direct product, metabelian, supersoluble, monomial, A-group
Aliases: Dic3×C3×C12, C3⋊C122, C33⋊7C42, C62.162D6, (C3×C12)⋊9C12, C12⋊2(C3×C12), C6.3(C6×C12), C6.41(S3×C12), (C6×C12).60S3, (C6×C12).43C6, (C32×C12)⋊8C4, C32⋊5(C4×C12), C62.59(C2×C6), (C2×C6).14C62, C6.37(C6×Dic3), (C6×Dic3).18C6, (C3×C62).41C22, C2.2(S3×C3×C12), (C3×C6×C12).14C2, C22.3(S3×C3×C6), C2.2(Dic3×C3×C6), (C3×C6).95(C4×S3), (C2×C6).89(S3×C6), (C2×C12).52(C3×S3), (C2×C12).12(C3×C6), (C3×C6).58(C2×C12), (C2×C4).6(S3×C32), (Dic3×C3×C6).13C2, (C32×C6).51(C2×C4), (C2×Dic3).4(C3×C6), (C3×C6).78(C2×Dic3), SmallGroup(432,471)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C3×C12 |
Generators and relations for Dic3×C3×C12
G = < a,b,c,d | a3=b12=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 392 in 244 conjugacy classes, 138 normal (18 characteristic)
C1, C2, 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, C42, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C2×C12, C33, C3×Dic3, C3×C12, C3×C12, C62, C62, C62, C4×Dic3, C4×C12, C32×C6, C32×C6, C6×Dic3, C6×C12, C6×C12, C6×C12, C32×Dic3, C32×C12, C3×C62, Dic3×C12, C122, Dic3×C3×C6, C3×C6×C12, Dic3×C3×C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C32, Dic3, C12, D6, C2×C6, C42, C3×S3, C3×C6, C4×S3, C2×Dic3, C2×C12, C3×Dic3, C3×C12, S3×C6, C62, C4×Dic3, C4×C12, S3×C32, S3×C12, C6×Dic3, C6×C12, C32×Dic3, S3×C3×C6, Dic3×C12, C122, S3×C3×C12, Dic3×C3×C6, Dic3×C3×C12
►On 144 points
Generators in S144
(1 48 128)(2 37 129)(3 38 130)(4 39 131)(5 40 132)(6 41 121)(7 42 122)(8 43 123)(9 44 124)(10 45 125)(11 46 126)(12 47 127)(13 85 26)(14 86 27)(15 87 28)(16 88 29)(17 89 30)(18 90 31)(19 91 32)(20 92 33)(21 93 34)(22 94 35)(23 95 36)(24 96 25)(49 104 135)(50 105 136)(51 106 137)(52 107 138)(53 108 139)(54 97 140)(55 98 141)(56 99 142)(57 100 143)(58 101 144)(59 102 133)(60 103 134)(61 82 113)(62 83 114)(63 84 115)(64 73 116)(65 74 117)(66 75 118)(67 76 119)(68 77 120)(69 78 109)(70 79 110)(71 80 111)(72 81 112)
(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 32 5 36 9 28)(2 33 6 25 10 29)(3 34 7 26 11 30)(4 35 8 27 12 31)(13 46 17 38 21 42)(14 47 18 39 22 43)(15 48 19 40 23 44)(16 37 20 41 24 45)(49 77 57 73 53 81)(50 78 58 74 54 82)(51 79 59 75 55 83)(52 80 60 76 56 84)(61 136 69 144 65 140)(62 137 70 133 66 141)(63 138 71 134 67 142)(64 139 72 135 68 143)(85 126 89 130 93 122)(86 127 90 131 94 123)(87 128 91 132 95 124)(88 129 92 121 96 125)(97 113 105 109 101 117)(98 114 106 110 102 118)(99 115 107 111 103 119)(100 116 108 112 104 120)
(1 59 36 83)(2 60 25 84)(3 49 26 73)(4 50 27 74)(5 51 28 75)(6 52 29 76)(7 53 30 77)(8 54 31 78)(9 55 32 79)(10 56 33 80)(11 57 34 81)(12 58 35 82)(13 116 38 104)(14 117 39 105)(15 118 40 106)(16 119 41 107)(17 120 42 108)(18 109 43 97)(19 110 44 98)(20 111 45 99)(21 112 46 100)(22 113 47 101)(23 114 48 102)(24 115 37 103)(61 127 144 94)(62 128 133 95)(63 129 134 96)(64 130 135 85)(65 131 136 86)(66 132 137 87)(67 121 138 88)(68 122 139 89)(69 123 140 90)(70 124 141 91)(71 125 142 92)(72 126 143 93)
