direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C6×D39, C78⋊5C6, C78⋊2S3, C39⋊7D6, C32⋊5D26, C6⋊(C3×D13), C13⋊5(S3×C6), C26⋊3(C3×S3), C39⋊7(C2×C6), (C3×C78)⋊2C2, (C3×C6)⋊1D13, C3⋊2(C6×D13), (C3×C39)⋊7C22, SmallGroup(468,52)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — C6×D39 |
Generators and relations for C6×D39
G = < a,b,c | a6=b39=c2=1, ab=ba, ac=ca, cbc=b-1 >
39C2
39C2
2C3
39C22
2C6
13S3
13S3
39C6
39C6
3D13
3D13
2C39
13D6
39C2×C6
13C3×S3
13C3×S3
3D26
2C78
13S3×C6
Smallest permutation representation of C6×D39
►On 156 points
Generators in S156
(1 65 27 52 14 78)(2 66 28 53 15 40)(3 67 29 54 16 41)(4 68 30 55 17 42)(5 69 31 56 18 43)(6 70 32 57 19 44)(7 71 33 58 20 45)(8 72 34 59 21 46)(9 73 35 60 22 47)(10 74 36 61 23 48)(11 75 37 62 24 49)(12 76 38 63 25 50)(13 77 39 64 26 51)(79 142 92 155 105 129)(80 143 93 156 106 130)(81 144 94 118 107 131)(82 145 95 119 108 132)(83 146 96 120 109 133)(84 147 97 121 110 134)(85 148 98 122 111 135)(86 149 99 123 112 136)(87 150 100 124 113 137)(88 151 101 125 114 138)(89 152 102 126 115 139)(90 153 103 127 116 140)(91 154 104 128 117 141)
(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)
(1 134)(2 133)(3 132)(4 131)(5 130)(6 129)(7 128)(8 127)(9 126)(10 125)(11 124)(12 123)(13 122)(14 121)(15 120)(16 119)(17 118)(18 156)(19 155)(20 154)(21 153)(22 152)(23 151)(24 150)(25 149)(26 148)(27 147)(28 146)(29 145)(30 144)(31 143)(32 142)(33 141)(34 140)(35 139)(36 138)(37 137)(38 136)(39 135)(40 109)(41 108)(42 107)(43 106)(44 105)(45 104)(46 103)(47 102)(48 101)(49 100)(50 99)(51 98)(52 97)(53 96)(54 95)(55 94)(56 93)(57 92)(58 91)(59 90)(60 89)(61 88)(62 87)(63 86)(64 85)(65 84)(66 83)(67 82)(68 81)(69 80)(70 79)(71 117)(72 116)(73 115)(74 114)(75 113)(76 112)(77 111)(78 110)
G:=sub<Sym(156)| (1,65,27,52,14,78)(2,66,28,53,15,40)(3,67,29,54,16,41)(4,68,30,55,17,42)(5,69,31,56,18,43)(6,70,32,57,19,44)(7,71,33,58,20,45)(8,72,34,59,21,46)(9,73,35,60,22,47)(10,74,36,61,23,48)(11,75,37,62,24,49)(12,76,38,63,25,50)(13,77,39,64,26,51)(79,142,92,155,105,129)(80,143,93,156,106,130)(81,144,94,118,107,131)(82,145,95,119,108,132)(83,146,96,120,109,133)(84,147,97,121,110,134)(85,148,98,122,111,135)(86,149,99,123,112,136)(87,150,100,124,113,137)(88,151,101,125,114,138)(89,152,102,126,115,139)(90,153,103,127,116,140)(91,154,104,128,117,141), (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), (1,134)(2,133)(3,132)(4,131)(5,130)(6,129)(7,128)(8,127)(9,126)(10,125)(11,124)(12,123)(13,122)(14,121)(15,120)(16,119)(17,118)(18,156)(19,155)(20,154)(21,153)(22,152)(23,151)(24,150)(25,149)(26,148)(27,147)(28,146)(29,145)(30,144)(31,143)(32,142)(33,141)(34,140)(35,139)(36,138)(37,137)(38,136)(39,135)(40,109)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,117)(72,116)(73,115)(74,114)(75,113)(76,112)(77,111)(78,110)>;
