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## G = Dic3×C3×C12order 432 = 24·33

### Direct product of C3×C12 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C3×C12
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C3×C62 — Dic3×C3×C6 — Dic3×C3×C12
 Lower central C3 — Dic3×C3×C12
 Upper central C1 — C6×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

Smallest permutation representation of 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

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