G:=sub<Sym(144)| (1,48,128)(2,37,129)(3,38,130)(4,39,131)(5,40,132)(6,41,121)(7,42,122)(8,43,123)(9,44,124)(10,45,125)(11,46,126)(12,47,127)(13,85,26)(14,86,27)(15,87,28)(16,88,29)(17,89,30)(18,90,31)(19,91,32)(20,92,33)(21,93,34)(22,94,35)(23,95,36)(24,96,25)(49,104,135)(50,105,136)(51,106,137)(52,107,138)(53,108,139)(54,97,140)(55,98,141)(56,99,142)(57,100,143)(58,101,144)(59,102,133)(60,103,134)(61,82,113)(62,83,114)(63,84,115)(64,73,116)(65,74,117)(66,75,118)(67,76,119)(68,77,120)(69,78,109)(70,79,110)(71,80,111)(72,81,112), (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), (1,32,5,36,9,28)(2,33,6,25,10,29)(3,34,7,26,11,30)(4,35,8,27,12,31)(13,46,17,38,21,42)(14,47,18,39,22,43)(15,48,19,40,23,44)(16,37,20,41,24,45)(49,77,57,73,53,81)(50,78,58,74,54,82)(51,79,59,75,55,83)(52,80,60,76,56,84)(61,136,69,144,65,140)(62,137,70,133,66,141)(63,138,71,134,67,142)(64,139,72,135,68,143)(85,126,89,130,93,122)(86,127,90,131,94,123)(87,128,91,132,95,124)(88,129,92,121,96,125)(97,113,105,109,101,117)(98,114,106,110,102,118)(99,115,107,111,103,119)(100,116,108,112,104,120), (1,59,36,83)(2,60,25,84)(3,49,26,73)(4,50,27,74)(5,51,28,75)(6,52,29,76)(7,53,30,77)(8,54,31,78)(9,55,32,79)(10,56,33,80)(11,57,34,81)(12,58,35,82)(13,116,38,104)(14,117,39,105)(15,118,40,106)(16,119,41,107)(17,120,42,108)(18,109,43,97)(19,110,44,98)(20,111,45,99)(21,112,46,100)(22,113,47,101)(23,114,48,102)(24,115,37,103)(61,127,144,94)(62,128,133,95)(63,129,134,96)(64,130,135,85)(65,131,136,86)(66,132,137,87)(67,121,138,88)(68,122,139,89)(69,123,140,90)(70,124,141,91)(71,125,142,92)(72,126,143,93)>;
G:=Group( (1,48,128)(2,37,129)(3,38,130)(4,39,131)(5,40,132)(6,41,121)(7,42,122)(8,43,123)(9,44,124)(10,45,125)(11,46,126)(12,47,127)(13,85,26)(14,86,27)(15,87,28)(16,88,29)(17,89,30)(18,90,31)(19,91,32)(20,92,33)(21,93,34)(22,94,35)(23,95,36)(24,96,25)(49,104,135)(50,105,136)(51,106,137)(52,107,138)(53,108,139)(54,97,140)(55,98,141)(56,99,142)(57,100,143)(58,101,144)(59,102,133)(60,103,134)(61,82,113)(62,83,114)(63,84,115)(64,73,116)(65,74,117)(66,75,118)(67,76,119)(68,77,120)(69,78,109)(70,79,110)(71,80,111)(72,81,112), (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), (1,32,5,36,9,28)(2,33,6,25,10,29)(3,34,7,26,11,30)(4,35,8,27,12,31)(13,46,17,38,21,42)(14,47,18,39,22,43)(15,48,19,40,23,44)(16,37,20,41,24,45)(49,77,57,73,53,81)(50,78,58,74,54,82)(51,79,59,75,55,83)(52,80,60,76,56,84)(61,136,69,144,65,140)(62,137,70,133,66,141)(63,138,71,134,67,142)(64,139,72,135,68,143)(85,126,89,130,93,122)(86,127,90,131,94,123)(87,128,91,132,95,124)(88,129,92,121,96,125)(97,113,105,109,101,117)(98,114,106,110,102,118)(99,115,107,111,103,119)(100,116,108,112,104,120), (1,59,36,83)(2,60,25,84)(3,49,26,73)(4,50,27,74)(5,51,28,75)(6,52,29,76)(7,53,30,77)(8,54,31,78)(9,55,32,79)(10,56,33,80)(11,57,34,81)(12,58,35,82)(13,116,38,104)(14,117,39,105)(15,118,40,106)(16,119,41,107)(17,120,42,108)(18,109,43,97)(19,110,44,98)(20,111,45,99)(21,112,46,100)(22,113,47,101)(23,114,48,102)(24,115,37,103)(61,127,144,94)(62,128,133,95)(63,129,134,96)(64,130,135,85)(65,131,136,86)(66,132,137,87)(67,121,138,88)(68,122,139,89)(69,123,140,90)(70,124,141,91)(71,125,142,92)(72,126,143,93) );
