direct product, metabelian, soluble, monomial, A-group
Aliases: C3×C13⋊A4, C39⋊A4, C13⋊2(C3×A4), (C2×C78)⋊2C3, (C2×C26)⋊3C32, (C2×C6)⋊(C13⋊C3), C22⋊2(C3×C13⋊C3), SmallGroup(468,49)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C26 — C3×C13⋊A4 |
Generators and relations for C3×C13⋊A4
G = < a,b,c,d,e | a3=b13=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b9, ece-1=cd=dc, ede-1=c >
Smallest permutation representation of C3×C13⋊A4
►On 156 points
Generators in S156
(1 105 53)(2 106 54)(3 107 55)(4 108 56)(5 109 57)(6 110 58)(7 111 59)(8 112 60)(9 113 61)(10 114 62)(11 115 63)(12 116 64)(13 117 65)(14 118 66)(15 119 67)(16 120 68)(17 121 69)(18 122 70)(19 123 71)(20 124 72)(21 125 73)(22 126 74)(23 127 75)(24 128 76)(25 129 77)(26 130 78)(27 131 79)(28 132 80)(29 133 81)(30 134 82)(31 135 83)(32 136 84)(33 137 85)(34 138 86)(35 139 87)(36 140 88)(37 141 89)(38 142 90)(39 143 91)(40 144 92)(41 145 93)(42 146 94)(43 147 95)(44 148 96)(45 149 97)(46 150 98)(47 151 99)(48 152 100)(49 153 101)(50 154 102)(51 155 103)(52 156 104)
(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 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(61 74)(62 75)(63 76)(64 77)(65 78)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(91 104)(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)(131 144)(132 145)(133 146)(134 147)(135 148)(136 149)(137 150)(138 151)(139 152)(140 153)(141 154)(142 155)(143 156)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(53 92)(54 93)(55 94)(56 95)(57 96)(58 97)(59 98)(60 99)(61 100)(62 101)(63 102)(64 103)(65 104)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(105 144)(106 145)(107 146)(108 147)(109 148)(110 149)(111 150)(112 151)(113 152)(114 153)(115 154)(116 155)(117 156)(118 131)(119 132)(120 133)(121 134)(122 135)(123 136)(124 137)(125 138)(126 139)(127 140)(128 141)(129 142)(130 143)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(14 40 27)(15 43 36)(16 46 32)(17 49 28)(18 52 37)(19 42 33)(20 45 29)(21 48 38)(22 51 34)(23 41 30)(24 44 39)(25 47 35)(26 50 31)(54 56 62)(55 59 58)(57 65 63)(60 61 64)(66 92 79)(67 95 88)(68 98 84)(69 101 80)(70 104 89)(71 94 85)(72 97 81)(73 100 90)(74 103 86)(75 93 82)(76 96 91)(77 99 87)(78 102 83)(106 108 114)(107 111 110)(109 117 115)(112 113 116)(118 144 131)(119 147 140)(120 150 136)(121 153 132)(122 156 141)(123 146 137)(124 149 133)(125 152 142)(126 155 138)(127 145 134)(128 148 143)(129 151 139)(130 154 135)
G:=sub<Sym(156)| (1,105,53)(2,106,54)(3,107,55)(4,108,56)(5,109,57)(6,110,58)(7,111,59)(8,112,60)(9,113,61)(10,114,62)(11,115,63)(12,116,64)(13,117,65)(14,118,66)(15,119,67)(16,120,68)(17,121,69)(18,122,70)(19,123,71)(20,124,72)(21,125,73)(22,126,74)(23,127,75)(24,128,76)(25,129,77)(26,130,78)(27,131,79)(28,132,80)(29,133,81)(30,134,82)(31,135,83)(32,136,84)(33,137,85)(34,138,86)(35,139,87)(36,140,88)(37,141,89)(38,142,90)(39,143,91)(40,144,92)(41,145,93)(42,146,94)(43,147,95)(44,148,96)(45,149,97)(46,150,98)(47,151,99)(48,152,100)(49,153,101)(50,154,102)(51,155,103)(52,156,104), (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), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(91,104)(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)(131,144)(132,145)(133,146)(134,147)(135,148)(136,149)(137,150)(138,151)(139,152