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G = Dic3×C13order 156 = 22·3·13

Direct product of C13 and Dic3

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic3×C13, C3⋊C52, C395C4, C6.C26, C78.3C2, C26.2S3, C2.(S3×C13), SmallGroup(156,3)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3×C13
C1C3C6C78 — Dic3×C13
C3 — Dic3×C13
C1C26

Generators and relations for Dic3×C13
 G = < a,b,c | a13=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

3C4
3C52

Smallest permutation representation of Dic3×C13
Regular action 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 72 102 155 29 42)(2 73 103 156 30 43)(3 74 104 144 31 44)(4 75 92 145 32 45)(5 76 93 146 33 46)(6 77 94 147 34 47)(7 78 95 148 35 48)(8 66 96 149 36 49)(9 67 97 150 37 50)(10 68 98 151 38 51)(11 69 99 152 39 52)(12 70 100 153 27 40)(13 71 101 154 28 41)(14 80 113 138 57 128)(15 81 114 139 58 129)(16 82 115 140 59 130)(17 83 116 141 60 118)(18 84 117 142 61 119)(19 85 105 143 62 120)(20 86 106 131 63 121)(21 87 107 132 64 122)(22 88 108 133 65 123)(23 89 109 134 53 124)(24 90 110 135 54 125)(25 91 111 136 55 126)(26 79 112 137 56 127)
(1 14 155 138)(2 15 156 139)(3 16 144 140)(4 17 145 141)(5 18 146 142)(6 19 147 143)(7 20 148 131)(8 21 149 132)(9 22 150 133)(10 23 151 134)(11 24 152 135)(12 25 153 136)(13 26 154 137)(27 111 70 126)(28 112 71 127)(29 113 72 128)(30 114 73 129)(31 115 74 130)(32 116 75 118)(33 117 76 119)(34 105 77 120)(35 106 78 121)(36 107 66 122)(37 108 67 123)(38 109 68 124)(39 110 69 125)(40 91 100 55)(41 79 101 56)(42 80 102 57)(43 81 103 58)(44 82 104 59)(45 83 92 60)(46 84 93 61)(47 85 94 62)(48 86 95 63)(49 87 96 64)(50 88 97 65)(51 89 98 53)(52 90 99 54)

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

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

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

Dic3×C13 is a maximal subgroup of   D78.C2  C3⋊D52  C39⋊Q8  S3×C52

78 conjugacy classes

class 1  2  3 4A4B 6 13A···13L26A···26L39A···39L52A···52X78A···78L
order12344613···1326···2639···3952···5278···78
size1123321···11···12···23···32···2

78 irreducible representations

dim1111112222
type+++-
imageC1C2C4C13C26C52S3Dic3S3×C13Dic3×C13
kernelDic3×C13C78C39Dic3C6C3C26C13C2C1
# reps112121224111212

Matrix representation of Dic3×C13 in GL2(𝔽157) generated by

1010
0101
,
1156
10
,
149122
1148
G:=sub<GL(2,GF(157))| [101,0,0,101],[1,1,156,0],[149,114,122,8] >;

Dic3×C13 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{13}
% in TeX

G:=Group("Dic3xC13");
// GroupNames label

G:=SmallGroup(156,3);
// by ID

G=gap.SmallGroup(156,3);
# by ID

G:=PCGroup([4,-2,-13,-2,-3,104,1667]);
// Polycyclic

G:=Group<a,b,c|a^13=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Dic3×C13 in TeX

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