6).30S32, C6.3(C4×C3⋊S3), C3⋊1(C4×C3⋊Dic3), (C3×C6).48(C4×S3), (C3×C3⋊Dic3)⋊6C4, C22.4(S3×C3⋊S3), C6.3(C2×C3⋊Dic3), C2.2(S3×C3⋊Dic3), C2.2(Dic3×C3⋊S3), (Dic3×C3×C6).11C2, C2.2(C33⋊8(C2×C4)), (C6×C3⋊Dic3).12C2, (C2×C3⋊Dic3).14S3, (C32×C6).37(C2×C4), (C2×C33⋊5C4).1C2, (C3×C6).37(C2×Dic3), (C2×Dic3).5(C3⋊S3), (C2×C6).12(C2×C3⋊S3), SmallGroup(432,448)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — Dic3×C3⋊Dic3 |
Generators and relations for Dic3×C3⋊Dic3
G = < a,b,c,d,e | a6=c3=d6=1, b2=a3, e2=d3, bab-1=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: 984 in 244 conjugacy classes, 88 normal (26 characteristic)
C1, C2, C3, C3, C3, C4, C22, C6, C6, C6, C2×C4, C32, C32, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C62, C62, C4×Dic3, C32×C6, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, C32×Dic3, C3×C3⋊Dic3, C33⋊5C4, C3×C62, Dic32, C4×C3⋊Dic3, Dic3×C3×C6, C6×C3⋊Dic3, C2×C33⋊5C4, Dic3×C3⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, C3⋊S3, C4×S3, C2×Dic3, C3⋊Dic3, S32, C2×C3⋊S3, C4×Dic3, S3×Dic3, C6.D6, C4×C3⋊S3, C2×C3⋊Dic3, S3×C3⋊S3, Dic32, C4×C3⋊Dic3, S3×C3⋊Dic3, Dic3×C3⋊S3, C33⋊8(C2×C4), Dic3×C3⋊Dic3
►On 144 points
Generators in S144
(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)
(1 98 4 101)(2 97 5 100)(3 102 6 99)(7 91 10 94)(8 96 11 93)(9 95 12 92)(13 89 16 86)(14 88 17 85)(15 87 18 90)(19 78 22 75)(20 77 23 74)(21 76 24 73)(25 79 28 82)(26 84 29 81)(27 83 30 80)(31 104 34 107)(32 103 35 106)(33 108 36 105)(37 113 40 110)(38 112 41 109)(39 111 42 114)(43 117 46 120)(44 116 47 119)(45 115 48 118)(49 125 52 122)(50 124 53 121)(51 123 54 126)(55 127 58 130)(56 132 59 129)(57 131 60 128)(61 138 64 135)(62 137 65 134)(63 136 66 133)(67 140 70 143)(68 139 71 142)(69 144 72 141)
(1 9 16)(2 10 17)(3 11 18)(4 12 13)(5 7 14)(6 8 15)(19 142 117)(20 143 118)(21 144 119)(22 139 120)(23 140 115)(24 141 116)(25 31 42)(26 32 37)(27 33 38)(28 34 39)(29 35 40)(30 36 41)(43 75 71)(44 76 72)(45 77 67)(46 78 68)(47 73 69)(48 74 70)(49 60 62)(50 55 63)(51 56 64)(52 57 65)(53 58 66)(54 59 61)(79 104 114)(80 105 109)(81 106 110)(82 107 111)(83 108 112)(84 103 113)(85 97 94)(86 98 95)(87 99 96)(88 100 91)(89 101 92)(90 102 93)(121 130 133)(122 131 134)(123 132 135)(124 127 136)(125 128 137)(126 129 138)
(1 42 14 35 11 27)(2 37 15 36 12 28)(3 38 16 31 7 29)(4 39 17 32 8 30)(5 40 18 33 9 25)(6 41 13 34 10 26)(19 136 115 131 144 126)(20 137 116 132 139 121)(21 138 117 127 140 122)(22 133 118 128 141 123)(23 134 119 129 142 124)(24 135 120 130 143 125)(43 55 67 52 73 61)(44 56 68 53 74 62)(45 57 69 54 75 63)(46 58 70 49 76 64)(47 59 71 50 77 65)(48 60 72 51 78 66)(79 100 110 90 108 95)(80 101 111 85 103 96)(81 102 112 86 104 91)(82 97 113 87 105 92)(83 98 114 88 106 93)(84 99 109 89 107 94)
