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## G = C13×D12order 312 = 23·3·13

### Direct product of C13 and D12

Aliases: C13×D12, C396D4, C523S3, C1565C2, C121C26, D61C26, C26.15D6, C78.20C22, C4⋊(S3×C13), C31(D4×C13), (S3×C26)⋊4C2, C2.4(S3×C26), C6.3(C2×C26), SmallGroup(312,34)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C13×D12
 Chief series C1 — C3 — C6 — C78 — S3×C26 — C13×D12
 Lower central C3 — C6 — C13×D12
 Upper central C1 — C26 — C52

Generators and relations for C13×D12
G = < a,b,c | a13=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C13×D12
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 62 135 68 31 147 98 19 47 111 89 124)(2 63 136 69 32 148 99 20 48 112 90 125)(3 64 137 70 33 149 100 21 49 113 91 126)(4 65 138 71 34 150 101 22 50 114 79 127)(5 53 139 72 35 151 102 23 51 115 80 128)(6 54 140 73 36 152 103 24 52 116 81 129)(7 55 141 74 37 153 104 25 40 117 82 130)(8 56 142 75 38 154 92 26 41 105 83 118)(9 57 143 76 39 155 93 14 42 106 84 119)(10 58 131 77 27 156 94 15 43 107 85 120)(11 59 132 78 28 144 95 16 44 108 86 121)(12 60 133 66 29 145 96 17 45 109 87 122)(13 61 134 67 30 146 97 18 46 110 88 123)
(1 124)(2 125)(3 126)(4 127)(5 128)(6 129)(7 130)(8 118)(9 119)(10 120)(11 121)(12 122)(13 123)(14 39)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(40 74)(41 75)(42 76)(43 77)(44 78)(45 66)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)(52 73)(53 80)(54 81)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 88)(62 89)(63 90)(64 91)(65 79)(92 154)(93 155)(94 156)(95 144)(96 145)(97 146)(98 147)(99 148)(100 149)(101 150)(102 151)(103 152)(104 153)(105 142)(106 143)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 137)(114 138)(115 139)(116 140)(117 141)

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

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

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

117 conjugacy classes

 class 1 2A 2B 2C 3 4 6 12A 12B 13A ··· 13L 26A ··· 26L 26M ··· 26AJ 39A ··· 39L 52A ··· 52L 78A ··· 78L 156A ··· 156X order 1 2 2 2 3 4 6 12 12 13 ··· 13 26 ··· 26 26 ··· 26 39 ··· 39 52 ··· 52 78 ··· 78 156 ··· 156 size 1 1 6 6 2 2 2 2 2 1 ··· 1 1 ··· 1 6 ··· 6 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

117 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C13 C26 C26 S3 D4 D6 D12 S3×C13 D4×C13 S3×C26 C13×D12 kernel C13×D12 C156 S3×C26 D12 C12 D6 C52 C39 C26 C13 C4 C3 C2 C1 # reps 1 1 2 12 12 24 1 1 1 2 12 12 12 24

Matrix representation of C13×D12 in GL4(𝔽157) generated by

 39 0 0 0 0 39 0 0 0 0 1 0 0 0 0 1
,
 156 156 0 0 1 0 0 0 0 0 24 133 0 0 24 48
,
 156 156 0 0 0 1 0 0 0 0 24 133 0 0 109 133
G:=sub<GL(4,GF(157))| [39,0,0,0,0,39,0,0,0,0,1,0,0,0,0,1],[156,1,0,0,156,0,0,0,0,0,24,24,0,0,133,48],[156,0,0,0,156,1,0,0,0,0,24,109,0,0,133,133] >;

C13×D12 in GAP, Magma, Sage, TeX

C_{13}\times D_{12}
% in TeX

G:=Group("C13xD12");
// GroupNames label

G:=SmallGroup(312,34);
// by ID

G=gap.SmallGroup(312,34);
# by ID

G:=PCGroup([5,-2,-2,-13,-2,-3,541,266,5204]);
// Polycyclic

G:=Group<a,b,c|a^13=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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