direct product, metabelian, supersoluble, monomial
Aliases: C3×C12⋊Dic3, C62.142D6, (C3×C12)⋊7C12, C33⋊19(C4⋊C4), (C6×C12).28C6, (C6×C12).44S3, (C32×C12)⋊7C4, (C3×C12)⋊8Dic3, C12⋊1(C3×Dic3), (C3×C6).64D12, C6.22(C3×D12), C12⋊3(C3⋊Dic3), C62.66(C2×C6), (C3×C6).29Dic6, C6.27(C6×Dic3), C6.13(C3×Dic6), (C32×C6).60D4, (C32×C6).13Q8, C6.25(C12⋊S3), C32⋊11(C4⋊Dic3), (C3×C62).48C22, C6.13(C32⋊4Q8), C4⋊(C3×C3⋊Dic3), (C3×C6×C12).10C2, C3⋊2(C3×C4⋊Dic3), C32⋊12(C3×C4⋊C4), (C2×C6).69(S3×C6), (C3×C6).52(C3×D4), C22.5(C6×C3⋊S3), C2.4(C6×C3⋊Dic3), (C3×C6).15(C3×Q8), C2.1(C3×C12⋊S3), (C2×C12).20(C3×S3), (C3×C6).64(C2×C12), (C2×C3⋊Dic3).9C6, (C6×C3⋊Dic3).5C2, C6.23(C2×C3⋊Dic3), (C2×C12).26(C3⋊S3), C2.2(C3×C32⋊4Q8), (C32×C6).70(C2×C4), (C3×C6).69(C2×Dic3), (C2×C4).3(C3×C3⋊S3), (C2×C6).63(C2×C3⋊S3), SmallGroup(432,489)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C12⋊Dic3
G = < a,b,c,d | a3=b12=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 532 in 220 conjugacy classes, 102 normal (26 characteristic)
C1, C2, C3, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, C32, C32, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C4⋊C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C62, C62, C62, C4⋊Dic3, C3×C4⋊C4, C32×C6, C6×Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C6×C12, C3×C3⋊Dic3, C32×C12, C3×C62, C3×C4⋊Dic3, C12⋊Dic3, C6×C3⋊Dic3, C3×C6×C12, C3×C12⋊Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C4⋊C4, C3×S3, C3⋊S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C4⋊Dic3, C3×C4⋊C4, C3×C3⋊S3, C3×Dic6, C3×D12, C6×Dic3, C32⋊4Q8, C12⋊S3, C2×C3⋊Dic3, C3×C3⋊Dic3, C6×C3⋊S3, C3×C4⋊Dic3, C12⋊Dic3, C3×C32⋊4Q8, C3×C12⋊S3, C6×C3⋊Dic3, C3×C12⋊Dic3
►On 144 points
Generators in S144
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 93 89)(86 94 90)(87 95 91)(88 96 92)(97 101 105)(98 102 106)(99 103 107)(100 104 108)(109 117 113)(110 118 114)(111 119 115)(112 120 116)(121 125 129)(122 126 130)(123 127 131)(124 128 132)(133 137 141)(134 138 142)(135 139 143)(136 140 144)
(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 70 41 18 109 95)(2 71 42 19 110 96)(3 72 43 20 111 85)(4 61 44 21 112 86)(5 62 45 22 113 87)(6 63 46 23 114 88)(7 64 47 24 115 89)(8 65 48 13 116 90)(9 66 37 14 117 91)(10 67 38 15 118 92)(11 68 39 16 119 93)(12 69 40 17 120 94)(25 129 54 136 77 98)(26 130 55 137 78 99)(27 131 56 138 79 100)(28 132 57 139 80 101)(29 121 58 140 81 102)(30 122 59 141 82 103)(31 123 60 142 83 104)(32 124 49 143 84 105)(33 125 50 144 73 106)(34 126 51 133 74 107)(35 127 52 134 75 108)(36 128 53 135 76 97)
(1 80 18 132)(2 79 19 131)(3 78 20 130)(4 77 21 129)(5 76 22 128)(6 75 23 127)(7 74 24 126)(8 73 13 125)(9 84 14 124)(10 83 15 123)(11 82 16 122)(12 81 17 121)(25 61 136 112)(26 72 137 111)(27 71 138 110)(28 70 139 109)(29 69 140 120)(30 68 141 119)(31 67 142 118)(32 66 143 117)(33 65 144 116)(34 64 133 115)(35 63 134 114)(36 62 135 113)(37 49 91 105)(38 60 92 104)(39 59 93 103)(40 58 94 102)(41 57 95 101)(42 56 96 100)(43 55 85 99)(44 54 86 98)(45 53 87 97)(46 52 88 108)(47 51 89 107)(48 50 90 106)
