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## G = Dic3×C62order 432 = 24·33

### Direct product of C62 and Dic3

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic3×C62
G = < a,b,c,d | a6=b6=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 712 in 452 conjugacy classes, 258 normal (14 characteristic)
C1, C2, C2 [×6], C3, C3 [×4], C3 [×4], C4 [×4], C22 [×7], C6, C6 [×34], C6 [×28], C2×C4 [×6], C23, C32, C32 [×4], C32 [×4], Dic3 [×4], C12 [×16], C2×C6 [×35], C2×C6 [×28], C22×C4, C3×C6, C3×C6 [×34], C3×C6 [×28], C2×Dic3 [×6], C2×C12 [×24], C22×C6, C22×C6 [×4], C22×C6 [×4], C33, C3×Dic3 [×16], C3×C12 [×4], C62 [×35], C62 [×28], C22×Dic3, C22×C12 [×4], C32×C6, C32×C6 [×6], C6×Dic3 [×24], C6×C12 [×6], C2×C62, C2×C62 [×4], C2×C62 [×4], C32×Dic3 [×4], C3×C62 [×7], Dic3×C2×C6 [×4], C2×C6×C12, Dic3×C3×C6 [×6], C63, Dic3×C62
Quotients: C1, C2 [×7], C3 [×4], C4 [×4], C22 [×7], S3, C6 [×28], C2×C4 [×6], C23, C32, Dic3 [×4], C12 [×16], D6 [×3], C2×C6 [×28], C22×C4, C3×S3 [×4], C3×C6 [×7], C2×Dic3 [×6], C2×C12 [×24], C22×S3, C22×C6 [×4], C3×Dic3 [×16], C3×C12 [×4], S3×C6 [×12], C62 [×7], C22×Dic3, C22×C12 [×4], S3×C32, C6×Dic3 [×24], C6×C12 [×6], S3×C2×C6 [×4], C2×C62, C32×Dic3 [×4], S3×C3×C6 [×3], Dic3×C2×C6 [×4], C2×C6×C12, Dic3×C3×C6 [×6], S3×C62, Dic3×C62

Smallest permutation representation of Dic3×C62
On 144 points
Generators in S144
(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)
(1 115 114 86 69 75)(2 116 109 87 70 76)(3 117 110 88 71 77)(4 118 111 89 72 78)(5 119 112 90 67 73)(6 120 113 85 68 74)(7 42 144 46 49 14)(8 37 139 47 50 15)(9 38 140 48 51 16)(10 39 141 43 52 17)(11 40 142 44 53 18)(12 41 143 45 54 13)(19 66 59 31 30 84)(20 61 60 32 25 79)(21 62 55 33 26 80)(22 63 56 34 27 81)(23 64 57 35 28 82)(24 65 58 36 29 83)(91 100 137 130 108 121)(92 101 138 131 103 122)(93 102 133 132 104 123)(94 97 134 127 105 124)(95 98 135 128 106 125)(96 99 136 129 107 126)
(1 28 114 23 69 57)(2 29 109 24 70 58)(3 30 110 19 71 59)(4 25 111 20 72 60)(5 26 112 21 67 55)(6 27 113 22 68 56)(7 123 49 132 144 102)(8 124 50 127 139 97)(9 125 51 128 140 98)(10 126 52 129 141 99)(11 121 53 130 142 100)(12 122 54 131 143 101)(13 103 45 138 41 92)(14 104 46 133 42 93)(15 105 47 134 37 94)(16 106 48 135 38 95)(17 107 43 136 39 96)(18 108 44 137 40 91)(31 117 84 88 66 77)(32 118 79 89 61 78)(33 119 80 90 62 73)(34 120 81 85 63 74)(35 115 82 86 64 75)(36 116 83 87 65 76)
(1 49 23 102)(2 50 24 97)(3 51 19 98)(4 52 20 99)(5 53 21 100)(6 54 22 101)(7 57 132 114)(8 58 127 109)(9 59 128 110)(10 60 129 111)(11 55 130 112)(12 56 131 113)(13 63 138 120)(14 64 133 115)(15 65 134 116)(16 66 135 117)(17 61 136 118)(18 62 137 119)(25 126 72 141)(26 121 67 142)(27 122 68 143)(28 123 69 144)(29 124 70 139)(30 125 71 140)(31 106 88 38)(32 107 89 39)(33 108 90 40)(34 103 85 41)(35 104 86 42)(36 105 87 37)(43 79 96 78)(44 80 91 73)(45 81 92 74)(46 82 93 75)(47 83 94 76)(48 84 95 77)

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

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

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

216 conjugacy classes

 class 1 2A ··· 2G 3A ··· 3H 3I ··· 3Q 4A ··· 4H 6A ··· 6BD 6BE ··· 6DO 12A ··· 12BL order 1 2 ··· 2 3 ··· 3 3 ··· 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

216 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 Dic3 D6 C3×S3 C3×Dic3 S3×C6 kernel Dic3×C62 Dic3×C3×C6 C63 Dic3×C2×C6 C3×C62 C6×Dic3 C2×C62 C62 C2×C62 C62 C62 C22×C6 C2×C6 C2×C6 # reps 1 6 1 8 8 48 8 64 1 4 3 8 32 24

Matrix representation of Dic3×C62 in GL4(𝔽13) generated by

 1 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 12 0 0 0 0 1 0 0 0 0 9 0 0 0 0 3
,
 8 0 0 0 0 12 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[12,0,0,0,0,1,0,0,0,0,9,0,0,0,0,3],[8,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0] >;

Dic3×C62 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_6^2
% in TeX

G:=Group("Dic3xC6^2");
// GroupNames label

G:=SmallGroup(432,708);
// by ID

G=gap.SmallGroup(432,708);
# by ID

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

G:=Group<a,b,c,d|a^6=b^6=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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