direct product, metacyclic, supersoluble, monomial, A-group
Aliases: Dic3×C13⋊C3, C39⋊5C12, C78.5C6, C26.2(C3×S3), (Dic3×C13)⋊C3, C13⋊4(C3×Dic3), C3⋊(C4×C13⋊C3), C6.(C2×C13⋊C3), C2.(S3×C13⋊C3), (C3×C13⋊C3)⋊5C4, (C6×C13⋊C3).3C2, (C2×C13⋊C3).2S3, SmallGroup(468,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — Dic3×C13⋊C3 |
Generators and relations for Dic3×C13⋊C3
G = < a,b,c,d | a6=c13=d3=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c9 >
Smallest permutation representation of Dic3×C13⋊C3
►On 156 points
Generators in S156
(1 66 14 40 27 53)(2 67 15 41 28 54)(3 68 16 42 29 55)(4 69 17 43 30 56)(5 70 18 44 31 57)(6 71 19 45 32 58)(7 72 20 46 33 59)(8 73 21 47 34 60)(9 74 22 48 35 61)(10 75 23 49 36 62)(11 76 24 50 37 63)(12 77 25 51 38 64)(13 78 26 52 39 65)(79 131 105 118 92 144)(80 132 106 119 93 145)(81 133 107 120 94 146)(82 134 108 121 95 147)(83 135 109 122 96 148)(84 136 110 123 97 149)(85 137 111 124 98 150)(86 138 112 125 99 151)(87 139 113 126 100 152)(88 140 114 127 101 153)(89 141 115 128 102 154)(90 142 116 129 103 155)(91 143 117 130 104 156)
(1 118 40 79)(2 119 41 80)(3 120 42 81)(4 121 43 82)(5 122 44 83)(6 123 45 84)(7 124 46 85)(8 125 47 86)(9 126 48 87)(10 127 49 88)(11 128 50 89)(12 129 51 90)(13 130 52 91)(14 131 53 92)(15 132 54 93)(16 133 55 94)(17 134 56 95)(18 135 57 96)(19 136 58 97)(20 137 59 98)(21 138 60 99)(22 139 61 100)(23 140 62 101)(24 141 63 102)(25 142 64 103)(26 143 65 104)(27 144 66 105)(28 145 67 106)(29 146 68 107)(30 147 69 108)(31 148 70 109)(32 149 71 110)(33 150 72 111)(34 151 73 112)(35 152 74 113)(36 153 75 114)(37 154 76 115)(38 155 77 116)(39 156 78 117)
(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)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(15 17 23)(16 20 19)(18 26 24)(21 22 25)(28 30 36)(29 33 32)(31 39 37)(34 35 38)(41 43 49)(42 46 45)(44 52 50)(47 48 51)(54 56 62)(55 59 58)(57 65 63)(60 61 64)(67 69 75)(68 72 71)(70 78 76)(73 74 77)(80 82 88)(81 85 84)(83 91 89)(86 87 90)(93 95 101)(94 98 97)(96 104 102)(99 100 103)(106 108 114)(107 111 110)(109 117 115)(112 113 116)(119 121 127)(120 124 123)(122 130 128)(125 126 129)(132 134 140)(133 137 136)(135 143 141)(138 139 142)(145 147 153)(146 150 149)(148 156 154)(151 152 155)
G:=sub<Sym(156)| (1,66,14,40,27,53)(2,67,15,41,28,54)(3,68,16,42,29,55)(4,69,17,43,30,56)(5,70,18,44,31,57)(6,71,19,45,32,58)(7,72,20,46,33,59)(8,73,21,47,34,60)(9,74,22,48,35,61)(10,75,23,49,36,62)(11,76,24,50,37,63)(12,77,25,51,38,64)(13,78,26,52,39,65)(79,131,105,118,92,144)(80,132,106,119,93,145)(81,133,107,120,94,146)(82,134,108,121,95,147)(83,135,109,122,96,148)(84,136,110,123,97,149)(85,137,111,124,98,150)(86,138,112,125,99,151)(87,139,113,126,100,152)(88,140,114,127,101,153)(89,141,115,128,102,154)(90,142,116,129,103,155)(91,143,117,130,104,156), (1,118,40,79)(2,119,41,80)(3,120,42,81)(4,121,43,82)(5,122,44,83)(6,123,45,84)(7,124,46,85)(8,125,47,86)(9,126,48,87)(10,127,49,88)(11,128,50,89)(12,129,51,90)(13,130,52,91)(14,131,53,92)(15,132,54,93)(16,133,55,94)(17,134,56,95)(18,135,57,96)(19,136,58,97)(20,137,59,98)(21,138,60,99)(22,139,61,100)(23,140,62,101)(24,141,63,102)(25,142,64,103)(26,143,65,104)(27,144,66,105)(28,145,67,106)(29,146,68,107)(30,147,69,108)(31,148,70,109)(32,149,71,110)(33,150,72,111)(34,151,73,112)(35,152,74,113)(36,153,75,114)(37,154,76,115)(38,155,77,116)(39,156,78,117), (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), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)(41,43,49)(42,46,45)(44,52,50)(47,48,51)(54,56,62)(55,59,58)(57,65,63)(60,61,64)(67,69,75)(68,72,71)(70,78,76)(73,74,77)(80,82,88)(81,85,84)(83,91,89)(86,87,90)(93,95,101)(94,98,97)(96,104,102)(99,100,103)(106,108,114)(107,111,110)(109,117,115)(112,113,116)(119,121,127)(120,124,123)(122,130,128)(125,126,129)(132,134,140)(133,137,136)(135,143,141)(138,139,142)(145,147,153)(146,150,149)(148,156,154)(151,152,155)>;
G:=Group( (1,66,14,40,27,53)(2,67,15,41,28,54)(3,68,16,42,29,55)(4,69,17,43,30,56)(5,70,18,44,31,57)(6,71,19,45,32,58)(7,72,20,46,33,59)(8,73,21,47,34,60)(9,74,22,48,35,61)(10,75,23,49,36,62)(11,76,24,50,37,63)(12,77,25,51,38,64)(13,78,26,52,39,65)(79,131,105,118,92,144)(80,132,106,119,93,145)(81,133,107,120,94,146)(82,134,108,121,95,147)(83,135,109,122,96,148)(84,136,110,123,97,149)(85,137,111,124,98,150)(86,138,112,125,99,151)(87,139,113,126,100,152)(88,140,114,127,101,153)(89,141,115,128,102,154)(90,142,116,129,103,155)(91,143,117,130,104,156), (1,118,40,79)(2,119,41,80)(3,120,42,81)(4,121,43,82)(5,122,44,83)(6,123,45,84)(7,124,46,85)(8,125,47,86)(9,126,48,87)(10,127,49,88)(11,128,50,89)(12,129,51,90)(13,130,52,91)(14,131,53,92)(15,132,54,93)(16,133,55,94)(17,134,56,95)(18,135,57,96)(19,136,58,97)(20,137,59,98)(21,138,60,99)(22,139,61,100)(23,140,62,101)(24,141,63,102)(25,142,64,103)(26,143,65,104)(27,144,66,105)(28,145,67,106)(29,146,68,107)(30,147,69,108)(31,148,70,109)(32,149,71,110)(33,150,72,111)(34,151,73,112)(35,152,74,113)(36,153,75,114)(37,154,76,115)(38,155,77,116)(39,156,78,117), (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), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)(41,43,49)(42,46,45)(44,52,50)(47,48,51)(54,56,62)(55,59,58)(57,65,63)(60,61,64)(67,69,75)(68,72,71)(70,78,76)(73,74,77)(80,82,88)(81,85,84)(83,91,89)(86,87,90)(93,95,101)(94,98,97)(96,104,102)(99,100,103)(106,108,114)(107,111,110)(109,117,115)(112,113,116)(119,121,127)(120,124,123)(122,130,128)(125,126,129)(132,134,140)(133,137,136)(135,143,141)(138,139,142)(145,147,153)(146,150,149)(148,156,154)(151,152,155) );
G=PermutationGroup([[(1,66,14,40,27,53),(2,67,15,41,28,54),(3,68,16,42,29,55),(4,69,17,43,30,56),(5,70,18,44,31,57),(6,71,19,45,32,58),(7,72,20,46,33,59),(8,73,21,47,34,60),(9,74,22,48,35,61),(10,75,23,49,36,62),(11,76,24,50,37,63),(12,77,25,51,38,64),(13,78,26,52,39,65),(79,131,105,118,92,144),(80,132,106,119,93,145),(81,133,107,120,94,146),(82,134,108,121,95,147),(83,135,109,122,96,148),(84,136,110,123,97,149),(85,137,111,124,98,150),(86,138,112,125,99,151),(87,139,113,126,100,152),(88,140,114,127,101,153),(89,141,115,128,102,154),(90,142,116,129,103,155),(91,143,117,130,104,156)], [(1,118,40,79),(2,119,41,80),(3,120,42,81),(4,121,43,82),(5,122,44,83),(6,123,45,84),