(1 71 35 65)(2 72 36 66)(3 67 31 61)(4 68 32 62)(5 69 33 63)(6 70 34 64)(7 73 38 55)(8 74 39 56)(9 75 40 57)(10 76 41 58)(11 77 42 59)(12 78 37 60)(13 46 26 49)(14 47 27 50)(15 48 28 51)(16 43 29 52)(17 44 30 53)(18 45 25 54)(19 110 131 95)(20 111 132 96)(21 112 127 91)(22 113 128 92)(23 114 129 93)(24 109 130 94)(79 126 90 115)(80 121 85 116)(81 122 86 117)(82 123 87 118)(83 124 88 119)(84 125 89 120)(97 141 105 133)(98 142 106 134)(99 143 107 135)(100 144 108 136)(101 139 103 137)(102 140 104 138)
G:=sub<Sym(144)| (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), (1,98,4,101)(2,97,5,100)(3,102,6,99)(7,91,10,94)(8,96,11,93)(9,95,12,92)(13,89,16,86)(14,88,17,85)(15,87,18,90)(19,78,22,75)(20,77,23,74)(21,76,24,73)(25,79,28,82)(26,84,29,81)(27,83,30,80)(31,104,34,107)(32,103,35,106)(33,108,36,105)(37,113,40,110)(38,112,41,109)(39,111,42,114)(43,117,46,120)(44,116,47,119)(45,115,48,118)(49,125,52,122)(50,124,53,121)(51,123,54,126)(55,127,58,130)(56,132,59,129)(57,131,60,128)(61,138,64,135)(62,137,65,134)(63,136,66,133)(67,140,70,143)(68,139,71,142)(69,144,72,141), (1,9,16)(2,10,17)(3,11,18)(4,12,13)(5,7,14)(6,8,15)(19,142,117)(20,143,118)(21,144,119)(22,139,120)(23,140,115)(24,141,116)(25,31,42)(26,32,37)(27,33,38)(28,34,39)(29,35,40)(30,36,41)(43,75,71)(44,76,72)(45,77,67)(46,78,68)(47,73,69)(48,74,70)(49,60,62)(50,55,63)(51,56,64)(52,57,65)(53,58,66)(54,59,61)(79,104,114)(80,105,109)(81,106,110)(82,107,111)(83,108,112)(84,103,113)(85,97,94)(86,98,95)(87,99,96)(88,100,91)(89,101,92)(90,102,93)(121,130,133)(122,131,134)(123,132,135)(124,127,136)(125,128,137)(126,129,138), (1,42,14,35,11,27)(2,37,15,36,12,28)(3,38,16,31,7,29)(4,39,17,32,8,30)(5,40,18,33,9,25)(6,41,13,34,10,26)(19,136,115,131,144,126)(20,137,116,132,139,121)(21,138,117,127,140,122)(22,133,118,128,141,123)(23,134,119,129,142,124)(24,135,120,130,143,125)(43,55,67,52,73,61)(44,56,68,53,74,62)(45,57,69,54,75,63)(46,58,70,49,76,64)(47,59,71,50,77,65)(48,60,72,51,78,66)(79,100,110,90,108,95)(80,101,111,85,103,96)(81,102,112,86,104,91)(82,97,113,87,105,92)(83,98,114,88,106,93)(84,99,109,89,107,94), (1,71,35,65)(2,72,36,66)(3,67,31,61)(4,68,32,62)(5,69,33,63)(6,70,34,64)(7,73,38,55)(8,74,39,56)(9,75,40,57)(10,76,41,58)(11,77,42,59)(12,78,37,60)(13,46,26,49)(14,47,27,50)(15,48,28,51)(16,43,29,52)(17,44,30,53)(18,45,25,54)(19,110,131,95)(20,111,132,96)(21,112,127,91)(22,113,128,92)(23,114,129,93)(24,109,130,94)(79,126,90,115)(80,121,85,116)(81,122,86,117)(82,123,87,118)(83,124,88,119)(84,125,89,120)(97,141,105,133)(98,142,106,134)(99,143,107,135)(100,144,108,136)(101,139,103,137)(102,140,104,138)>;