G:=sub<Sym(144)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,93,89)(86,94,90)(87,95,91)(88,96,92)(97,101,105)(98,102,106)(99,103,107)(100,104,108)(109,117,113)(110,118,114)(111,119,115)(112,120,116)(121,125,129)(122,126,130)(123,127,131)(124,128,132)(133,137,141)(134,138,142)(135,139,143)(136,140,144), (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), (1,70,41,18,109,95)(2,71,42,19,110,96)(3,72,43,20,111,85)(4,61,44,21,112,86)(5,62,45,22,113,87)(6,63,46,23,114,88)(7,64,47,24,115,89)(8,65,48,13,116,90)(9,66,37,14,117,91)(10,67,38,15,118,92)(11,68,39,16,119,93)(12,69,40,17,120,94)(25,129,54,136,77,98)(26,130,55,137,78,99)(27,131,56,138,79,100)(28,132,57,139,80,101)(29,121,58,140,81,102)(30,122,59,141,82,103)(31,123,60,142,83,104)(32,124,49,143,84,105)(33,125,50,144,73,106)(34,126,51,133,74,107)(35,127,52,134,75,108)(36,128,53,135,76,97), (1,80,18,132)(2,79,19,131)(3,78,20,130)(4,77,21,129)(5,76,22,128)(6,75,23,127)(7,74,24,126)(8,73,13,125)(9,84,14,124)(10,83,15,123)(11,82,16,122)(12,81,17,121)(25,61,136,112)(26,72,137,111)(27,71,138,110)(28,70,139,109)(29,69,140,120)(30,68,141,119)(31,67,142,118)(32,66,143,117)(33,65,144,116)(34,64,133,115)(35,63,134,114)(36,62,135,113)(37,49,91,105)(38,60,92,104)(39,59,93,103)(40,58,94,102)(41,57,95,101)(42,56,96,100)(43,55,85,99)(44,54,86,98)(45,53,87,97)(46,52,88,108)(47,51,89,107)(48,50,90,106)>;
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,93,89)(86,94,90)(87,95,91)(88,96,92)(97,101,105)(98,102,106)(99,103,107)(100,104,108)(109,117,113)(110,118,114)(111,119,115)(112,120,116)(121,125,129)(122,126,130)(123,127,131)(124,128,132)(133,137,141)(134,138,142)(135,139,143)(136,140,144), (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), (1,70,41,18,109,95)(2,71,42,19,110,96)(3,72,43,20,111,85)(4,61,44,21,112,86)(5,62,45,22,113,87)(6,63,46,23,114,88)(7,64,47,24,115,89)(8,65,48,13,116,90)(9,66,37,14,117,91)(10,67,38,15,118,92)(11,68,39,16,119,93)(12,69,40,17,120,94)(25,129,54,136,77,98)(26,130,55,137,78,99)(27,131,56,138,79,100)(28,132,57,139,80,101)(29,121,58,140,81,102)(30,122,59,141,82,103)(31,123,60,142,83,104)(32,124,49,143,84,105)(33,125,50,144,73,106)(34,126,51,133,74,107)(35,127,52,134,75,108)(36,128,53,135,76,97), (1,80,18,132)(2,79,19,131)(3,78,20,130)(4,77,21,129)(5,76,22,128)(6,75,23,127)(7,74,24,126)(8,73,13,125)(9,84,14,124)(10,83,15,123)(11,82,16,122)(12,81,17,121)(25,61,136,112)(26,72,137,111)(27,71,138,110)(28,70,139,109)(29,69,140,120)(30,68,141,119)(31,67,142,118)(32,66,143,117)(33,65,144,116)(34,64,133,115)(35,63,134,114)(36,62,135,113)(37,49,91,105)(38,60,92,104)(39,59,93,103)(40,58,94,102)(41,57,95,101)(42,56,96,100)(43,55,85,99)(44,54,86,98)(45,53,87,97)(46,52,88,108)(47,51,89,107)(48,50,90,106) );
