direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: Dic3×D13, C6.1D26, C26.1D6, D26.2S3, Dic39⋊2C2, C78.1C22, C39⋊4(C2×C4), C3⋊3(C4×D13), (C3×D13)⋊1C4, C2.1(S3×D13), C13⋊2(C2×Dic3), (C6×D13).1C2, (Dic3×C13)⋊1C2, SmallGroup(312,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — Dic3×D13 |
Generators and relations for Dic3×D13
G = < a,b,c,d | a6=c13=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of Dic3×D13
►On 156 points
Generators in S156
(1 70 17 46 39 61)(2 71 18 47 27 62)(3 72 19 48 28 63)(4 73 20 49 29 64)(5 74 21 50 30 65)(6 75 22 51 31 53)(7 76 23 52 32 54)(8 77 24 40 33 55)(9 78 25 41 34 56)(10 66 26 42 35 57)(11 67 14 43 36 58)(12 68 15 44 37 59)(13 69 16 45 38 60)(79 136 106 123 92 147)(80 137 107 124 93 148)(81 138 108 125 94 149)(82 139 109 126 95 150)(83 140 110 127 96 151)(84 141 111 128 97 152)(85 142 112 129 98 153)(86 143 113 130 99 154)(87 131 114 118 100 155)(88 132 115 119 101 156)(89 133 116 120 102 144)(90 134 117 121 103 145)(91 135 105 122 104 146)
(1 125 46 81)(2 126 47 82)(3 127 48 83)(4 128 49 84)(5 129 50 85)(6 130 51 86)(7 118 52 87)(8 119 40 88)(9 120 41 89)(10 121 42 90)(11 122 43 91)(12 123 44 79)(13 124 45 80)(14 135 58 104)(15 136 59 92)(16 137 60 93)(17 138 61 94)(18 139 62 95)(19 140 63 96)(20 141 64 97)(21 142 65 98)(22 143 53 99)(23 131 54 100)(24 132 55 101)(25 133 56 102)(26 134 57 103)(27 150 71 109)(28 151 72 110)(29 152 73 111)(30 153 74 112)(31 154 75 113)(32 155 76 114)(33 156 77 115)(34 144 78 116)(35 145 66 117)(36 146 67 105)(37 147 68 106)(38 148 69 107)(39 149 70 108)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 19)(15 18)(16 17)(20 26)(21 25)(22 24)(27 37)(28 36)(29 35)(30 34)(31 33)(38 39)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(53 55)(56 65)(57 64)(58 63)(59 62)(60 61)(66 73)(67 72)(68 71)(69 70)(74 78)(75 77)(79 82)(80 81)(83 91)(84 90)(85 89)(86 88)(92 95)(93 94)(96 104)(97 103)(98 102)(99 101)(105 110)(106 109)(107 108)(111 117)(112 116)(113 115)(119 130)(120 129)(121 128)(122 127)(123 126)(124 125)(132 143)(133 142)(134 141)(135 140)(136 139)(137 138)(144 153)(145 152)(146 151)(147 150)(148 149)(154 156)
G:=sub<Sym(156)| (1,70,17,46,39,61)(2,71,18,47,27,62)(3,72,19,48,28,63)(4,73,20,49,29,64)(5,74,21,50,30,65)(6,75,22,51,31,53)(7,76,23,52,32,54)(8,77,24,40,33,55)(9,78,25,41,34,56)(10,66,26,42,35,57)(11,67,14,43,36,58)(12,68,15,44,37,59)(13,69,16,45,38,60)(79,136,106,123,92,147)(80,137,107,124,93,148)(81,138,108,125,94,149)(82,139,109,126,95,150)(83,140,110,127,96,151)(84,141,111,128,97,152)(85,142,112,129,98,153)(86,143,113,130,99,154)(87,131,114,118,100,155)(88,132,115,119,101,156)(89,133,116,120,102,144)(90,134,117,121,103,145)(91,135,105,122,104,146), (1,125,46,81)(2,126,47,82)(3,127,48,83)(4,128,49,84)(5,129,50,85)(6,130,51,86)(7,118,52,87)(8,119,40,88)(9,120,41,89)(10,121,42,90)(11,122,43,91)(12,123,44,79)(13,124,45,80)(14,135,58,104)(15,136,59,92)(16,137,60,93)(17,138,61