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), (1,98,4,101)(2,97,5,100)(3,102,6,99)(7,91,10,94)(8,96,11,93)(9,95,12,92)(13,89,16,86)(14,88,17,85)(15,87,18,90)(19,78,22,75)(20,77,23,74)(21,76,24,73)(25,79,28,82)(26,84,29,81)(27,83,30,80)(31,104,34,107)(32,103,35,106)(33,108,36,105)(37,113,40,110)(38,112,41,109)(39,111,42,114)(43,117,46,120)(44,116,47,119)(45,115,48,118)(49,125,52,122)(50,124,53,121)(51,123,54,126)(55,127,58,130)(56,132,59,129)(57,131,60,128)(61,138,64,135)(62,137,65,134)(63,136,66,133)(67,140,70,143)(68,139,71,142)(69,144,72,141), (1,9,16)(2,10,17)(3,11,18)(4,12,13)(5,7,14)(6,8,15)(19,142,117)(20,143,118)(21,144,119)(22,139,120)(23,140,115)(24,141,116)(25,31,42)(26,32,37)(27,33,38)(28,34,39)(29,35,40)(30,36,41)(43,75,71)(44,76,72)(45,77,67)(46,78,68)(47,73,69)(48,74,70)(49,60,62)(50,55,63)(51,56,64)(52,57,65)(53,58,66)(54,59,61)(79,104,114)(80,105,109)(81,106,110)(82,107,111)(83,108,112)(84,103,113)(85,97,94)(86,98,95)(87,99,96)(88,100,91)(89,101,92)(90,102,93)(121,130,133)(122,131,134)(123,132,135)(124,127,136)(125,128,137)(126,129,138), (1,42,14,35,11,27)(2,37,15,36,12,28)(3,38,16,31,7,29)(4,39,17,32,8,30)(5,40,18,33,9,25)(6,41,13,34,10,26)(19,136,115,131,144,126)(20,137,116,132,139,121)(21,138,117,127,140,122)(22,133,118,128,141,123)(23,134,119,129,142,124)(24,135,120,130,143,125)(43,55,67,52,73,61)(44,56,68,53,74,62)(45,57,69,54,75,63)(46,58,70,49,76,64)(47,59,71,50,77,65)(48,60,72,51,78,66)(79,100,110,90,108,95)(80,101,111,85,103,96)(81,102,112,86,104,91)(82,97,113,87,105,92)(83,98,114,88,106,93)(84,99,109,89,107,94), (1,71,35,65)(2,72,36,66)(3,67,31,61)(4,68,32,62)(5,69,33,63)(6,70,34,64)(7,73,38,55)(8,74,39,56)(9,75,40,57)(10,76,41,58)(11,77,42,59)(12,78,37,60)(13,46,26,49)(14,47,27,50)(15,48,28,51)(16,43,29,52)(17,44,30,53)(18,45,25,54)(19,110,131,95)(20,111,132,96)(21,112,127,91)(22,113,128,92)(23,114,129,93)(24,109,130,94)(79,126,90,115)(80,121,85,116)(81,122,86,117)(82,123,87,118)(83,124,88,119)(84,125,89,120)(97,141,105,133)(98,142,106,134)(99,143,107,135)(100,144,108,136)(101,139,103,137)(102,140,104,138) );
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)], [(1,98,4,101),(2,97,5,100),(3,102,6,99),(7,91,10,94),(8,96,11,93),(9,95,12,92),(13,89,16,86),(14,88,17,85),(15,87,18,90),(19,78,22,75),(20,77,23,74),(21,76,24,73),(25,79,28,82),(26,84,29,81),(27,83,30,80),(31,104,34,107),(32,103,35,106),(33,108,36,105),(37,113,40,110),(38,112,41,109),(39,111,42,114),(43,117,46,120),(44,116,47,119),(45,115,48,118),(49,125,52,122),(50,124,53,121),(51,123,54,126),(55,127,58,130),(56,132,59,129),(57,131,60,128),(61,138,64,135),(62,137,65,134),(63,136,66,133),(67,140,70,143),(68,139,71,142),(69,144,72,141)], [(1,9,16),(2,10,17),(3,11,18),(4,12,13),(5,7,14),(6,8,15),(19,142,117),(20,143,118),(21,144,119),(22,139,120),(23,140,115),(24,141,116),(25,31,42),(26,32,37),(27,33,38),(28,34,39),(29,35,40),(30,36,41),(43,75,71),(44,76,72),(45,77,67),(46,78,68),(47,73,69),(48,74,70),(49,60,62),(50,55,63),(51,56,64),(52,57,65),(53,58,66),(54,59,61),(79,104,114),(80,105,109),(81,106,110),(82,107,111),(83,108,112),(84,103,113),(85,97,94),(86,98,95),(87,99,96),(88,100,91),(89,101,92),(90,102,93),(121,130,133),(122,131,134),(123,132,135),(124,127,136),(125,128,137),(126,129,138)], [(1,42,14,35,11,27),(2,37,15,36,12,28),(3,38,16,31,7,29),(4,39,17,32,8,30),(5,40,18,33,9,25),(6,41,13,34,10,26),(19,136,115,131,144,126),(20,137,116,132,139,121),(21,138,117,127,140,122),(22,133,118,128,141,123),(23,134,119,129,142,124),(24,135,120,130,143,125),(43,55,67,52,73,61),(44,56,68,53,74,62),(45,57,69,54,75,63),(46,58,70,49,76,64),(47,59,71,50,77,65),(48,60,72,51,78,66),(79,100,110,90,108,95),(80,101,111,85,103,96),(81,102,112,86,104,91),(82,97,113,87,105,92),(83,98,114,88,106,93),(84,99,109,89,107,94)], [(1,71,35,65),(2,72,36,66),(3,67,31,61),(4,68,32,62),(5,69,33,63),(6,70,34,64),(7,73,38,55),(8,74,39,56),(9,75,40,57),(10,76,41,58),(11,77,42,59),(12,78,37,60),(13,46,26,49),(14,47,27,50),(15,48,28,51),(16,43,29,52),(17,44,30,53),(18,45,25,54),(19,110,131,95),(20,111,132,96),(21,112,127,91),(22,113,128,92),(23,114,129,93),(24,109,130,94),(79,126,90,115),(80,121,85,116),(81,122,86,117),(82,123,87,118),(83,124,88,119),(84,125,89,120),(97,141,105,133),(98,142,106,134),(99,143,107,135),(100,144,108,136),(101,139,103,137),(102,140,104,138)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3E | 3F | 3G | 3H | 3I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6O | 6P | ··· | 6AA | 12A | ··· | 12P | 12Q | 12R | 12S | 12T |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 27 | 27 | 27 | 27 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 18 | 18 | 18 | 18 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | S3 | Dic3 | Dic3 | D6 | C4×S3 | S32 | S3×Dic3 | C6.D6 |
| kernel | Dic3×C3⋊Dic3 | Dic3×C3×C6 | C6×C3⋊Dic3 | C2×C33⋊5C4 | C32×Dic3 | C3×C3⋊Dic3 | C33⋊5C4 | C6×Dic3 | C2×C3⋊Dic3 | C3×Dic3 | C3⋊Dic3 | C62 | C3×C6 | C2×C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 8 | 2 | 5 | 20 | 4 | 8 | 4 |
Matrix representation of Dic3×C3⋊Dic3 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 11 | 3 | 0 | 0 | 0 | 0 |
| 12 | 1 | 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 |
| 2 | 10 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 8 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[11,12,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,1,0,0,0,0,10,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,8,0,0,0,0,0,5,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
Dic3×C3⋊Dic3 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_3\rtimes {\rm Dic}_3 % in TeX
G:=Group("Dic3xC3:Dic3"); // GroupNames label
G:=SmallGroup(432,448);
// by ID
G=gap.SmallGroup(432,448);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,64,571,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=c^3=d^6=1,b^2=a^3,e^2=d^3,b*a*b^-1=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