G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,45,41),(38,46,42),(39,47,43),(40,48,44),(49,53,57),(50,54,58),(51,55,59),(52,56,60),(61,69,65),(62,70,66),(63,71,67),(64,72,68),(73,77,81),(74,78,82),(75,79,83),(76,80,84),(85,93,89),(86,94,90),(87,95,91),(88,96,92),(97,101,105),(98,102,106),(99,103,107),(100,104,108),(109,117,113),(110,118,114),(111,119,115),(112,120,116),(121,125,129),(122,126,130),(123,127,131),(124,128,132),(133,137,141),(134,138,142),(135,139,143),(136,140,144)], [(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)], [(1,70,41,18,109,95),(2,71,42,19,110,96),(3,72,43,20,111,85),(4,61,44,21,112,86),(5,62,45,22,113,87),(6,63,46,23,114,88),(7,64,47,24,115,89),(8,65,48,13,116,90),(9,66,37,14,117,91),(10,67,38,15,118,92),(11,68,39,16,119,93),(12,69,40,17,120,94),(25,129,54,136,77,98),(26,130,55,137,78,99),(27,131,56,138,79,100),(28,132,57,139,80,101),(29,121,58,140,81,102),(30,122,59,141,82,103),(31,123,60,142,83,104),(32,124,49,143,84,105),(33,125,50,144,73,106),(34,126,51,133,74,107),(35,127,52,134,75,108),(36,128,53,135,76,97)], [(1,80,18,132),(2,79,19,131),(3,78,20,130),(4,77,21,129),(5,76,22,128),(6,75,23,127),(7,74,24,126),(8,73,13,125),(9,84,14,124),(10,83,15,123),(11,82,16,122),(12,81,17,121),(25,61,136,112),(26,72,137,111),(27,71,138,110),(28,70,139,109),(29,69,140,120),(30,68,141,119),(31,67,142,118),(32,66,143,117),(33,65,144,116),(34,64,133,115),(35,63,134,114),(36,62,135,113),(37,49,91,105),(38,60,92,104),(39,59,93,103),(40,58,94,102),(41,57,95,101),(42,56,96,100),(43,55,85,99),(44,54,86,98),(45,53,87,97),(46,52,88,108),(47,51,89,107),(48,50,90,106)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6AP | 12A | ··· | 12AZ | 12BA | ··· | 12BH |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 18 | ··· | 18 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | D4 | Q8 | Dic3 | D6 | C3×S3 | Dic6 | D12 | C3×D4 | C3×Q8 | C3×Dic3 | S3×C6 | C3×Dic6 | C3×D12 |
| kernel | C3×C12⋊Dic3 | C6×C3⋊Dic3 | C3×C6×C12 | C12⋊Dic3 | C32×C12 | C2×C3⋊Dic3 | C6×C12 | C3×C12 | C6×C12 | C32×C6 | C32×C6 | C3×C12 | C62 | C2×C12 | C3×C6 | C3×C6 | C3×C6 | C3×C6 | C12 | C2×C6 | C6 | C6 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 4 | 1 | 1 | 8 | 4 | 8 | 8 | 8 | 2 | 2 | 16 | 8 | 16 | 16 |
Matrix representation of C3×C12⋊Dic3 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 3 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 6 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[3,0,0,0,0,9,0,0,0,0,11,0,0,0,0,6],[12,0,0,0,0,12,0,0,0,0,10,0,0,0,0,4],[0,12,0,0,1,0,0,0,0,0,0,12,0,0,1,0] >;
C3×C12⋊Dic3 in GAP, Magma, Sage, TeX
C_3\times C_{12}\rtimes {\rm Dic}_3 % in TeX
G:=Group("C3xC12:Dic3"); // GroupNames label
G:=SmallGroup(432,489);
// by ID
G=gap.SmallGroup(432,489);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,176,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations