direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D39, C52⋊2S3, C156⋊2C2, C12⋊2D13, C2.1D78, C6.9D26, C26.9D6, D78.2C2, Dic39⋊5C2, C78.9C22, C13⋊3(C4×S3), C39⋊7(C2×C4), C3⋊2(C4×D13), SmallGroup(312,38)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — C4×D39 |
Generators and relations for C4×D39
G = < a,b,c | a4=b39=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D39
►On 156 points
Generators in S156
(1 125 66 94)(2 126 67 95)(3 127 68 96)(4 128 69 97)(5 129 70 98)(6 130 71 99)(7 131 72 100)(8 132 73 101)(9 133 74 102)(10 134 75 103)(11 135 76 104)(12 136 77 105)(13 137 78 106)(14 138 40 107)(15 139 41 108)(16 140 42 109)(17 141 43 110)(18 142 44 111)(19 143 45 112)(20 144 46 113)(21 145 47 114)(22 146 48 115)(23 147 49 116)(24 148 50 117)(25 149 51 79)(26 150 52 80)(27 151 53 81)(28 152 54 82)(29 153 55 83)(30 154 56 84)(31 155 57 85)(32 156 58 86)(33 118 59 87)(34 119 60 88)(35 120 61 89)(36 121 62 90)(37 122 63 91)(38 123 64 92)(39 124 65 93)
(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 65)(2 64)(3 63)(4 62)(5 61)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 53)(14 52)(15 51)(16 50)(17 49)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(25 41)(26 40)(27 78)(28 77)(29 76)(30 75)(31 74)(32 73)(33 72)(34 71)(35 70)(36 69)(37 68)(38 67)(39 66)(79 139)(80 138)(81 137)(82 136)(83 135)(84 134)(85 133)(86 132)(87 131)(88 130)(89 129)(90 128)(91 127)(92 126)(93 125)(94 124)(95 123)(96 122)(97 121)(98 120)(99 119)(100 118)(101 156)(102 155)(103 154)(104 153)(105 152)(106 151)(107 150)(108 149)(109 148)(110 147)(111 146)(112 145)(113 144)(114 143)(115 142)(116 141)(117 140)
G:=sub<Sym(156)| (1,125,66,94)(2,126,67,95)(3,127,68,96)(4,128,69,97)(5,129,70,98)(6,130,71,99)(7,131,72,100)(8,132,73,101)(9,133,74,102)(10,134,75,103)(11,135,76,104)(12,136,77,105)(13,137,78,106)(14,138,40,107)(15,139,41,108)(16,140,42,109)(17,141,43,110)(18,142,44,111)(19,143,45,112)(20,144,46,113)(21,145,47,114)(22,146,48,115)(23,147,49,116)(24,148,50,117)(25,149,51,79)(26,150,52,80)(27,151,53,81)(28,152,54,82)(29,153,55,83)(30,154,56,84)(31,155,57,85)(32,156,58,86)(33,118,59,87)(34,119,60,88)(35,120,61,89)(36,121,62,90)(37,122,63,91)(38,123,64,92)(39,124,65,93), (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), (1,65)(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(79,139)(80,138)(81,137)(82,136)(83,135)(84,134)(85,133)(86,132)(87,131)(88,130)(89,129)(90,128)(91,127)(92,126)(93,125)(94,124)(95,123)(96,122)(97,121)(98,120)(99,119)(100,118)(101,156)(102,155)(103,154)(104,153)(105,152)(106,151)(107,150)(108,149)(109,148)(110,147)(111,146)(112,145)(113,144)(114,143)(115,142)(116,141)(117,140)>;