G=PermutationGroup([[(1,48,128),(2,37,129),(3,38,130),(4,39,131),(5,40,132),(6,41,121),(7,42,122),(8,43,123),(9,44,124),(10,45,125),(11,46,126),(12,47,127),(13,85,26),(14,86,27),(15,87,28),(16,88,29),(17,89,30),(18,90,31),(19,91,32),(20,92,33),(21,93,34),(22,94,35),(23,95,36),(24,96,25),(49,104,135),(50,105,136),(51,106,137),(52,107,138),(53,108,139),(54,97,140),(55,98,141),(56,99,142),(57,100,143),(58,101,144),(59,102,133),(60,103,134),(61,82,113),(62,83,114),(63,84,115),(64,73,116),(65,74,117),(66,75,118),(67,76,119),(68,77,120),(69,78,109),(70,79,110),(71,80,111),(72,81,112)], [(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)], [(1,32,5,36,9,28),(2,33,6,25,10,29),(3,34,7,26,11,30),(4,35,8,27,12,31),(13,46,17,38,21,42),(14,47,18,39,22,43),(15,48,19,40,23,44),(16,37,20,41,24,45),(49,77,57,73,53,81),(50,78,58,74,54,82),(51,79,59,75,55,83),(52,80,60,76,56,84),(61,136,69,144,65,140),(62,137,70,133,66,141),(63,138,71,134,67,142),(64,139,72,135,68,143),(85,126,89,130,93,122),(86,127,90,131,94,123),(87,128,91,132,95,124),(88,129,92,121,96,125),(97,113,105,109,101,117),(98,114,106,110,102,118),(99,115,107,111,103,119),(100,116,108,112,104,120)], [(1,59,36,83),(2,60,25,84),(3,49,26,73),(4,50,27,74),(5,51,28,75),(6,52,29,76),(7,53,30,77),(8,54,31,78),(9,55,32,79),(10,56,33,80),(11,57,34,81),(12,58,35,82),(13,116,38,104),(14,117,39,105),(15,118,40,106),(16,119,41,107),(17,120,42,108),(18,109,43,97),(19,110,44,98),(20,111,45,99),(21,112,46,100),(22,113,47,101),(23,114,48,102),(24,115,37,103),(61,127,144,94),(62,128,133,95),(63,129,134,96),(64,130,135,85),(65,131,136,86),(66,132,137,87),(67,121,138,88),(68,122,139,89),(69,123,140,90),(70,124,141,91),(71,125,142,92),(72,126,143,93)]])
216 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6X | 6Y | ··· | 6AY | 12A | ··· | 12AF | 12AG | ··· | 12BP | 12BQ | ··· | 12EB |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
216 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | S3 | Dic3 | D6 | C3×S3 | C4×S3 | C3×Dic3 | S3×C6 | S3×C12 |
| kernel | Dic3×C3×C12 | Dic3×C3×C6 | C3×C6×C12 | Dic3×C12 | C32×Dic3 | C32×C12 | C6×Dic3 | C6×C12 | C3×Dic3 | C3×C12 | C6×C12 | C3×C12 | C62 | C2×C12 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 2 | 1 | 8 | 8 | 4 | 16 | 8 | 64 | 32 | 1 | 2 | 1 | 8 | 4 | 16 | 8 | 32 |
Matrix representation of Dic3×C3×C12 ►in GL3(𝔽13) generated by
| 3 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 7 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 7 |
| 12 | 0 | 0 |
| 0 | 3 | 5 |
| 0 | 0 | 9 |
| 5 | 0 | 0 |
| 0 | 2 | 2 |
| 0 | 5 | 11 |
G:=sub<GL(3,GF(13))| [3,0,0,0,1,0,0,0,1],[7,0,0,0,7,0,0,0,7],[12,0,0,0,3,0,0,5,9],[5,0,0,0,2,5,0,2,11] >;
Dic3×C3×C12 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_3\times C_{12} % in TeX
G:=Group("Dic3xC3xC12"); // GroupNames label
G:=SmallGroup(432,471);
// by ID
G=gap.SmallGroup(432,471);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,252,512,14118]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations