direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×Dic13, C39⋊4C4, C13⋊5C12, C26.3C6, C78.2C2, C6.2D13, C2.(C3×D13), SmallGroup(156,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C3×Dic13 |
Generators and relations for C3×Dic13
G = < a,b,c | a3=b26=1, c2=b13, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic13
►Regular action on 156 points
Generators in S156
(1 69 31)(2 70 32)(3 71 33)(4 72 34)(5 73 35)(6 74 36)(7 75 37)(8 76 38)(9 77 39)(10 78 40)(11 53 41)(12 54 42)(13 55 43)(14 56 44)(15 57 45)(16 58 46)(17 59 47)(18 60 48)(19 61 49)(20 62 50)(21 63 51)(22 64 52)(23 65 27)(24 66 28)(25 67 29)(26 68 30)(79 131 118)(80 132 119)(81 133 120)(82 134 121)(83 135 122)(84 136 123)(85 137 124)(86 138 125)(87 139 126)(88 140 127)(89 141 128)(90 142 129)(91 143 130)(92 144 105)(93 145 106)(94 146 107)(95 147 108)(96 148 109)(97 149 110)(98 150 111)(99 151 112)(100 152 113)(101 153 114)(102 154 115)(103 155 116)(104 156 117)
(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 92 14 79)(2 91 15 104)(3 90 16 103)(4 89 17 102)(5 88 18 101)(6 87 19 100)(7 86 20 99)(8 85 21 98)(9 84 22 97)(10 83 23 96)(11 82 24 95)(12 81 25 94)(13 80 26 93)(27 109 40 122)(28 108 41 121)(29 107 42 120)(30 106 43 119)(31 105 44 118)(32 130 45 117)(33 129 46 116)(34 128 47 115)(35 127 48 114)(36 126 49 113)(37 125 50 112)(38 124 51 111)(39 123 52 110)(53 134 66 147)(54 133 67 146)(55 132 68 145)(56 131 69 144)(57 156 70 143)(58 155 71 142)(59 154 72 141)(60 153 73 140)(61 152 74 139)(62 151 75 138)(63 150 76 137)(64 149 77 136)(65 148 78 135)
G:=sub<Sym(156)| (1,69,31)(2,70,32)(3,71,33)(4,72,34)(5,73,35)(6,74,36)(7,75,37)(8,76,38)(9,77,39)(10,78,40)(11,53,41)(12,54,42)(13,55,43)(14,56,44)(15,57,45)(16,58,46)(17,59,47)(18,60,48)(19,61,49)(20,62,50)(21,63,51)(22,64,52)(23,65,27)(24,66,28)(25,67,29)(26,68,30)(79,131,118)(80,132,119)(81,133,120)(82,134,121)(83,135,122)(84,136,123)(85,137,124)(86,138,125)(87,139,126)(88,140,127)(89,141,128)(90,142,129)(91,143,130)(92,144,105)(93,145,106)(94,146,107)(95,147,108)(96,148,109)(97,149,110)(98,150,111)(99,151,112)(100,152,113)(101,153,114)(102,154,115)(103,155,116)(104,156,117), (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), (1,92,14,79)(2,91,15,104)(3,90,16,103)(4,89,17,102)(5,88,18,101)(6,87,19,100)(7,86,20,99)(8,85,21,98)(9,84,22,97)(10,83,23,96)(11,82,24,95)(12,81,25,94)(13,80,26,93)(27,109,40,122)(28,108,41,121)(29,107,42,120)(30,106,43,119)(31,105,44,118)(32,130,45,117)(33,129,46,116)(34,128,47,115)(35,127,48,114)(36,126,49,113)(37,125,50,112)(38,124,51,111)(39,123,52,110)(53,134,66,147)(54,133,67,146)(55,132,68,145)(56,131,69,144)(57,156,70,143)(58,155,71,142)(59,154,72,141)(60,153,73,140)(61,152,74,139)(62,151,75,138)(63,150,76,137)(64,149,77,136)(65,148,78,135)>;
