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## G = C3×C6×Dic6order 432 = 24·33

### Direct product of C3×C6 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×C6×Dic6
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — C32×Dic3 — Dic3×C3×C6 — C3×C6×Dic6
 Lower central C3 — C6 — C3×C6×Dic6
 Upper central C1 — C62 — C6×C12

Generators and relations for C3×C6×Dic6
G = < a,b,c,d | a3=b6=c12=1, d2=c6, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 488 in 292 conjugacy classes, 162 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C32, C32, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C2×Q8, C3×C6, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, C62, C62, C62, C2×Dic6, C6×Q8, C32×C6, C32×C6, C3×Dic6, C6×Dic3, C6×C12, C6×C12, C6×C12, Q8×C32, C32×Dic3, C32×C12, C3×C62, C6×Dic6, Q8×C3×C6, C32×Dic6, Dic3×C3×C6, C3×C6×C12, C3×C6×Dic6
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, C32, D6, C2×C6, C2×Q8, C3×S3, C3×C6, Dic6, C3×Q8, C22×S3, C22×C6, S3×C6, C62, C2×Dic6, C6×Q8, S3×C32, C3×Dic6, Q8×C32, S3×C2×C6, C2×C62, S3×C3×C6, C6×Dic6, Q8×C3×C6, C32×Dic6, S3×C62, C3×C6×Dic6

Smallest permutation representation of C3×C6×Dic6
On 144 points
Generators in S144
(1 131 16)(2 132 17)(3 121 18)(4 122 19)(5 123 20)(6 124 21)(7 125 22)(8 126 23)(9 127 24)(10 128 13)(11 129 14)(12 130 15)(25 42 96)(26 43 85)(27 44 86)(28 45 87)(29 46 88)(30 47 89)(31 48 90)(32 37 91)(33 38 92)(34 39 93)(35 40 94)(36 41 95)(49 72 107)(50 61 108)(51 62 97)(52 63 98)(53 64 99)(54 65 100)(55 66 101)(56 67 102)(57 68 103)(58 69 104)(59 70 105)(60 71 106)(73 136 116)(74 137 117)(75 138 118)(76 139 119)(77 140 120)(78 141 109)(79 142 110)(80 143 111)(81 144 112)(82 133 113)(83 134 114)(84 135 115)
(1 99 9 107 5 103)(2 100 10 108 6 104)(3 101 11 97 7 105)(4 102 12 98 8 106)(13 61 21 69 17 65)(14 62 22 70 18 66)(15 63 23 71 19 67)(16 64 24 72 20 68)(25 117 29 109 33 113)(26 118 30 110 34 114)(27 119 31 111 35 115)(28 120 32 112 36 116)(37 81 41 73 45 77)(38 82 42 74 46 78)(39 83 43 75 47 79)(40 84 44 76 48 80)(49 123 57 131 53 127)(50 124 58 132 54 128)(51 125 59 121 55 129)(52 126 60 122 56 130)(85 138 89 142 93 134)(86 139 90 143 94 135)(87 140 91 144 95 136)(88 141 92 133 96 137)
(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 93 7 87)(2 92 8 86)(3 91 9 85)(4 90 10 96)(5 89 11 95)(6 88 12 94)(13 42 19 48)(14 41 20 47)(15 40 21 46)(16 39 22 45)(17 38 23 44)(18 37 24 43)(25 122 31 128)(26 121 32 127)(27 132 33 126)(28 131 34 125)(29 130 35 124)(30 129 36 123)(49 118 55 112)(50 117 56 111)(51 116 57 110)(52 115 58 109)(53 114 59 120)(54 113 60 119)(61 74 67 80)(62 73 68 79)(63 84 69 78)(64 83 70 77)(65 82 71 76)(66 81 72 75)(97 136 103 142)(98 135 104 141)(99 134 105 140)(100 133 106 139)(101 144 107 138)(102 143 108 137)

