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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, C3, C3, C3, C4, C22, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, C12, C2×C6, C2×C6, C22×C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C22×C6, C33, C3×Dic3, C3×C12, C62, C62, C22×Dic3, C22×C12, C32×C6, C32×C6, C6×Dic3, C6×C12, C2×C62, C2×C62, C2×C62, C32×Dic3, C3×C62, Dic3×C2×C6, C2×C6×C12, Dic3×C3×C6, C63, Dic3×C62
Quotients:

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