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## G = Dic3×C39order 468 = 22·32·13

### Direct product of C39 and Dic3

Aliases: Dic3×C39, C3⋊C156, C6.C78, C78.8S3, C322C52, C3911C12, C78.13C6, C2.(S3×C39), (C3×C39)⋊10C4, C26.4(C3×S3), C6.4(S3×C13), (C3×C6).1C26, (C3×C78).4C2, SmallGroup(468,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C39
 Chief series C1 — C3 — C6 — C78 — C3×C78 — Dic3×C39
 Lower central C3 — Dic3×C39
 Upper central C1 — C78

Generators and relations for Dic3×C39
G = < a,b,c | a39=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of Dic3×C39
On 156 points
Generators in S156
(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)(118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156)
(1 43 14 56 27 69)(2 44 15 57 28 70)(3 45 16 58 29 71)(4 46 17 59 30 72)(5 47 18 60 31 73)(6 48 19 61 32 74)(7 49 20 62 33 75)(8 50 21 63 34 76)(9 51 22 64 35 77)(10 52 23 65 36 78)(11 53 24 66 37 40)(12 54 25 67 38 41)(13 55 26 68 39 42)(79 135 105 122 92 148)(80 136 106 123 93 149)(81 137 107 124 94 150)(82 138 108 125 95 151)(83 139 109 126 96 152)(84 140 110 127 97 153)(85 141 111 128 98 154)(86 142 112 129 99 155)(87 143 113 130 100 156)(88 144 114 131 101 118)(89 145 115 132 102 119)(90 146 116 133 103 120)(91 147 117 134 104 121)
(1 146 56 103)(2 147 57 104)(3 148 58 105)(4 149 59 106)(5 150 60 107)(6 151 61 108)(7 152 62 109)(8 153 63 110)(9 154 64 111)(10 155 65 112)(11 156 66 113)(12 118 67 114)(13 119 68 115)(14 120 69 116)(15 121 70 117)(16 122 71 79)(17 123 72 80)(18 124 73 81)(19 125 74 82)(20 126 75 83)(21 127 76 84)(22 128 77 85)(23 129 78 86)(24 130 40 87)(25 131 41 88)(26 132 42 89)(27 133 43 90)(28 134 44 91)(29 135 45 92)(30 136 46 93)(31 137 47 94)(32 138 48 95)(33 139 49 96)(34 140 50 97)(35 141 51 98)(36 142 52 99)(37 143 53 100)(38 144 54 101)(39 145 55 102)

G:=sub<Sym(156)| (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)(118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156), (1,43,14,56,27,69)(2,44,15,57,28,70)(3,45,16,58,29,71)(4,46,17,59,30,72)(5,47,18,60,31,73)(6,48,19,61,32,74)(7,49,20,62,33,75)(8,50,21,63,34,76)(9,51,22,64,35,77)(10,52,23,65,36,78)(11,53,24,66,37,40)(12,54,25,67,38,41)(13,55,26,68,39,42)(79,135,105,122,92,148)(80,136,106,123,93,149)(81,137,107,124,94,150)(82,138,108,125,95,151)(83,139,109,126,96,152)(84,140,110,127,97,153)(85,141,111,128,98,154)(86,142,112,129,99,155)(87,143,113,130,100,156)(88,144,114,131,101,118)(89,145,115,132,102,119)(90,146,116,133,103,120)(91,147,117,134,104,121), (1,146,56,103)(2,147,57,104)(3,148,58,105)(4,149,59,106)(5,150,60,107)(6,151,61,108)(7,152,62,109)(8,153,63,110)(9,154,64,111)(10,155,65,112)(11,156,66,113)(12,118,67,114)(13,119,68,115)(14,120,69,116)(15,121,70,117)(16,122,71,79)(17,123,72,80)(18,124,73,81)(19,125,74,82)(20,126,75,83)(21,127,76,84)(22,128,77,85)(23,129,78,86)(24,130,40,87)(25,131,41,88)(26,132,42,89)(27,133,43,90)(28,134,44,91)(29,135,45,92)(30,136,46,93)(31,137,47,94)(32,138,48,95)(33,139,49,96)(34,140,50,97)(35,141,51,98)(36,142,52,99)(37,143,53,100)(38,144,54,101)(39,145,55,102)>;