G:=sub<Sym(144)| (1,131,16)(2,132,17)(3,121,18)(4,122,19)(5,123,20)(6,124,21)(7,125,22)(8,126,23)(9,127,24)(10,128,13)(11,129,14)(12,130,15)(25,42,96)(26,43,85)(27,44,86)(28,45,87)(29,46,88)(30,47,89)(31,48,90)(32,37,91)(33,38,92)(34,39,93)(35,40,94)(36,41,95)(49,72,107)(50,61,108)(51,62,97)(52,63,98)(53,64,99)(54,65,100)(55,66,101)(56,67,102)(57,68,103)(58,69,104)(59,70,105)(60,71,106)(73,136,116)(74,137,117)(75,138,118)(76,139,119)(77,140,120)(78,141,109)(79,142,110)(80,143,111)(81,144,112)(82,133,113)(83,134,114)(84,135,115), (1,99,9,107,5,103)(2,100,10,108,6,104)(3,101,11,97,7,105)(4,102,12,98,8,106)(13,61,21,69,17,65)(14,62,22,70,18,66)(15,63,23,71,19,67)(16,64,24,72,20,68)(25,117,29,109,33,113)(26,118,30,110,34,114)(27,119,31,111,35,115)(28,120,32,112,36,116)(37,81,41,73,45,77)(38,82,42,74,46,78)(39,83,43,75,47,79)(40,84,44,76,48,80)(49,123,57,131,53,127)(50,124,58,132,54,128)(51,125,59,121,55,129)(52,126,60,122,56,130)(85,138,89,142,93,134)(86,139,90,143,94,135)(87,140,91,144,95,136)(88,141,92,133,96,137), (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), (1,93,7,87)(2,92,8,86)(3,91,9,85)(4,90,10,96)(5,89,11,95)(6,88,12,94)(13,42,19,48)(14,41,20,47)(15,40,21,46)(16,39,22,45)(17,38,23,44)(18,37,24,43)(25,122,31,128)(26,121,32,127)(27,132,33,126)(28,131,34,125)(29,130,35,124)(30,129,36,123)(49,118,55,112)(50,117,56,111)(51,116,57,110)(52,115,58,109)(53,114,59,120)(54,113,60,119)(61,74,67,80)(62,73,68,79)(63,84,69,78)(64,83,70,77)(65,82,71,76)(66,81,72,75)(97,136,103,142)(98,135,104,141)(99,134,105,140)(100,133,106,139)(101,144,107,138)(102,143,108,137)>;

G:=Group( (1,131,16)(2,132,17)(3,121,18)(4,122,19)(5,123,20)(6,124,21)(7,125,22)(8,126,23)(9,127,24)(10,128,13)(11,129,14)(12,130,15)(25,42,96)(26,43,85)(27,44,86)(28,45,87)(29,46,88)(30,47,89)(31,48,90)(32,37,91)(33,38,92)(34,39,93)(35,40,94)(36,41,95)(49,72,107)(50,61,108)(51,62,97)(52,63,98)(53,64,99)(54,65,100)(55,66,101)(56,67,102)(57,68,103)(58,69,104)(59,70,105)(60,71,106)(73,136,116)(74,137,117)(75,138,118)(76,139,119)(77,140,120)(78,141,109)(79,142,110)(80,143,111)(81,144,112)(82,133,113)(83,134,114)(84,135,115), (1,99,9,107,5,103)(2,100,10,108,6,104)(3,101,11,97,7,105)(4,102,12,98,8,106)(13,61,21,69,17,65)(14,62,22,70,18,66)(15,63,23,71,19,67)(16,64,24,72,20,68)(25,117,29,109,33,113)(26,118,30,110,34,114)(27,119,31,111,35,115)(28,120,32,112,36,116)(37,81,41,73,45,77)(38,82,42,74,46,78)(39,83,43,75,47,79)(40,84,44,76,48,80)(49,123,57,131,53,127)(50,124,58,132,54,128)(51,125,59,121,55,129)(52,126,60,122,56,130)(85,138,89,142,93,134)(86,139,90,143,94,135)(87,140,91,144,95,136)(88,141,92,133,96,137), (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), (1,93,7,87)(2,92,8,86)(3,91,9,85)(4,90,10,96)(5,89,11,95)(6,88,12,94)(13,42,19,48)(14,41,20,47)(15,40,21,46)(16,39,22,45)(17,38,23,44)(18,37,24,43)(25,122,31,128)(26,121,32,127)(27,132,33,126)(28,131,34,125)(29,130,35,124)(30,129,36,123)(49,118,55,112)(50,117,56,111)(51,116,57,110)(52,115,58,109)(53,114,59,120)(54,113,60,119)(61,74,67,80)(62,73,68,79)(63,84,69,78)(64,83,70,77)(65,82,71,76)(66,81,72,75)(97,136,103,142)(98,135,104,141)(99,134,105,140)(100,133,106,139)(101,144,107,138)(102,143,108,137) );