)(140,153)(141,154)(142,155)(143,156), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(105,144)(106,145)(107,146)(108,147)(109,148)(110,149)(111,150)(112,151)(113,152)(114,153)(115,154)(116,155)(117,156)(118,131)(119,132)(120,133)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(127,140)(128,141)(129,142)(130,143), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(14,40,27)(15,43,36)(16,46,32)(17,49,28)(18,52,37)(19,42,33)(20,45,29)(21,48,38)(22,51,34)(23,41,30)(24,44,39)(25,47,35)(26,50,31)(54,56,62)(55,59,58)(57,65,63)(60,61,64)(66,92,79)(67,95,88)(68,98,84)(69,101,80)(70,104,89)(71,94,85)(72,97,81)(73,100,90)(74,103,86)(75,93,82)(76,96,91)(77,99,87)(78,102,83)(106,108,114)(107,111,110)(109,117,115)(112,113,116)(118,144,131)(119,147,140)(120,150,136)(121,153,132)(122,156,141)(123,146,137)(124,149,133)(125,152,142)(126,155,138)(127,145,134)(128,148,143)(129,151,139)(130,154,135)>;
G:=Group( (1,105,53)(2,106,54)(3,107,55)(4,108,56)(5,109,57)(6,110,58)(7,111,59)(8,112,60)(9,113,61)(10,114,62)(11,115,63)(12,116,64)(13,117,65)(14,118,66)(15,119,67)(16,120,68)(17,121,69)(18,122,70)(19,123,71)(20,124,72)(21,125,73)(22,126,74)(23,127,75)(24,128,76)(25,129,77)(26,130,78)(27,131,79)(28,132,80)(29,133,81)(30,134,82)(31,135,83)(32,136,84)(33,137,85)(34,138,86)(35,139,87)(36,140,88)(37,141,89)(38,142,90)(39,143,91)(40,144,92)(41,145,93)(42,146,94)(43,147,95)(44,148,96)(45,149,97)(46,150,98)(47,151,99)(48,152,100)(49,153,101)(50,154,102)(51,155,103)(52,156,104), (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), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(91,104)(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)(131,144)(132,145)(133,146)(134,147)(135,148)(136,149)(137,150)(138,151)(139,152)(140,153)(141,154)(142,155)(143,156), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(105,144)(106,145)(107,146)(108,147)(109,148)(110,149)(111,150)(112,151)(113,152)(114,153)(115,154)(116,155)(117,156)(118,131)(119,132)(120,133)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(127,140)(128,141)(129,142)(130,143), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(14,40,27)(15,43,36)(16,46,32)(17,49,28)(18,52,37)(19,42,33)(20,45,29)(21,48,38)(22,51,34)(23,41,30)(24,44,39)(25,47,35)(26,50,31)(54,56,62)(55,59,58)(57,65,63)(60,61,64)(66,92,79)(67,95,88)(68,98,84)(69,101,80)(70,104,89)(71,94,85)(72,97,81)(73,100,90)(74,103,86)(75,93,82)(76,96,91)(77,99,87)(78,102,83)(106,108,114)(107,111,110)(109,117,115)(112,113,116)(118,144,131)(119,147,140)(120,150,136)(121,153,132)(122,156,141)(123,146,137)(124,149,133)(125,152,142)(126,155,138)(127,145,134)(128,148,143)(129,151,139)(130,154,135) );
G=PermutationGroup([[(1,105,53),(2,106,54),(3,107,55),(4,108,56),(5,109,57),(6,110,58),(7,111,59),(8,112,60),(9,113,61),(10,114,62),(11,115,63),(12,116,64),(13,117,65),(14,118,66),(15,119,67),(16,120,68),(17,121,69),(18,122,70),(19,123,71),(20,124,72),(21,125,73),(22,126,74),(23,127,75),(24,128,76),(25,129,77),(26,130,78),(27,131,79),(28,132,80),(29,133,81),(30,134,82),(31,135,83),(32,136,84),(33,137,85),(34,138,86),(35,139,87),(36,140,88),(37,141,89),(38,142,90),(39,143,91),(40,144,92),(41,145,93),(42,146,94),(43,147,95),(44,148,96),(45,149,97),(46,150,98),(47,151,99),(48,152,100),(49,153,101),(50,154,102),(51,155,103),(52