(7,124,46,85),(8,125,47,86),(9,126,48,87),(10,127,49,88),(11,128,50,89),(12,129,51,90),(13,130,52,91),(14,131,53,92),(15,132,54,93),(16,133,55,94),(17,134,56,95),(18,135,57,96),(19,136,58,97),(20,137,59,98),(21,138,60,99),(22,139,61,100),(23,140,62,101),(24,141,63,102),(25,142,64,103),(26,143,65,104),(27,144,66,105),(28,145,67,106),(29,146,68,107),(30,147,69,108),(31,148,70,109),(32,149,71,110),(33,150,72,111),(34,151,73,112),(35,152,74,113),(36,153,75,114),(37,154,76,115),(38,155,77,116),(39,156,78,117)], [(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)], [(2,4,10),(3,7,6),(5,13,11),(8,9,12),(15,17,23),(16,20,19),(18,26,24),(21,22,25),(28,30,36),(29,33,32),(31,39,37),(34,35,38),(41,43,49),(42,46,45),(44,52,50),(47,48,51),(54,56,62),(55,59,58),(57,65,63),(60,61,64),(67,69,75),(68,72,71),(70,78,76),(73,74,77),(80,82,88),(81,85,84),(83,91,89),(86,87,90),(93,95,101),(94,98,97),(96,104,102),(99,100,103),(106,108,114),(107,111,110),(109,117,115),(112,113,116),(119,121,127),(120,124,123),(122,130,128),(125,126,129),(132,134,140),(133,137,136),(135,143,141),(138,139,142),(145,147,153),(146,150,149),(148,156,154),(151,152,155)]])
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 13A | 13B | 13C | 13D | 26A | 26B | 26C | 26D | 39A | 39B | 39C | 39D | 52A | ··· | 52H | 78A | 78B | 78C | 78D |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 13 | 13 | 13 | 13 | 26 | 26 | 26 | 26 | 39 | 39 | 39 | 39 | 52 | ··· | 52 | 78 | 78 | 78 | 78 |
| size | 1 | 1 | 2 | 13 | 13 | 26 | 26 | 3 | 3 | 2 | 13 | 13 | 26 | 26 | 39 | 39 | 39 | 39 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 6 | 6 | 6 | 6 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 | 6 |
| type | + | + | + | - | |||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | S3 | Dic3 | C3×S3 | C3×Dic3 | C13⋊C3 | C2×C13⋊C3 | C4×C13⋊C3 | S3×C13⋊C3 | Dic3×C13⋊C3 |
| kernel | Dic3×C13⋊C3 | C6×C13⋊C3 | Dic3×C13 | C3×C13⋊C3 | C78 | C39 | C2×C13⋊C3 | C13⋊C3 | C26 | C13 | Dic3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 4 | 4 |
Matrix representation of Dic3×C13⋊C3 ►in GL5(𝔽157)
| 1 | 156 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 13 | 105 | 0 | 0 | 0 |
| 118 | 144 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 90 | 1 |
| 0 | 0 | 82 | 34 | 119 |
| 0 | 0 | 9 | 92 | 83 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 135 | 113 | 121 |
| 0 | 0 | 27 | 141 | 34 |
| 0 | 0 | 130 | 122 | 38 |
G:=sub<GL(5,GF(157))| [1,1,0,0,0,156,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[13,118,0,0,0,105,144,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,2,82,9,0,0,90,34,92,0,0,1,119,83],[1,0,0,0,0,0,1,0,0,0,0,0,135,27,130,0,0,113,141,122,0,0,121,34,38] >;
Dic3×C13⋊C3 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{13}\rtimes C_3 % in TeX
G:=Group("Dic3xC13:C3"); // GroupNames label
G:=SmallGroup(468,20);
// by ID
G=gap.SmallGroup(468,20);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-13,30,483,1359]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^13=d^3=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^-1=c^9>;
// generators/relations
Export