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## G = C3×Dic39order 468 = 22·32·13

### Direct product of C3 and Dic39

Aliases: C3×Dic39, C399C12, C78.9C6, C78.4S3, C6.4D39, C394Dic3, C322Dic13, (C3×C39)⋊8C4, C6.(C3×D13), C2.(C3×D39), C3⋊(C3×Dic13), C26.3(C3×S3), (C3×C78).2C2, (C3×C6).1D13, C135(C3×Dic3), SmallGroup(468,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C39 — C3×Dic39
 Chief series C1 — C13 — C39 — C78 — C3×C78 — C3×Dic39
 Lower central C39 — C3×Dic39
 Upper central C1 — C6

Generators and relations for C3×Dic39
G = < a,b,c | a3=b78=1, c2=b39, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×Dic39
On 156 points
Generators in S156
(1 27 53)(2 28 54)(3 29 55)(4 30 56)(5 31 57)(6 32 58)(7 33 59)(8 34 60)(9 35 61)(10 36 62)(11 37 63)(12 38 64)(13 39 65)(14 40 66)(15 41 67)(16 42 68)(17 43 69)(18 44 70)(19 45 71)(20 46 72)(21 47 73)(22 48 74)(23 49 75)(24 50 76)(25 51 77)(26 52 78)(79 131 105)(80 132 106)(81 133 107)(82 134 108)(83 135 109)(84 136 110)(85 137 111)(86 138 112)(87 139 113)(88 140 114)(89 141 115)(90 142 116)(91 143 117)(92 144 118)(93 145 119)(94 146 120)(95 147 121)(96 148 122)(97 149 123)(98 150 124)(99 151 125)(100 152 126)(101 153 127)(102 154 128)(103 155 129)(104 156 130)
(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)
(1 144 40 105)(2 143 41 104)(3 142 42 103)(4 141 43 102)(5 140 44 101)(6 139 45 100)(7 138 46 99)(8 137 47 98)(9 136 48 97)(10 135 49 96)(11 134 50 95)(12 133 51 94)(13 132 52 93)(14 131 53 92)(15 130 54 91)(16 129 55 90)(17 128 56 89)(18 127 57 88)(19 126 58 87)(20 125 59 86)(21 124 60 85)(22 123 61 84)(23 122 62 83)(24 121 63 82)(25 120 64 81)(26 119 65 80)(27 118 66 79)(28 117 67 156)(29 116 68 155)(30 115 69 154)(31 114 70 153)(32 113 71 152)(33 112 72 151)(34 111 73 150)(35 110 74 149)(36 109 75 148)(37 108 76 147)(38 107 77 146)(39 106 78 145)

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

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

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

126 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 12A 12B 12C 12D 13A ··· 13F 26A ··· 26F 39A ··· 39AV 78A ··· 78AV 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 78 ··· 78 size 1 1 1 1 2 2 2 39 39 1 1 2 2 2 39 39 39 39 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + - + - + - image C1 C2 C3 C4 C6 C12 S3 Dic3 C3×S3 D13 C3×Dic3 Dic13 C3×D13 D39 C3×Dic13 Dic39 C3×D39 C3×Dic39 kernel C3×Dic39 C3×C78 Dic39 C3×C39 C78 C39 C78 C39 C26 C3×C6 C13 C32 C6 C6 C3 C3 C2 C1 # reps 1 1 2 2 2 4 1 1 2 6 2 6 12 12 12 12 24 24

Matrix representation of C3×Dic39 in GL3(𝔽157) generated by

 12 0 0 0 12 0 0 0 12
,
 156 0 0 0 71 0 0 0 115
,
 129 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(157))| [12,0,0,0,12,0,0,0,12],[156,0,0,0,71,0,0,0,115],[129,0,0,0,0,1,0,1,0] >;

C3×Dic39 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{39}
% in TeX

G:=Group("C3xDic39");
// GroupNames label

G:=SmallGroup(468,25);
// by ID

G=gap.SmallGroup(468,25);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-13,30,483,10804]);
// Polycyclic

G:=Group<a,b,c|a^3=b^78=1,c^2=b^39,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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