direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2×C6×C13⋊C3, C78⋊4C6, C13⋊2C62, (C2×C78)⋊3C3, C26⋊2(C3×C6), C39⋊6(C2×C6), (C2×C26)⋊4C32, SmallGroup(468,47)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C2×C6×C13⋊C3 |
Generators and relations for C2×C6×C13⋊C3
G = < a,b,c,d | a2=b6=c13=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c9 >
Subgroups: 300 in 60 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C3, C3, C22, C6, C6, C32, C2×C6, C2×C6, C13, C3×C6, C26, C62, C13⋊C3, C39, C2×C26, C2×C13⋊C3, C78, C3×C13⋊C3, C22×C13⋊C3, C2×C78, C6×C13⋊C3, C2×C6×C13⋊C3
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, C3×C6, C62, C13⋊C3, C2×C13⋊C3, C3×C13⋊C3, C22×C13⋊C3, C6×C13⋊C3, C2×C6×C13⋊C3
►On 156 points
Generators in S156
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 99)(22 100)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 107)(30 108)(31 109)(32 110)(33 111)(34 112)(35 113)(36 114)(37 115)(38 116)(39 117)(40 118)(41 119)(42 120)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 127)(50 128)(51 129)(52 130)(53 131)(54 132)(55 133)(56 134)(57 135)(58 136)(59 137)(60 138)(61 139)(62 140)(63 141)(64 142)(65 143)(66 144)(67 145)(68 146)(69 147)(70 148)(71 149)(72 150)(73 151)(74 152)(75 153)(76 154)(77 155)(78 156)
(1 53 27 40 14 66)(2 54 28 41 15 67)(3 55 29 42 16 68)(4 56 30 43 17 69)(5 57 31 44 18 70)(6 58 32 45 19 71)(7 59 33 46 20 72)(8 60 34 47 21 73)(9 61 35 48 22 74)(10 62 36 49 23 75)(11 63 37 50 24 76)(12 64 38 51 25 77)(13 65 39 52 26 78)(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 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 27)(2 17 36)(3 20 32)(4 23 28)(5 26 37)(6 16 33)(7 19 29)(8 22 38)(9 25 34)(10 15 30)(11 18 39)(12 21 35)(13 24 31)(40 53 66)(41 56 75)(42 59 71)(43 62 67)(44 65 76)(45 55 72)(46 58 68)(47 61 77)(48 64 73)(49 54 69)(50 57 78)(51 60 74)(52 63 70)(79 92 105)(80 95 114)(81 98 110)(82 101 106)(83 104 115)(84 94 111)(85 97 107)(86 100 116)(87 103 112)(88 93 108)(89 96 117)(90 99 113)(91 102 109)(118 131 144)(119 134 153)(120 137 149)(121 140 145)(122 143 154)(123 133 150)(124 136 146)(125 139 155)(126 142 151)(127 132 147)(128 135 156)(129 138 152)(130 141 148)
G:=sub<Sym(156)| (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,119)(42,120)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,127)(50,128)(51,129)(52,130)(53,131)(54,132)(55,133)(56,134)(57,135)(58,136)(59,137)(60,138)(61,139)(62,140)(63,141)(64,142)(65,143)(66,144)(67,145)(68,146)(69,147)(70,148)(71,149)(72,150)(73,151)(74,152)(75,153)(76,154)(77,155)(78,156), (1,53,27,40,14,66)(2,54,28,41,15,67)(3,55,29,42,16,68)(4,56,30,43,17,69)(5,57,31,44,18,70)(6,58,32,45,19,71)(7,59,33,46,20,72)(8,60,34,47,21,73)(9,61,35,48,22,74)(10,62,36,49,23,75)(11,63,37,50,24,76)(12,64,38,51,25,77)(13,65,39,52,26,78)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13)(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), (1,14,27)(2,17,36)(3,20,32)(4,23,28)(5,26,37)(6,16,33)(7,19,29)(8,22,38)(9,25,34)(10,15,30)(11,18,39)(12,21,35)(13,24,31)(40,53,66)(41,56,75)(42,59,71)(43,62,67)(44,65,76)(45,55,72)(46,58,68)(47,61,77)(48,64,73)(49,54,69)(50,57,78)(51,60,74)(52,63,70)(79,92,105)(80,95,114)(81,98,110)(82,101,106)(83,104,115)(84,94,111)(85,97,107)(86,100,116)(87,103,112)(88,93,108)(89,96,117)(90,99,113)(91,102,109)(118,131,144)(119,134,153)(120,137,149)(121,140,145)(122,143,154)(123,133,150)(124,136,146)(125,139,155)(126,142,151)(127,132,147)(128,135,156)(129,138,152)(130,141,148)>;