G:=Group( (1,65,27,52,14,78)(2,66,28,53,15,40)(3,67,29,54,16,41)(4,68,30,55,17,42)(5,69,31,56,18,43)(6,70,32,57,19,44)(7,71,33,58,20,45)(8,72,34,59,21,46)(9,73,35,60,22,47)(10,74,36,61,23,48)(11,75,37,62,24,49)(12,76,38,63,25,50)(13,77,39,64,26,51)(79,142,92,155,105,129)(80,143,93,156,106,130)(81,144,94,118,107,131)(82,145,95,119,108,132)(83,146,96,120,109,133)(84,147,97,121,110,134)(85,148,98,122,111,135)(86,149,99,123,112,136)(87,150,100,124,113,137)(88,151,101,125,114,138)(89,152,102,126,115,139)(90,153,103,127,116,140)(91,154,104,128,117,141), (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), (1,134)(2,133)(3,132)(4,131)(5,130)(6,129)(7,128)(8,127)(9,126)(10,125)(11,124)(12,123)(13,122)(14,121)(15,120)(16,119)(17,118)(18,156)(19,155)(20,154)(21,153)(22,152)(23,151)(24,150)(25,149)(26,148)(27,147)(28,146)(29,145)(30,144)(31,143)(32,142)(33,141)(34,140)(35,139)(36,138)(37,137)(38,136)(39,135)(40,109)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,117)(72,116)(73,115)(74,114)(75,113)(76,112)(77,111)(78,110) );
G=PermutationGroup([[(1,65,27,52,14,78),(2,66,28,53,15,40),(3,67,29,54,16,41),(4,68,30,55,17,42),(5,69,31,56,18,43),(6,70,32,57,19,44),(7,71,33,58,20,45),(8,72,34,59,21,46),(9,73,35,60,22,47),(10,74,36,61,23,48),(11,75,37,62,24,49),(12,76,38,63,25,50),(13,77,39,64,26,51),(79,142,92,155,105,129),(80,143,93,156,106,130),(81,144,94,118,107,131),(82,145,95,119,108,132),(83,146,96,120,109,133),(84,147,97,121,110,134),(85,148,98,122,111,135),(86,149,99,123,112,136),(87,150,100,124,113,137),(88,151,101,125,114,138),(89,152,102,126,115,139),(90,153,103,127,116,140),(91,154,104,128,117,141)], [(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)], [(1,134),(2,133),(3,132),(4,131),(5,130),(6,129),(7,128),(8,127),(9,126),(10,125),(11,124),(12,123),(13,122),(14,121),(15,120),(16,119),(17,118),(18,156),(19,155),(20,154),(21,153),(22,152),(23,151),(24,150),(25,149),(26,148),(27,147),(28,146),(29,145),(30,144),(31,143),(32,142),(33,141),(34,140),(35,139),(36,138),(37,137),(38,136),(39,135),(40,109),(41,108),(42,107),(43,106),(44,105),(45,104),(46,103),(47,102),(48,101),(49,100),(50,99),(51,98),(52,97),(53,96),(54,95),(55,94),(56,93),(57,92),(58,91),(59,90),(60,89),(61,88),(62,87),(63,86),(64,85),(65,84),(66,83),(67,82),(68,81),(69,80),(70,79),(71,117),(72,116),(73,115),(74,114),(75,113),(76,112),(77,111),(78,110)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39AV | 78A | ··· | 78AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 39 | 39 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 39 | 39 | 39 | 39 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D6 | C3×S3 | D13 | S3×C6 | D26 | C3×D13 | D39 | C6×D13 | D78 | C3×D39 | C6×D39 |
| kernel | C6×D39 | C3×D39 | C3×C78 | D78 | D39 | C78 | C78 | C39 | C26 | C3×C6 | C13 | C32 | C6 | C6 | C3 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 6 | 2 | 6 | 12 | 12 | 12 | 12 | 24 | 24 |
Matrix representation of C6×D39 ►in GL3(𝔽79) generated by
| 24 | 0 | 0 |
| 0 | 23 | 0 |
| 0 | 0 | 23 |
| 1 | 0 | 0 |
| 0 | 72 | 23 |
| 0 | 0 | 45 |
| 1 | 0 | 0 |
| 0 | 67 | 53 |
| 0 | 45 | 12 |
G:=sub<GL(3,GF(79))| [24,0,0,0,23,0,0,0,23],[1,0,0,0,72,0,0,23,45],[1,0,0,0,67,45,0,53,12] >;
C6×D39 in GAP, Magma, Sage, TeX
C_6\times D_{39} % in TeX
G:=Group("C6xD39"); // GroupNames label
G:=SmallGroup(468,52);
// by ID
G=gap.SmallGroup(468,52);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-13,483,10804]);
// Polycyclic
G:=Group<a,b,c|a^6=b^39=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export