,156,104)], [(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)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(39,52),(53,66),(54,67),(55,68),(56,69),(57,70),(58,71),(59,72),(60,73),(61,74),(62,75),(63,76),(64,77),(65,78),(79,92),(80,93),(81,94),(82,95),(83,96),(84,97),(85,98),(86,99),(87,100),(88,101),(89,102),(90,103),(91,104),(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),(131,144),(132,145),(133,146),(134,147),(135,148),(136,149),(137,150),(138,151),(139,152),(140,153),(141,154),(142,155),(143,156)], [(1,40),(2,41),(3,42),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,49),(11,50),(12,51),(13,52),(14,27),(15,28),(16,29),(17,30),(18,31),(19,32),(20,33),(21,34),(22,35),(23,36),(24,37),(25,38),(26,39),(53,92),(54,93),(55,94),(56,95),(57,96),(58,97),(59,98),(60,99),(61,100),(62,101),(63,102),(64,103),(65,104),(66,79),(67,80),(68,81),(69,82),(70,83),(71,84),(72,85),(73,86),(74,87),(75,88),(76,89),(77,90),(78,91),(105,144),(106,145),(107,146),(108,147),(109,148),(110,149),(111,150),(112,151),(113,152),(114,153),(115,154),(116,155),(117,156),(118,131),(119,132),(120,133),(121,134),(122,135),(123,136),(124,137),(125,138),(126,139),(127,140),(128,141),(129,142),(130,143)], [(2,4,10),(3,7,6),(5,13,11),(8,9,12),(14,40,27),(15,43,36),(16,46,32),(17,49,28),(18,52,37),(19,42,33),(20,45,29),(21,48,38),(22,51,34),(23,41,30),(24,44,39),(25,47,35),(26,50,31),(54,56,62),(55,59,58),(57,65,63),(60,61,64),(66,92,79),(67,95,88),(68,98,84),(69,101,80),(70,104,89),(71,94,85),(72,97,81),(73,100,90),(74,103,86),(75,93,82),(76,96,91),(77,99,87),(78,102,83),(106,108,114),(107,111,110),(109,117,115),(112,113,116),(118,144,131),(119,147,140),(120,150,136),(121,153,132),(122,156,141),(123,146,137),(124,149,133),(125,152,142),(126,155,138),(127,145,134),(128,148,143),(129,151,139),(130,154,135)]])
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3H | 6A | 6B | 13A | 13B | 13C | 13D | 26A | ··· | 26L | 39A | ··· | 39H | 78A | ··· | 78X |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 13 | 13 | 13 | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 3 | 1 | 1 | 52 | ··· | 52 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
60 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | |||||||
| image | C1 | C3 | C3 | A4 | C3×A4 | C13⋊C3 | C3×C13⋊C3 | C13⋊A4 | C3×C13⋊A4 |
| kernel | C3×C13⋊A4 | C13⋊A4 | C2×C78 | C39 | C13 | C2×C6 | C22 | C3 | C1 |
| # reps | 1 | 6 | 2 | 1 | 2 | 4 | 8 | 12 | 24 |
Matrix representation of C3×C13⋊A4 ►in GL3(𝔽79) generated by
| 55 | 0 | 0 |
| 0 | 55 | 0 |
| 0 | 0 | 55 |
| 0 | 15 | 74 |
| 1 | 4 | 31 |
| 0 | 0 | 21 |
| 10 | 4 | 68 |
| 74 | 69 | 5 |
| 0 | 0 | 78 |
| 69 | 75 | 0 |
| 5 | 10 | 0 |
| 0 | 0 | 78 |
| 78 | 75 | 1 |
| 0 | 1 | 0 |
| 78 | 61 | 0 |
G:=sub<GL(3,GF(79))| [55,0,0,0,55,0,0,0,55],[0,1,0,15,4,0,74,31,21],[10,74,0,4,69,0,68,5,78],[69,5,0,75,10,0,0,0,78],[78,0,78,75,1,61,1,0,0] >;
C3×C13⋊A4 in GAP, Magma, Sage, TeX
C_3\times C_{13}\rtimes A_4 % in TeX
G:=Group("C3xC13:A4"); // GroupNames label
G:=SmallGroup(468,49);
// by ID
G=gap.SmallGroup(468,49);
# by ID
G:=PCGroup([5,-3,-3,-2,2,-13,272,543,2704]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^13=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^9,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
Export