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G = C3×Dic13order 156 = 22·3·13

Direct product of C3 and Dic13

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

Aliases: C3×Dic13, C394C4, C135C12, C26.3C6, C78.2C2, C6.2D13, C2.(C3×D13), SmallGroup(156,4)

Series: Derived Chief Lower central Upper central

C1C13 — C3×Dic13
C1C13C26C78 — C3×Dic13
C13 — C3×Dic13
C1C6

Generators and relations for C3×Dic13
 G = < a,b,c | a3=b26=1, c2=b13, ab=ba, ac=ca, cbc-1=b-1 >

13C4
13C12

Smallest permutation representation of C3×Dic13
Regular action on 156 points
Generators in S156
(1 57 35)(2 58 36)(3 59 37)(4 60 38)(5 61 39)(6 62 40)(7 63 41)(8 64 42)(9 65 43)(10 66 44)(11 67 45)(12 68 46)(13 69 47)(14 70 48)(15 71 49)(16 72 50)(17 73 51)(18 74 52)(19 75 27)(20 76 28)(21 77 29)(22 78 30)(23 53 31)(24 54 32)(25 55 33)(26 56 34)(79 144 118)(80 145 119)(81 146 120)(82 147 121)(83 148 122)(84 149 123)(85 150 124)(86 151 125)(87 152 126)(88 153 127)(89 154 128)(90 155 129)(91 156 130)(92 131 105)(93 132 106)(94 133 107)(95 134 108)(96 135 109)(97 136 110)(98 137 111)(99 138 112)(100 139 113)(101 140 114)(102 141 115)(103 142 116)(104 143 117)
(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 79 14 92)(2 104 15 91)(3 103 16 90)(4 102 17 89)(5 101 18 88)(6 100 19 87)(7 99 20 86)(8 98 21 85)(9 97 22 84)(10 96 23 83)(11 95 24 82)(12 94 25 81)(13 93 26 80)(27 126 40 113)(28 125 41 112)(29 124 42 111)(30 123 43 110)(31 122 44 109)(32 121 45 108)(33 120 46 107)(34 119 47 106)(35 118 48 105)(36 117 49 130)(37 116 50 129)(38 115 51 128)(39 114 52 127)(53 148 66 135)(54 147 67 134)(55 146 68 133)(56 145 69 132)(57 144 70 131)(58 143 71 156)(59 142 72 155)(60 141 73 154)(61 140 74 153)(62 139 75 152)(63 138 76 151)(64 137 77 150)(65 136 78 149)

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

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

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

C3×Dic13 is a maximal subgroup of   C39⋊C8  D78.C2  C13⋊D12  C39⋊Q8  C12×D13  C132C36

48 conjugacy classes

class 1  2 3A3B4A4B6A6B12A12B12C12D13A···13F26A···26F39A···39L78A···78L
order123344661212121213···1326···2639···3978···78
size1111131311131313132···22···22···22···2

48 irreducible representations

dim1111112222
type+++-
imageC1C2C3C4C6C12D13Dic13C3×D13C3×Dic13
kernelC3×Dic13C78Dic13C39C26C13C6C3C2C1
# reps112224661212

Matrix representation of C3×Dic13 in GL2(𝔽157) generated by

120
012
,
0156
191
,
10394
14154
G:=sub<GL(2,GF(157))| [12,0,0,12],[0,1,156,91],[103,141,94,54] >;

C3×Dic13 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{13}
% in TeX

G:=Group("C3xDic13");
// GroupNames label

G:=SmallGroup(156,4);
// by ID

G=gap.SmallGroup(156,4);
# by ID

G:=PCGroup([4,-2,-3,-2,-13,24,2307]);
// Polycyclic

G:=Group<a,b,c|a^3=b^26=1,c^2=b^13,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×Dic13 in TeX

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