metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic39, C39⋊3C4, C26.S3, C6.D13, C2.D39, C3⋊Dic13, C78.1C2, C13⋊2Dic3, SmallGroup(156,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — Dic39 |
Generators and relations for Dic39
G = < a,b | a78=1, b2=a39, bab-1=a-1 >
Smallest permutation representation of Dic39
►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 95 40 134)(2 94 41 133)(3 93 42 132)(4 92 43 131)(5 91 44 130)(6 90 45 129)(7 89 46 128)(8 88 47 127)(9 87 48 126)(10 86 49 125)(11 85 50 124)(12 84 51 123)(13 83 52 122)(14 82 53 121)(15 81 54 120)(16 80 55 119)(17 79 56 118)(18 156 57 117)(19 155 58 116)(20 154 59 115)(21 153 60 114)(22 152 61 113)(23 151 62 112)(24 150 63 111)(25 149 64 110)(26 148 65 109)(27 147 66 108)(28 146 67 107)(29 145 68 106)(30 144 69 105)(31 143 70 104)(32 142 71 103)(33 141 72 102)(34 140 73 101)(35 139 74 100)(36 138 75 99)(37 137 76 98)(38 136 77 97)(39 135 78 96)
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), (1,95,40,134)(2,94,41,133)(3,93,42,132)(4,92,43,131)(5,91,44,130)(6,90,45,129)(7,89,46,128)(8,88,47,127)(9,87,48,126)(10,86,49,125)(11,85,50,124)(12,84,51,123)(13,83,52,122)(14,82,53,121)(15,81,54,120)(16,80,55,119)(17,79,56,118)(18,156,57,117)(19,155,58,116)(20,154,59,115)(21,153,60,114)(22,152,61,113)(23,151,62,112)(24,150,63,111)(25,149,64,110)(26,148,65,109)(27,147,66,108)(28,146,67,107)(29,145,68,106)(30,144,69,105)(31,143,70,104)(32,142,71,103)(33,141,72,102)(34,140,73,101)(35,139,74,100)(36,138,75,99)(37,137,76,98)(38,136,77,97)(39,135,78,96)>;
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), (1,95,40,134)(2,94,41,133)(3,93,42,132)(4,92,43,131)(5,91,44,130)(6,90,45,129)(7,89,46,128)(8,88,47,127)(9,87,48,126)(10,86,49,125)(11,85,50,124)(12,84,51,123)(13,83,52,122)(14,82,53,121)(15,81,54,120)(16,80,55,119)(17,79,56,118)(18,156,57,117)(19,155,58,116)(20,154,59,115)(21,153,60,114)(22,152,61,113)(23,151,62,112)(24,150,63,111)(25,149,64,110)(26,148,65,109)(27,147,66,108)(28,146,67,107)(29,145,68,106)(30,144,69,105)(31,143,70,104)(32,142,71,103)(33,141,72,102)(34,140,73,101)(35,139,74,100)(36,138,75,99)(37,137,76,98)(38,136,77,97)(39,135,78,96) );
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)], [(1,95,40,134),(2,94,41,133),(3,93,42,132),(4,92,43,131),(5,91,44,130),(6,90,45,129),(7,89,46,128),(8,88,47,127),(9,87,48,126),(10,86,49,125),(11,85,50,124),(12,84,51,123),(13,83,52,122),(14,82,53,121),(15,81,54,120),(16,80,55,119),(17,79,56,118),(18,156,57,117),(19,155,58,116),(20,154,59,115),(21,153,60,114),(22,152,61,113),(23,151,62,112),(24,150,63,111),(25,149,64,110),(26,148,65,109),(27,147,66,108),(28,146,67,107),(29,145,68,106),(30,144,69,105),(31,143,70,104),(32,142,71,103),(33,141,72,102),(34,140,73,101),(35,139,74,100),(36,138,75,99),(37,137,76,98),(38,136,77,97),(39,135,78,96)]])
Dic39 is a maximal subgroup of
Dic3×D13 S3×Dic13 C39⋊D4 C39⋊Q8 Dic78 C4×D39 C39⋊7D4 Dic117 C39⋊3C12 C3⋊Dic39
Dic39 is a maximal quotient of
C39⋊3C8 Dic117 C3⋊Dic39
42 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39L | 78A | ··· | 78L |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 2 | 39 | 39 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C4 | S3 | Dic3 | D13 | Dic13 | D39 | Dic39 |
| kernel | Dic39 | C78 | C39 | C26 | C13 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 6 | 6 | 12 | 12 |
Matrix representation of Dic39 ►in GL2(𝔽157) generated by
| 47 | 126 |
| 138 | 66 |
| 25 | 23 |
| 82 | 132 |
G:=sub<GL(2,GF(157))| [47,138,126,66],[25,82,23,132] >;
Dic39 in GAP, Magma, Sage, TeX
{\rm Dic}_{39} % in TeX
G:=Group("Dic39"); // GroupNames label
G:=SmallGroup(156,5);
// by ID
G=gap.SmallGroup(156,5);
# by ID
G:=PCGroup([4,-2,-2,-3,-13,8,98,2307]);
// Polycyclic
G:=Group<a,b|a^78=1,b^2=a^39,b*a*b^-1=a^-1>;
// generators/relations
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