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)(118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156), (1,43,14,56,27,69)(2,44,15,57,28,70)(3,45,16,58,29,71)(4,46,17,59,30,72)(5,47,18,60,31,73)(6,48,19,61,32,74)(7,49,20,62,33,75)(8,50,21,63,34,76)(9,51,22,64,35,77)(10,52,23,65,36,78)(11,53,24,66,37,40)(12,54,25,67,38,41)(13,55,26,68,39,42)(79,135,105,122,92,148)(80,136,106,123,93,149)(81,137,107,124,94,150)(82,138,108,125,95,151)(83,139,109,126,96,152)(84,140,110,127,97,153)(85,141,111,128,98,154)(86,142,112,129,99,155)(87,143,113,130,100,156)(88,144,114,131,101,118)(89,145,115,132,102,119)(90,146,116,133,103,120)(91,147,117,134,104,121), (1,146,56,103)(2,147,57,104)(3,148,58,105)(4,149,59,106)(5,150,60,107)(6,151,61,108)(7,152,62,109)(8,153,63,110)(9,154,64,111)(10,155,65,112)(11,156,66,113)(12,118,67,114)(13,119,68,115)(14,120,69,116)(15,121,70,117)(16,122,71,79)(17,123,72,80)(18,124,73,81)(19,125,74,82)(20,126,75,83)(21,127,76,84)(22,128,77,85)(23,129,78,86)(24,130,40,87)(25,131,41,88)(26,132,42,89)(27,133,43,90)(28,134,44,91)(29,135,45,92)(30,136,46,93)(31,137,47,94)(32,138,48,95)(33,139,49,96)(34,140,50,97)(35,141,51,98)(36,142,52,99)(37,143,53,100)(38,144,54,101)(39,145,55,102) );

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),(118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)], [(1,43,14,56,27,69),(2,44,15,57,28,70),(3,45,16,58,29,71),(4,46,17,59,30,72),(5,47,18,60,31,73),(6,48,19,61,32,74),(7,49,20,62,33,75),(8,50,21,63,34,76),(9,51,22,64,35,77),(10,52,23,65,36,78),(11,53,24,66,37,40),(12,54,25,67,38,41),(13,55,26,68,39,42),(79,135,105,122,92,148),(80,136,106,123,93,149),(81,137,107,124,94,150),(82,138,108,125,95,151),(83,139,109,126,96,152),(84,140,110,127,97,153),(85,141,111,128,98,154),(86,142,112,129,99,155),(87,143,113,130,100,156),(88,144,114,131,101,118),(89,145,115,132,102,119),(90,146,116,133,103,120),(91,147,117,134,104,121)], [(1,146,56,103),(2,147,57,104),(3,148,58,105),(4,149,59,106),(5,150,60,107),(6,151,61,108),(7,152,62,109),(8,153,63,110),(9,154,64,111),(10,155,65,112),(11,156,66,113),(12,118,67,114),(13,119,68,115),(14,120,69,116),(15,121,70,117),(16,122,71,79),(17,123,72,80),(18,124,73,81),(19,125,74,82),(20,126,75,83),(21,127,76,84),(22,128,77,85),(23,129,78,86),(24,130,40,87),(25,131,41,88),(26,132,42,89),(27,133,43,90),(28,134,44,91),(29,135,45,92),(30,136,46,93),(31,137,47,94),(32,138,48,95),(33,139,49,96),(34,140,50,97),(35,141,51,98),(36,142,52,99),(37,143,53,100),(38,144,54,101),(39,145,55,102)])

234 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 12A 12B 12C 12D 13A ··· 13L 26A ··· 26L 39A ··· 39X 39Y ··· 39BH 52A ··· 52X 78A ··· 78X 78Y ··· 78BH 156A ··· 156AV order 1 2 3 3 3 3 3 4 4 6 6 6 6 6 12 12 12 12 13 ··· 13 26 ··· 26 39 ··· 39 39 ··· 39 52 ··· 52 78 ··· 78 78 ··· 78 156 ··· 156 size 1 1 1 1 2 2 2 3 3 1 1 2 2 2 3 3 3 3 1 ··· 1 1 ··· 1 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

234 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C12 C13 C26 C39 C52 C78 C156 S3 Dic3 C3×S3 C3×Dic3 S3×C13 Dic3×C13 S3×C39 Dic3×C39 kernel Dic3×C39 C3×C78 Dic3×C13 C3×C39 C78 C39 C3×Dic3 C3×C6 Dic3 C32 C6 C3 C78 C39 C26 C13 C6 C3 C2 C1 # reps 1 1 2 2 2 4 12 12 24 24 24 48 1 1 2 2 12 12 24 24

Matrix representation of Dic3×C39 in GL3(𝔽157) generated by

 12 0 0 0 17 0 0 0 17
,
 156 0 0 0 144 0 0 101 12
,
 129 0 0 0 113 143 0 127 44
G:=sub<GL(3,GF(157))| [12,0,0,0,17,0,0,0,17],[156,0,0,0,144,101,0,0,12],[129,0,0,0,113,127,0,143,44] >;

Dic3×C39 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{39}
% in TeX

G:=Group("Dic3xC39");
// GroupNames label

G:=SmallGroup(468,24);
// by ID

G=gap.SmallGroup(468,24);
# by ID

G:=PCGroup([5,-2,-3,-13,-2,-3,390,7804]);
// Polycyclic

G:=Group<a,b,c|a^39=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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