G=PermutationGroup([[(1,131,16),(2,132,17),(3,121,18),(4,122,19),(5,123,20),(6,124,21),(7,125,22),(8,126,23),(9,127,24),(10,128,13),(11,129,14),(12,130,15),(25,42,96),(26,43,85),(27,44,86),(28,45,87),(29,46,88),(30,47,89),(31,48,90),(32,37,91),(33,38,92),(34,39,93),(35,40,94),(36,41,95),(49,72,107),(50,61,108),(51,62,97),(52,63,98),(53,64,99),(54,65,100),(55,66,101),(56,67,102),(57,68,103),(58,69,104),(59,70,105),(60,71,106),(73,136,116),(74,137,117),(75,138,118),(76,139,119),(77,140,120),(78,141,109),(79,142,110),(80,143,111),(81,144,112),(82,133,113),(83,134,114),(84,135,115)], [(1,99,9,107,5,103),(2,100,10,108,6,104),(3,101,11,97,7,105),(4,102,12,98,8,106),(13,61,21,69,17,65),(14,62,22,70,18,66),(15,63,23,71,19,67),(16,64,24,72,20,68),(25,117,29,109,33,113),(26,118,30,110,34,114),(27,119,31,111,35,115),(28,120,32,112,36,116),(37,81,41,73,45,77),(38,82,42,74,46,78),(39,83,43,75,47,79),(40,84,44,76,48,80),(49,123,57,131,53,127),(50,124,58,132,54,128),(51,125,59,121,55,129),(52,126,60,122,56,130),(85,138,89,142,93,134),(86,139,90,143,94,135),(87,140,91,144,95,136),(88,141,92,133,96,137)], [(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)], [(1,93,7,87),(2,92,8,86),(3,91,9,85),(4,90,10,96),(5,89,11,95),(6,88,12,94),(13,42,19,48),(14,41,20,47),(15,40,21,46),(16,39,22,45),(17,38,23,44),(18,37,24,43),(25,122,31,128),(26,121,32,127),(27,132,33,126),(28,131,34,125),(29,130,35,124),(30,129,36,123),(49,118,55,112),(50,117,56,111),(51,116,57,110),(52,115,58,109),(53,114,59,120),(54,113,60,119),(61,74,67,80),(62,73,68,79),(63,84,69,78),(64,83,70,77),(65,82,71,76),(66,81,72,75),(97,136,103,142),(98,135,104,141),(99,134,105,140),(100,133,106,139),(101,144,107,138),(102,143,108,137)]])

162 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3Q 4A 4B 4C 4D 4E 4F 6A ··· 6X 6Y ··· 6AY 12A ··· 12AZ 12BA ··· 12CF 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 6 6 6 6 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6

162 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + - image C1 C2 C2 C2 C3 C6 C6 C6 S3 Q8 D6 D6 C3×S3 Dic6 C3×Q8 S3×C6 S3×C6 C3×Dic6 kernel C3×C6×Dic6 C32×Dic6 Dic3×C3×C6 C3×C6×C12 C6×Dic6 C3×Dic6 C6×Dic3 C6×C12 C6×C12 C32×C6 C3×C12 C62 C2×C12 C3×C6 C3×C6 C12 C2×C6 C6 # reps 1 4 2 1 8 32 16 8 1 2 2 1 8 4 16 16 8 32

Matrix representation of C3×C6×Dic6 in GL3(𝔽13) generated by

 3 0 0 0 1 0 0 0 1
,
 4 0 0 0 4 0 0 0 4
,
 12 0 0 0 7 0 0 0 2
,
 1 0 0 0 0 1 0 12 0
G:=sub<GL(3,GF(13))| [3,0,0,0,1,0,0,0,1],[4,0,0,0,4,0,0,0,4],[12,0,0,0,7,0,0,0,2],[1,0,0,0,0,12,0,1,0] >;

C3×C6×Dic6 in GAP, Magma, Sage, TeX

C_3\times C_6\times {\rm Dic}_6
% in TeX

G:=Group("C3xC6xDic6");
// GroupNames label

G:=SmallGroup(432,700);
// by ID

G=gap.SmallGroup(432,700);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,504,1598,394,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^12=1,d^2=c^6,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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