G:=Group( (1,125,66,94)(2,126,67,95)(3,127,68,96)(4,128,69,97)(5,129,70,98)(6,130,71,99)(7,131,72,100)(8,132,73,101)(9,133,74,102)(10,134,75,103)(11,135,76,104)(12,136,77,105)(13,137,78,106)(14,138,40,107)(15,139,41,108)(16,140,42,109)(17,141,43,110)(18,142,44,111)(19,143,45,112)(20,144,46,113)(21,145,47,114)(22,146,48,115)(23,147,49,116)(24,148,50,117)(25,149,51,79)(26,150,52,80)(27,151,53,81)(28,152,54,82)(29,153,55,83)(30,154,56,84)(31,155,57,85)(32,156,58,86)(33,118,59,87)(34,119,60,88)(35,120,61,89)(36,121,62,90)(37,122,63,91)(38,123,64,92)(39,124,65,93), (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), (1,65)(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(79,139)(80,138)(81,137)(82,136)(83,135)(84,134)(85,133)(86,132)(87,131)(88,130)(89,129)(90,128)(91,127)(92,126)(93,125)(94,124)(95,123)(96,122)(97,121)(98,120)(99,119)(100,118)(101,156)(102,155)(103,154)(104,153)(105,152)(106,151)(107,150)(108,149)(109,148)(110,147)(111,146)(112,145)(113,144)(114,143)(115,142)(116,141)(117,140) );
G=PermutationGroup([[(1,125,66,94),(2,126,67,95),(3,127,68,96),(4,128,69,97),(5,129,70,98),(6,130,71,99),(7,131,72,100),(8,132,73,101),(9,133,74,102),(10,134,75,103),(11,135,76,104),(12,136,77,105),(13,137,78,106),(14,138,40,107),(15,139,41,108),(16,140,42,109),(17,141,43,110),(18,142,44,111),(19,143,45,112),(20,144,46,113),(21,145,47,114),(22,146,48,115),(23,147,49,116),(24,148,50,117),(25,149,51,79),(26,150,52,80),(27,151,53,81),(28,152,54,82),(29,153,55,83),(30,154,56,84),(31,155,57,85),(32,156,58,86),(33,118,59,87),(34,119,60,88),(35,120,61,89),(36,121,62,90),(37,122,63,91),(38,123,64,92),(39,124,65,93)], [(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)], [(1,65),(2,64),(3,63),(4,62),(5,61),(6,60),(7,59),(8,58),(9,57),(10,56),(11,55),(12,54),(13,53),(14,52),(15,51),(16,50),(17,49),(18,48),(19,47),(20,46),(21,45),(22,44),(23,43),(24,42),(25,41),(26,40),(27,78),(28,77),(29,76),(30,75),(31,74),(32,73),(33,72),(34,71),(35,70),(36,69),(37,68),(38,67),(39,66),(79,139),(80,138),(81,137),(82,136),(83,135),(84,134),(85,133),(86,132),(87,131),(88,130),(89,129),(90,128),(91,127),(92,126),(93,125),(94,124),(95,123),(96,122),(97,121),(98,120),(99,119),(100,118),(101,156),(102,155),(103,154),(104,153),(105,152),(106,151),(107,150),(108,149),(109,148),(110,147),(111,146),(112,145),(113,144),(114,143),(115,142),(116,141),(117,140)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 12A | 12B | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39L | 52A | ··· | 52L | 78A | ··· | 78L | 156A | ··· | 156X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 12 | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 | 156 | ··· | 156 |
| size | 1 | 1 | 39 | 39 | 2 | 1 | 1 | 39 | 39 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | C4×S3 | D13 | D26 | D39 | C4×D13 | D78 | C4×D39 |
| kernel | C4×D39 | Dic39 | C156 | D78 | D39 | C52 | C26 | C13 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 6 | 6 | 12 | 12 | 12 | 24 |
Matrix representation of C4×D39 ►in GL2(𝔽157) generated by
| 28 | 0 |
| 0 | 28 |
| 69 | 62 |
| 95 | 74 |
| 74 | 132 |
| 62 | 83 |
G:=sub<GL(2,GF(157))| [28,0,0,28],[69,95,62,74],[74,62,132,83] >;
C4×D39 in GAP, Magma, Sage, TeX
C_4\times D_{39} % in TeX
G:=Group("C4xD39"); // GroupNames label
G:=SmallGroup(312,38);
// by ID
G=gap.SmallGroup(312,38);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-13,26,323,7204]);
// Polycyclic
G:=Group<a,b,c|a^4=b^39=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export