,94)(18,139,62,95)(19,140,63,96)(20,141,64,97)(21,142,65,98)(22,143,53,99)(23,131,54,100)(24,132,55,101)(25,133,56,102)(26,134,57,103)(27,150,71,109)(28,151,72,110)(29,152,73,111)(30,153,74,112)(31,154,75,113)(32,155,76,114)(33,156,77,115)(34,144,78,116)(35,145,66,117)(36,146,67,105)(37,147,68,106)(38,148,69,107)(39,149,70,108), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,19)(15,18)(16,17)(20,26)(21,25)(22,24)(27,37)(28,36)(29,35)(30,34)(31,33)(38,39)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(53,55)(56,65)(57,64)(58,63)(59,62)(60,61)(66,73)(67,72)(68,71)(69,70)(74,78)(75,77)(79,82)(80,81)(83,91)(84,90)(85,89)(86,88)(92,95)(93,94)(96,104)(97,103)(98,102)(99,101)(105,110)(106,109)(107,108)(111,117)(112,116)(113,115)(119,130)(120,129)(121,128)(122,127)(123,126)(124,125)(132,143)(133,142)(134,141)(135,140)(136,139)(137,138)(144,153)(145,152)(146,151)(147,150)(148,149)(154,156)>;
G:=Group( (1,70,17,46,39,61)(2,71,18,47,27,62)(3,72,19,48,28,63)(4,73,20,49,29,64)(5,74,21,50,30,65)(6,75,22,51,31,53)(7,76,23,52,32,54)(8,77,24,40,33,55)(9,78,25,41,34,56)(10,66,26,42,35,57)(11,67,14,43,36,58)(12,68,15,44,37,59)(13,69,16,45,38,60)(79,136,106,123,92,147)(80,137,107,124,93,148)(81,138,108,125,94,149)(82,139,109,126,95,150)(83,140,110,127,96,151)(84,141,111,128,97,152)(85,142,112,129,98,153)(86,143,113,130,99,154)(87,131,114,118,100,155)(88,132,115,119,101,156)(89,133,116,120,102,144)(90,134,117,121,103,145)(91,135,105,122,104,146), (1,125,46,81)(2,126,47,82)(3,127,48,83)(4,128,49,84)(5,129,50,85)(6,130,51,86)(7,118,52,87)(8,119,40,88)(9,120,41,89)(10,121,42,90)(11,122,43,91)(12,123,44,79)(13,124,45,80)(14,135,58,104)(15,136,59,92)(16,137,60,93)(17,138,61,94)(18,139,62,95)(19,140,63,96)(20,141,64,97)(21,142,65,98)(22,143,53,99)(23,131,54,100)(24,132,55,101)(25,133,56,102)(26,134,57,103)(27,150,71,109)(28,151,72,110)(29,152,73,111)(30,153,74,112)(31,154,75,113)(32,155,76,114)(33,156,77,115)(34,144,78,116)(35,145,66,117)(36,146,67,105)(37,147,68,106)(38,148,69,107)(39,149,70,108), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,19)(15,18)(16,17)(20,26)(21,25)(22,24)(27,37)(28,36)(29,35)(30,34)(31,33)(38,39)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(53,55)(56,65)(57,64)(58,63)(59,62)(60,61)(66,73)(67,72)(68,71)(69,70)(74,78)(75,77)(79,82)(80,81)(83,91)(84,90)(85,89)(86,88)(92,95)(93,94)(96,104)(97,103)(98,102)(99,101)(105,110)(106,109)(107,108)(111,117)(112,116)(113,115)(119,130)(120,129)(121,128)(122,127)(123,126)(124,125)(132,143)(133,142)(134,141)(135,140)(136,139)(137,138)(144,153)(145,152)(146,151)(147,150)(148,149)(154,156) );
