direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C39, (C2×C6)⋊C39, C22⋊(C3×C39), (C2×C78)⋊1C3, (C2×C26)⋊2C32, SmallGroup(468,48)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C39 |
Generators and relations for A4×C39
G = < a,b,c,d | a39=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of A4×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)
(1 145)(2 146)(3 147)(4 148)(5 149)(6 150)(7 151)(8 152)(9 153)(10 154)(11 155)(12 156)(13 118)(14 119)(15 120)(16 121)(17 122)(18 123)(19 124)(20 125)(21 126)(22 127)(23 128)(24 129)(25 130)(26 131)(27 132)(28 133)(29 134)(30 135)(31 136)(32 137)(33 138)(34 139)(35 140)(36 141)(37 142)(38 143)(39 144)(40 102)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 79)(57 80)(58 81)(59 82)(60 83)(61 84)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 91)(69 92)(70 93)(71 94)(72 95)(73 96)(74 97)(75 98)(76 99)(77 100)(78 101)
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 101)(9 102)(10 103)(11 104)(12 105)(13 106)(14 107)(15 108)(16 109)(17 110)(18 111)(19 112)(20 113)(21 114)(22 115)(23 116)(24 117)(25 79)(26 80)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 153)(41 154)(42 155)(43 156)(44 118)(45 119)(46 120)(47 121)(48 122)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 129)(56 130)(57 131)(58 132)(59 133)(60 134)(61 135)(62 136)(63 137)(64 138)(65 139)(66 140)(67 141)(68 142)(69 143)(70 144)(71 145)(72 146)(73 147)(74 148)(75 149)(76 150)(77 151)(78 152)
(1 27 14)(2 28 15)(3 29 16)(4 30 17)(5 31 18)(6 32 19)(7 33 20)(8 34 21)(9 35 22)(10 36 23)(11 37 24)(12 38 25)(13 39 26)(40 140 115)(41 141 116)(42 142 117)(43 143 79)(44 144 80)(45 145 81)(46 146 82)(47 147 83)(48 148 84)(49 149 85)(50 150 86)(51 151 87)(52 152 88)(53 153 89)(54 154 90)(55 155 91)(56 156 92)(57 118 93)(58 119 94)(59 120 95)(60 121 96)(61 122 97)(62 123 98)(63 124 99)(64 125 100)(65 126 101)(66 127 102)(67 128 103)(68 129 104)(69 130 105)(70 131 106)(71 132 107)(72 133 108)(73 134 109)(74 135 110)(75 136 111)(76 137 112)(77 138 113)(78 139 114)
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), (1,145)(2,146)(3,147)(4,148)(5,149)(6,150)(7,151)(8,152)(9,153)(10,154)(11,155)(12,156)(13,118)(14,119)(15,120)(16,121)(17,122)(18,123)(19,124)(20,125)(21,126)(22,127)(23,128)(24,129)(25,130)(26,131)(27,132)(28,133)(29,134)(30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,141)(37,142)(38,143)(39,144)(40,102)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,79)(57,80)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95)(73,96)(74,97)(75,98)(76,99)(77,100)(78,101), (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,101)(9,102)(10,103)(11,104)(12,105)(13,106)(14,107)(15,108)(16,109)(17,110)(18,111)(19,112)(20,113)(21,114)(22,115)(23,116)(24,117)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,153)(41,154)(42,155)(43,156)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,145)(72,146)(73,147)(74,148)(75,149)(76,150)(77,151)(78,152), (1,27,14)(2,28,15)(3,29,16)(4,30,17)(5,31,18)(6,32,19)(7,33,20)(8,34,21)(9,35,22)(10,36,23)(11,37,24)(12,38,25)(13,39,26)(40,140,115)(41,141,116)(42,142,117)(43,143,79)(44,144,80)(45,145,81)(46,146,82)(47,147,83)(48,148,84)(49,149,85)(50,150,86)(51,151,87)(52,152,88)(53,153,89)(54,154,90)(55,155,91)(56,156,92)(57,118,93)(58,119,94)(59,120,95)(60,121,96)(61,122,97)(62,123,98)(63,124,99)(64,125,100)(65,126,101)(66,127,102)(67,128,103)(68,129,104)(69,130,105)(70,131,106)(71,132,107)(72,133,108)(73,134,109)(74,135,110)(75,136,111)(76,137,112)(77,138,113)(78,139,114)>;
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), (1,145)(2,146)(3,147)(4,148)(5,149)(6,150)(7,151)(8,152)(9,153)(10,154)(11,155)(12,156)(13,118)(14,119)(15,120)(16,121)(17,122)(18,123)(19,124)(20,125)(21,126)(22,127)(23,128)(24,129)(25,130)(26,131)(27,132)(28,133)(29,134)(30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,141)(37,142)(38,143)(39,144)(40,102)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,79)(57,80)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95)(73,96)(74,97)(75,98)(76,99)(77,100)(78,101), (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,101)(9,102)(10,103)(11,104)(12,105)(13,106)(14,107)(15,108)(16,109)(17,110)(18,111)(19,112)(20,113)(21,114)(22,115)