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## G = C3×C12⋊Dic3order 432 = 24·33

### Direct product of C3 and C12⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×C12⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C3×C62 — C6×C3⋊Dic3 — C3×C12⋊Dic3
 Lower central C32 — C3×C6 — C3×C12⋊Dic3
 Upper central C1 — C2×C6 — C2×C12

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:

Smallest permutation representation of 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

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