G:=Group( (1,69,31)(2,70,32)(3,71,33)(4,72,34)(5,73,35)(6,74,36)(7,75,37)(8,76,38)(9,77,39)(10,78,40)(11,53,41)(12,54,42)(13,55,43)(14,56,44)(15,57,45)(16,58,46)(17,59,47)(18,60,48)(19,61,49)(20,62,50)(21,63,51)(22,64,52)(23,65,27)(24,66,28)(25,67,29)(26,68,30)(79,131,118)(80,132,119)(81,133,120)(82,134,121)(83,135,122)(84,136,123)(85,137,124)(86,138,125)(87,139,126)(88,140,127)(89,141,128)(90,142,129)(91,143,130)(92,144,105)(93,145,106)(94,146,107)(95,147,108)(96,148,109)(97,149,110)(98,150,111)(99,151,112)(100,152,113)(101,153,114)(102,154,115)(103,155,116)(104,156,117), (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), (1,92,14,79)(2,91,15,104)(3,90,16,103)(4,89,17,102)(5,88,18,101)(6,87,19,100)(7,86,20,99)(8,85,21,98)(9,84,22,97)(10,83,23,96)(11,82,24,95)(12,81,25,94)(13,80,26,93)(27,109,40,122)(28,108,41,121)(29,107,42,120)(30,106,43,119)(31,105,44,118)(32,130,45,117)(33,129,46,116)(34,128,47,115)(35,127,48,114)(36,126,49,113)(37,125,50,112)(38,124,51,111)(39,123,52,110)(53,134,66,147)(54,133,67,146)(55,132,68,145)(56,131,69,144)(57,156,70,143)(58,155,71,142)(59,154,72,141)(60,153,73,140)(61,152,74,139)(62,151,75,138)(63,150,76,137)(64,149,77,136)(65,148,78,135) );
G=PermutationGroup([[(1,69,31),(2,70,32),(3,71,33),(4,72,34),(5,73,35),(6,74,36),(7,75,37),(8,76,38),(9,77,39),(10,78,40),(11,53,41),(12,54,42),(13,55,43),(14,56,44),(15,57,45),(16,58,46),(17,59,47),(18,60,48),(19,61,49),(20,62,50),(21,63,51),(22,64,52),(23,65,27),(24,66,28),(25,67,29),(26,68,30),(79,131,118),(80,132,119),(81,133,120),(82,134,121),(83,135,122),(84,136,123),(85,137,124),(86,138,125),(87,139,126),(88,140,127),(89,141,128),(90,142,129),(91,143,130),(92,144,105),(93,145,106),(94,146,107),(95,147,108),(96,148,109),(97,149,110),(98,150,111),(99,151,112),(100,152,113),(101,153,114),(102,154,115),(103,155,116),(104,156,117)], [(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)], [(1,92,14,79),(2,91,15,104),(3,90,16,103),(4,89,17,102),(5,88,18,101),(6,87,19,100),(7,86,20,99),(8,85,21,98),(9,84,22,97),(10,83,23,96),(11,82,24,95),(12,81,25,94),(13,80,26,93),(27,109,40,122),(28,108,41,121),(29,107,42,120),(30,106,43,119),(31,105,44,118),(32,130,45,117),(33,129,46,116),(34,128,47,115),(35,127,48,114),(36,126,49,113),(37,125,50,112),(38,124,51,111),(39,123,52,110),(53,134,66,147),(54,133,67,146),(55,132,68,145),(56,131,69,144),(57,156,70,143),(58,155,71,142),(59,154,72,141),(60,153,73,140),(61,152,74,139),(62,151,75,138),(63,150,76,137),(64,149,77,136),(65,148,78,135)]])
C3×Dic13 is a maximal subgroup of
C39⋊C8 D78.C2 C13⋊D12 C39⋊Q8 C12×D13 C13⋊2C36
48 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39L | 78A | ··· | 78L |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 1 | 1 | 13 | 13 | 1 | 1 | 13 | 13 | 13 | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | D13 | Dic13 | C3×D13 | C3×Dic13 |
| kernel | C3×Dic13 | C78 | Dic13 | C39 | C26 | C13 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 12 |
Matrix representation of C3×Dic13 ►in GL2(𝔽157) generated by
| 12 | 0 |
| 0 | 12 |
| 0 | 156 |
| 1 | 91 |
| 103 | 94 |
| 141 | 54 |
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
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