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## G = C6×D39order 468 = 22·32·13

### Direct product of C6 and D39

Aliases: C6×D39, C785C6, C782S3, C397D6, C325D26, C6⋊(C3×D13), C135(S3×C6), C263(C3×S3), C397(C2×C6), (C3×C78)⋊2C2, (C3×C6)⋊1D13, C32(C6×D13), (C3×C39)⋊7C22, SmallGroup(468,52)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C39 — C6×D39
 Chief series C1 — C13 — C39 — C3×C39 — C3×D39 — C6×D39
 Lower central C39 — C6×D39
 Upper central C1 — C6

Generators and relations for C6×D39
G = < a,b,c | a6=b39=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C6×D39
On 156 points
Generators in S156
(1 41 27 67 14 54)(2 42 28 68 15 55)(3 43 29 69 16 56)(4 44 30 70 17 57)(5 45 31 71 18 58)(6 46 32 72 19 59)(7 47 33 73 20 60)(8 48 34 74 21 61)(9 49 35 75 22 62)(10 50 36 76 23 63)(11 51 37 77 24 64)(12 52 38 78 25 65)(13 53 39 40 26 66)(79 136 92 149 105 123)(80 137 93 150 106 124)(81 138 94 151 107 125)(82 139 95 152 108 126)(83 140 96 153 109 127)(84 141 97 154 110 128)(85 142 98 155 111 129)(86 143 99 156 112 130)(87 144 100 118 113 131)(88 145 101 119 114 132)(89 146 102 120 115 133)(90 147 103 121 116 134)(91 148 104 122 117 135)
(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)
(1 148)(2 147)(3 146)(4 145)(5 144)(6 143)(7 142)(8 141)(9 140)(10 139)(11 138)(12 137)(13 136)(14 135)(15 134)(16 133)(17 132)(18 131)(19 130)(20 129)(21 128)(22 127)(23 126)(24 125)(25 124)(26 123)(27 122)(28 121)(29 120)(30 119)(31 118)(32 156)(33 155)(34 154)(35 153)(36 152)(37 151)(38 150)(39 149)(40 105)(41 104)(42 103)(43 102)(44 101)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)(61 84)(62 83)(63 82)(64 81)(65 80)(66 79)(67 117)(68 116)(69 115)(70 114)(71 113)(72 112)(73 111)(74 110)(75 109)(76 108)(77 107)(78 106)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 6A 6B 6C 6D 6E 6F 6G 6H 6I 13A ··· 13F 26A ··· 26F 39A ··· 39AV 78A ··· 78AV order 1 2 2 2 3 3 3 3 3 6 6 6 6 6 6 6 6 6 13 ··· 13 26 ··· 26 39 ··· 39 78 ··· 78 size 1 1 39 39 1 1 2 2 2 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 C2 C3 C6 C6 S3 D6 C3×S3 D13 S3×C6 D26 C3×D13 D39 C6×D13 D78 C3×D39 C6×D39 kernel C6×D39 C3×D39 C3×C78 D78 D39 C78 C78 C39 C26 C3×C6 C13 C32 C6 C6 C3 C3 C2 C1 # reps 1 2 1 2 4 2 1 1 2 6 2 6 12 12 12 12 24 24

Matrix representation of C6×D39 in GL3(𝔽79) generated by

 24 0 0 0 23 0 0 0 23
,
 1 0 0 0 72 23 0 0 45
,
 1 0 0 0 67 53 0 45 12
G:=sub<GL(3,GF(79))| [24,0,0,0,23,0,0,0,23],[1,0,0,0,72,0,0,23,45],[1,0,0,0,67,45,0,53,12] >;

C6×D39 in GAP, Magma, Sage, TeX

C_6\times D_{39}
% in TeX

G:=Group("C6xD39");
// GroupNames label

G:=SmallGroup(468,52);
// by ID

G=gap.SmallGroup(468,52);
# by ID

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

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

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