direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C39, C4⋊C78, C52⋊7C6, C12⋊3C26, C156⋊7C2, C22⋊2C78, C78.23C22, (C2×C6)⋊1C26, (C2×C78)⋊1C2, (C2×C26)⋊9C6, C2.1(C2×C78), C6.6(C2×C26), C26.14(C2×C6), SmallGroup(312,43)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C39
G = < a,b,c | a39=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C39
►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 98 150 50)(2 99 151 51)(3 100 152 52)(4 101 153 53)(5 102 154 54)(6 103 155 55)(7 104 156 56)(8 105 118 57)(9 106 119 58)(10 107 120 59)(11 108 121 60)(12 109 122 61)(13 110 123 62)(14 111 124 63)(15 112 125 64)(16 113 126 65)(17 114 127 66)(18 115 128 67)(19 116 129 68)(20 117 130 69)(21 79 131 70)(22 80 132 71)(23 81 133 72)(24 82 134 73)(25 83 135 74)(26 84 136 75)(27 85 137 76)(28 86 138 77)(29 87 139 78)(30 88 140 40)(31 89 141 41)(32 90 142 42)(33 91 143 43)(34 92 144 44)(35 93 145 45)(36 94 146 46)(37 95 147 47)(38 96 148 48)(39 97 149 49)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(79 131)(80 132)(81 133)(82 134)(83 135)(84 136)(85 137)(86 138)(87 139)(88 140)(89 141)(90 142)(91 143)(92 144)(93 145)(94 146)(95 147)(96 148)(97 149)(98 150)(99 151)(100 152)(101 153)(102 154)(103 155)(104 156)(105 118)(106 119)(107 120)(108 121)(109 122)(110 123)(111 124)(112 125)(113 126)(114 127)(115 128)(116 129)(117 130)
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), (1,98,150,50)(2,99,151,51)(3,100,152,52)(4,101,153,53)(5,102,154,54)(6,103,155,55)(7,104,156,56)(8,105,118,57)(9,106,119,58)(10,107,120,59)(11,108,121,60)(12,109,122,61)(13,110,123,62)(14,111,124,63)(15,112,125,64)(16,113,126,65)(17,114,127,66)(18,115,128,67)(19,116,129,68)(20,117,130,69)(21,79,131,70)(22,80,132,71)(23,81,133,72)(24,82,134,73)(25,83,135,74)(26,84,136,75)(27,85,137,76)(28,86,138,77)(29,87,139,78)(30,88,140,40)(31,89,141,41)(32,90,142,42)(33,91,143,43)(34,92,144,44)(35,93,145,45)(36,94,146,46)(37,95,147,47)(38,96,148,48)(39,97,149,49), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(79,131)(80,132)(81,133)(82,134)(83,135)(84,136)(85,137)(86,138)(87,139)(88,140)(89,141)(90,142)(91,143)(92,144)(93,145)(94,146)(95,147)(96,148)(97,149)(98,150)(99,151)(100,152)(101,153)(102,154)(103,155)(104,156)(105,118)(106,119)(107,120)(108,121)(109,122)(110,123)(111,124)(112,125)(113,126)(114,127)(115,128)(116,129)(117,130)>;
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), (1,98,150,50)(2,99,151,51)(3,100,152,52)(4,101,153,53)(5,102,154,54)(6,103,155,55)(7,104,156,56)(8,105,118,57)(9,106,119,58)(10,107,120,59)(11,108,121,60)(12,109,122,61)(13,110,123,62)(14,111,124,63)(15,112,125,64)(16,113,126,65)(17,114,127,66)(18,115,128,67)(19,116,129,68)(20,117,130,69)(21,79,131,70)(22,80,132,71)(23,81,133,72)(24,82,134,73)(25,83,135,74)(26,84,136,75)(27,85,137,76)(28,86,138,77)(29,87,139,78)(30,88,140,40)(31,89,141,41)(32,90,142,42)(33,91,143,43)(34,92,144,44)(35,93,145,45)(36,94,146,46)(37,95,147,47)(38,96,148,48)(39,97,149,49), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(79,131)(80,132)(81,133)(82,134)(83,135)(84,136)(85,137)(86,138)(87,139)(88,140)(89,141)(90,142)(91,143)(92,144)(93,145)(94,146)(95,147)(96,148)(97,149)(98,150)(99,151)(100,152)(101,153)(102,154)(103,155)(104,156)(105,118)(106,119)(107,120)(108,121)(109,122)(110,123)(111,124)(112,125)(113,126)(114,127)(115,128)(116,129)(117,130) );
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)], [(1,98,150,50),(2,99,151,51),(3,100,152,52),(4,101,153,53),(5,102,154,54),(6,103,155,55),(7,104,156,56),(8,105,118,57),(9,106,119,58),(10,107,120,59),(11,108,121,60),(12,109,122,61),(13,110,123,62),(14,111,124,63),(15,112,125,64),(16,113,126,65),(17,114,127,66),(18,115,128,67),(19,116,129,68),(20,117,130,69),(21,79,131,70),(22,80,132,71),(23,81,133,72),(24,82,134,73),(25,83,135,74),(26,84,136,75),(27,85,137,76),(28,86,138,77),(29,87,139,78),(30,88,140,40),(31,89,141,41),(32,90,142,42),(33,91,143,43),(34,92,144,44),(35,93,145,45),(36,94,146,46),(37,95,147,47),(38,96,148,48),(39,97,149,49)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,57),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,70),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,78),(30,40),(31,41),(32,42),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,49),(79,131),(80,132),(81,133),(82,134),(83,135),(84,136),(85,137),(86,138),(87,139),(88,140),(89,141),(90,142),(91,143),(92,144),(93,145),(94,146),(95,147),(96,148),(97,149),(98,150),(99,151),(100,152),(101,153),(102,154),(103,155),(104,156),(105,118),(106,119),(107,120),(108,121),(109,122),(110,123),(111,124),(112,125),(113,126),(114,127),(115,128),(116,129),(117,130)]])
195 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26AJ | 39A | ··· | 39X | 52A | ··· | 52L | 78A | ··· | 78X | 78Y | ··· | 78BT | 156A | ··· | 156X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 | 78 | ··· | 78 | 156 | ··· | 156 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
195 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C13 | C26 | C26 | C39 | C78 | C78 | D4 | C3×D4 | D4×C13 | D4×C39 |
| kernel | D4×C39 | C156 | C2×C78 | D4×C13 | C52 | C2×C26 | C3×D4 | C12 | C2×C6 | D4 | C4 | C22 | C39 | C13 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 12 | 12 | 24 | 24 | 24 | 48 | 1 | 2 | 12 | 24 |
Matrix representation of D4×C39 ►in GL2(𝔽157) generated by
| 132 | 0 |
| 0 | 132 |
| 109 | 155 |
| 132 | 48 |
| 109 | 155 |
| 131 | 48 |
G:=sub<GL(2,GF(157))| [132,0,0,132],[109,132,155,48],[109,131,155,48] >;
D4×C39 in GAP, Magma, Sage, TeX
D_4\times C_{39} % in TeX
G:=Group("D4xC39"); // GroupNames label
G:=SmallGroup(312,43);
// by ID
G=gap.SmallGroup(312,43);
# by ID
G:=PCGroup([5,-2,-2,-3,-13,-2,1581]);
// Polycyclic
G:=Group<a,b,c|a^39=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export