(23,116)(24,117)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,153)(41,154)(42,155)(43,156)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,145)(72,146)(73,147)(74,148)(75,149)(76,150)(77,151)(78,152), (1,27,14)(2,28,15)(3,29,16)(4,30,17)(5,31,18)(6,32,19)(7,33,20)(8,34,21)(9,35,22)(10,36,23)(11,37,24)(12,38,25)(13,39,26)(40,140,115)(41,141,116)(42,142,117)(43,143,79)(44,144,80)(45,145,81)(46,146,82)(47,147,83)(48,148,84)(49,149,85)(50,150,86)(51,151,87)(52,152,88)(53,153,89)(54,154,90)(55,155,91)(56,156,92)(57,118,93)(58,119,94)(59,120,95)(60,121,96)(61,122,97)(62,123,98)(63,124,99)(64,125,100)(65,126,101)(66,127,102)(67,128,103)(68,129,104)(69,130,105)(70,131,106)(71,132,107)(72,133,108)(73,134,109)(74,135,110)(75,136,111)(76,137,112)(77,138,113)(78,139,114) );
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)], [(1,145),(2,146),(3,147),(4,148),(5,149),(6,150),(7,151),(8,152),(9,153),(10,154),(11,155),(12,156),(13,118),(14,119),(15,120),(16,121),(17,122),(18,123),(19,124),(20,125),(21,126),(22,127),(23,128),(24,129),(25,130),(26,131),(27,132),(28,133),(29,134),(30,135),(31,136),(32,137),(33,138),(34,139),(35,140),(36,141),(37,142),(38,143),(39,144),(40,102),(41,103),(42,104),(43,105),(44,106),(45,107),(46,108),(47,109),(48,110),(49,111),(50,112),(51,113),(52,114),(53,115),(54,116),(55,117),(56,79),(57,80),(58,81),(59,82),(60,83),(61,84),(62,85),(63,86),(64,87),(65,88),(66,89),(67,90),(68,91),(69,92),(70,93),(71,94),(72,95),(73,96),(74,97),(75,98),(76,99),(77,100),(78,101)], [(1,94),(2,95),(3,96),(4,97),(5,98),(6,99),(7,100),(8,101),(9,102),(10,103),(11,104),(12,105),(13,106),(14,107),(15,108),(16,109),(17,110),(18,111),(19,112),(20,113),(21,114),(22,115),(23,116),(24,117),(25,79),(26,80),(27,81),(28,82),(29,83),(30,84),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(37,91),(38,92),(39,93),(40,153),(41,154),(42,155),(43,156),(44,118),(45,119),(46,120),(47,121),(48,122),(49,123),(50,124),(51,125),(52,126),(53,127),(54,128),(55,129),(56,130),(57,131),(58,132),(59,133),(60,134),(61,135),(62,136),(63,137),(64,138),(65,139),(66,140),(67,141),(68,142),(69,143),(70,144),(71,145),(72,146),(73,147),(74,148),(75,149),(76,150),(77,151),(78,152)], [(1,27,14),(2,28,15),(3,29,16),(4,30,17),(5,31,18),(6,32,19),(7,33,20),(8,34,21),(9,35,22),(10,36,23),(11,37,24),(12,38,25),(13,39,26),(40,140,115),(41,141,116),(42,142,117),(43,143,79),(44,144,80),(45,145,81),(46,146,82),(47,147,83),(48,148,84),(49,149,85),(50,150,86),(51,151,87),(52,152,88),(53,153,89),(54,154,90),(55,155,91),(56,156,92),(57,118,93),(58,119,94),(59,120,95),(60,121,96),(61,122,97),(62,123,98),(63,124,99),(64,125,100),(65,126,101),(66,127,102),(67,128,103),(68,129,104),(69,130,105),(70,131,106),(71,132,107),(72,133,108),(73,134,109),(74,135,110),(75,136,111),(76,137,112),(77,138,113),(78,139,114)]])
156 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3H | 6A | 6B | 13A | ··· | 13L | 26A | ··· | 26L | 39A | ··· | 39X | 39Y | ··· | 39CR | 78A | ··· | 78X |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 3 | 1 | 1 | 4 | ··· | 4 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 4 | ··· | 4 | 3 | ··· | 3 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C3 | C3 | C13 | C39 | C39 | A4 | C3×A4 | A4×C13 | A4×C39 |
| kernel | A4×C39 | A4×C13 | C2×C78 | C3×A4 | A4 | C2×C6 | C39 | C13 | C3 | C1 |
| # reps | 1 | 6 | 2 | 12 | 72 | 24 | 1 | 2 | 12 | 24 |
Matrix representation of A4×C39 ►in GL4(𝔽79) generated by
| 23 | 0 | 0 | 0 |
| 0 | 38 | 0 | 0 |
| 0 | 0 | 38 | 0 |
| 0 | 0 | 0 | 38 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 78 | 78 | 78 |
| 1 | 0 | 0 | 0 |
| 0 | 78 | 78 | 78 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 23 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 78 | 78 | 78 |
G:=sub<GL(4,GF(79))| [23,0,0,0,0,38,0,0,0,0,38,0,0,0,0,38],[1,0,0,0,0,0,1,78,0,1,0,78,0,0,0,78],[1,0,0,0,0,78,0,0,0,78,0,1,0,78,1,0],[23,0,0,0,0,1,0,78,0,0,0,78,0,0,1,78] >;
A4×C39 in GAP, Magma, Sage, TeX
A_4\times C_{39} % in TeX
G:=Group("A4xC39"); // GroupNames label
G:=SmallGroup(468,48);
// by ID
G=gap.SmallGroup(468,48);
# by ID
G:=PCGroup([5,-3,-3,-13,-2,2,4683,8779]);
// Polycyclic
G:=Group<a,b,c,d|a^39=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
Export