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G = Dic3xC13order 156 = 22·3·13

Direct product of C13 and Dic3

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

Aliases: Dic3xC13, C3:C52, C39:5C4, C6.C26, C78.3C2, C26.2S3, C2.(S3xC13), SmallGroup(156,3)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3xC13
C1C3C6C78 — Dic3xC13
C3 — Dic3xC13
C1C26

Generators and relations for Dic3xC13
 G = < a,b,c | a13=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 16 in 12 conjugacy classes, 10 normal (all characteristic)
Quotients: C1, C2, C4, S3, Dic3, C13, C26, C52, S3xC13, Dic3xC13
3C4
3C52

Smallest permutation representation of Dic3xC13
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)
(1 101 105 133 83 123)(2 102 106 134 84 124)(3 103 107 135 85 125)(4 104 108 136 86 126)(5 92 109 137 87 127)(6 93 110 138 88 128)(7 94 111 139 89 129)(8 95 112 140 90 130)(9 96 113 141 91 118)(10 97 114 142 79 119)(11 98 115 143 80 120)(12 99 116 131 81 121)(13 100 117 132 82 122)(14 59 49 33 77 150)(15 60 50 34 78 151)(16 61 51 35 66 152)(17 62 52 36 67 153)(18 63 40 37 68 154)(19 64 41 38 69 155)(20 65 42 39 70 156)(21 53 43 27 71 144)(22 54 44 28 72 145)(23 55 45 29 73 146)(24 56 46 30 74 147)(25 57 47 31 75 148)(26 58 48 32 76 149)
(1 44 133 145)(2 45 134 146)(3 46 135 147)(4 47 136 148)(5 48 137 149)(6 49 138 150)(7 50 139 151)(8 51 140 152)(9 52 141 153)(10 40 142 154)(11 41 143 155)(12 42 131 156)(13 43 132 144)(14 128 33 110)(15 129 34 111)(16 130 35 112)(17 118 36 113)(18 119 37 114)(19 120 38 115)(20 121 39 116)(21 122 27 117)(22 123 28 105)(23 124 29 106)(24 125 30 107)(25 126 31 108)(26 127 32 109)(53 82 71 100)(54 83 72 101)(55 84 73 102)(56 85 74 103)(57 86 75 104)(58 87 76 92)(59 88 77 93)(60 89 78 94)(61 90 66 95)(62 91 67 96)(63 79 68 97)(64 80 69 98)(65 81 70 99)

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

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

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

Dic3xC13 is a maximal subgroup of   D78.C2  C3:D52  C39:Q8  S3xC52

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+++-
imageC1C2C4C13C26C52S3Dic3S3xC13Dic3xC13
kernelDic3xC13C78C39Dic3C6C3C26C13C2C1
# reps112121224111212

Matrix representation of Dic3xC13 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] >;

Dic3xC13 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 Dic3xC13 in TeX

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