G=PermutationGroup([[(1,70,17,46,39,61),(2,71,18,47,27,62),(3,72,19,48,28,63),(4,73,20,49,29,64),(5,74,21,50,30,65),(6,75,22,51,31,53),(7,76,23,52,32,54),(8,77,24,40,33,55),(9,78,25,41,34,56),(10,66,26,42,35,57),(11,67,14,43,36,58),(12,68,15,44,37,59),(13,69,16,45,38,60),(79,136,106,123,92,147),(80,137,107,124,93,148),(81,138,108,125,94,149),(82,139,109,126,95,150),(83,140,110,127,96,151),(84,141,111,128,97,152),(85,142,112,129,98,153),(86,143,113,130,99,154),(87,131,114,118,100,155),(88,132,115,119,101,156),(89,133,116,120,102,144),(90,134,117,121,103,145),(91,135,105,122,104,146)], [(1,125,46,81),(2,126,47,82),(3,127,48,83),(4,128,49,84),(5,129,50,85),(6,130,51,86),(7,118,52,87),(8,119,40,88),(9,120,41,89),(10,121,42,90),(11,122,43,91),(12,123,44,79),(13,124,45,80),(14,135,58,104),(15,136,59,92),(16,137,60,93),(17,138,61,94),(18,139,62,95),(19,140,63,96),(20,141,64,97),(21,142,65,98),(22,143,53,99),(23,131,54,100),(24,132,55,101),(25,133,56,102),(26,134,57,103),(27,150,71,109),(28,151,72,110),(29,152,73,111),(30,153,74,112),(31,154,75,113),(32,155,76,114),(33,156,77,115),(34,144,78,116),(35,145,66,117),(36,146,67,105),(37,147,68,106),(38,148,69,107),(39,149,70,108)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,19),(15,18),(16,17),(20,26),(21,25),(22,24),(27,37),(28,36),(29,35),(30,34),(31,33),(38,39),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46),(53,55),(56,65),(57,64),(58,63),(59,62),(60,61),(66,73),(67,72),(68,71),(69,70),(74,78),(75,77),(79,82),(80,81),(83,91),(84,90),(85,89),(86,88),(92,95),(93,94),(96,104),(97,103),(98,102),(99,101),(105,110),(106,109),(107,108),(111,117),(112,116),(113,115),(119,130),(120,129),(121,128),(122,127),(123,126),(124,125),(132,143),(133,142),(134,141),(135,140),(136,139),(137,138),(144,153),(145,152),(146,151),(147,150),(148,149),(154,156)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39F | 52A | ··· | 52L | 78A | ··· | 78F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 |
| size | 1 | 1 | 13 | 13 | 2 | 3 | 3 | 39 | 39 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C4 | S3 | Dic3 | D6 | D13 | D26 | C4×D13 | S3×D13 | Dic3×D13 |
| kernel | Dic3×D13 | Dic3×C13 | Dic39 | C6×D13 | C3×D13 | D26 | D13 | C26 | Dic3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 6 | 6 | 12 | 6 | 6 |
Matrix representation of Dic3×D13 ►in GL4(𝔽157) generated by
| 156 | 0 | 0 | 0 |
| 0 | 156 | 0 | 0 |
| 0 | 0 | 156 | 1 |
| 0 | 0 | 156 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 156 |
| 0 | 0 | 0 | 156 |
| 102 | 62 | 0 | 0 |
| 156 | 121 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 20 | 101 | 0 | 0 |
| 66 | 137 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(157))| [156,0,0,0,0,156,0,0,0,0,156,156,0,0,1,0],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,156,156],[102,156,0,0,62,121,0,0,0,0,1,0,0,0,0,1],[20,66,0,0,101,137,0,0,0,0,1,0,0,0,0,1] >;
Dic3×D13 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times D_{13} % in TeX
G:=Group("Dic3xD13"); // GroupNames label
G:=SmallGroup(312,15);
// by ID
G=gap.SmallGroup(312,15);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-13,26,168,7204]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^13=d^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export