G:=Group( (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,119)(42,120)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,127)(50,128)(51,129)(52,130)(53,131)(54,132)(55,133)(56,134)(57,135)(58,136)(59,137)(60,138)(61,139)(62,140)(63,141)(64,142)(65,143)(66,144)(67,145)(68,146)(69,147)(70,148)(71,149)(72,150)(73,151)(74,152)(75,153)(76,154)(77,155)(78,156), (1,53,27,40,14,66)(2,54,28,41,15,67)(3,55,29,42,16,68)(4,56,30,43,17,69)(5,57,31,44,18,70)(6,58,32,45,19,71)(7,59,33,46,20,72)(8,60,34,47,21,73)(9,61,35,48,22,74)(10,62,36,49,23,75)(11,63,37,50,24,76)(12,64,38,51,25,77)(13,65,39,52,26,78)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13)(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), (1,14,27)(2,17,36)(3,20,32)(4,23,28)(5,26,37)(6,16,33)(7,19,29)(8,22,38)(9,25,34)(10,15,30)(11,18,39)(12,21,35)(13,24,31)(40,53,66)(41,56,75)(42,59,71)(43,62,67)(44,65,76)(45,55,72)(46,58,68)(47,61,77)(48,64,73)(49,54,69)(50,57,78)(51,60,74)(52,63,70)(79,92,105)(80,95,114)(81,98,110)(82,101,106)(83,104,115)(84,94,111)(85,97,107)(86,100,116)(87,103,112)(88,93,108)(89,96,117)(90,99,113)(91,102,109)(118,131,144)(119,134,153)(120,137,149)(121,140,145)(122,143,154)(123,133,150)(124,136,146)(125,139,155)(126,142,151)(127,132,147)(128,135,156)(129,138,152)(130,141,148) );
G=PermutationGroup([[(1,79),(2,80),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,97),(20,98),(21,99),(22,100),(23,101),(24,102),(25,103),(26,104),(27,105),(28,106),(29,107),(30,108),(31,109),(32,110),(33,111),(34,112),(35,113),(36,114),(37,115),(38,116),(39,117),(40,118),(41,119),(42,120),(43,121),(44,122),(45,123),(46,124),(47,125),(48,126),(49,127),(50,128),(51,129),(52,130),(53,131),(54,132),(55,133),(56,134),(57,135),(58,136),(59,137),(60,138),(61,139),(62,140),(63,141),(64,142),(65,143),(66,144),(67,145),(68,146),(69,147),(70,148),(71,149),(72,150),(73,151),(74,152),(75,153),(76,154),(77,155),(78,156)], [(1,53,27,40,14,66),(2,54,28,41,15,67),(3,55,29,42,16,68),(4,56,30,43,17,69),(5,57,31,44,18,70),(6,58,32,45,19,71),(7,59,33,46,20,72),(8,60,34,47,21,73),(9,61,35,48,22,74),(10,62,36,49,23,75),(11,63,37,50,24,76),(12,64,38,51,25,77),(13,65,39,52,26,78),(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)], [(1,2,3,4,5,6,7,8,9,10,11,12,13),(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)], [(1,14,27),(2,17,36),(3,20,32),(4,23,28),(5,26,37),(6,16,33),(7,19,29),(8,22,38),(9,25,34),(10,15,30),(11,18,39),(12,21,35),(13,24,31),(40,53,66),(41,56,75),(42,59,71),(43,62,67),(44,65,76),(45,55,72),(46,58,68),(47,61,77),(48,64,73),(49,54,69),(50,57,78),(51,60,74),(52,63,70),(79,92,105),(80,95,114),(81,98,110),(82,101,106),(83,104,115),(84,94,111),(85,97,107),(86,100,116),(87,103,112),(88,93,108),(89,96,117),(90,99,113),(91,102,109),(118,131,144),(119,134,153),(120,137,149),(121,140,145),(122,143,154),(123,133,150),(124,136,146),(125,139,155),(126,142,151),(127,132,147),(128,135,156),(129,138,152),(130,141,148)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 6A | ··· | 6F | 6G | ··· | 6X | 13A | 13B | 13C | 13D | 26A | ··· | 26L | 39A | ··· | 39H | 78A | ··· | 78X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 13 | 13 | 13 | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 13 | ··· | 13 | 1 | ··· | 1 | 13 | ··· | 13 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C13⋊C3 | C2×C13⋊C3 | C3×C13⋊C3 | C6×C13⋊C3 |
| kernel | C2×C6×C13⋊C3 | C6×C13⋊C3 | C22×C13⋊C3 | C2×C78 | C2×C13⋊C3 | C78 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 3 | 6 | 2 | 18 | 6 | 4 | 12 | 8 | 24 |
Matrix representation of C2×C6×C13⋊C3 ►in GL4(𝔽79) generated by
| 78 | 0 | 0 | 0 |
| 0 | 78 | 0 | 0 |
| 0 | 0 | 78 | 0 |
| 0 | 0 | 0 | 78 |
| 23 | 0 | 0 | 0 |
| 0 | 56 | 0 | 0 |
| 0 | 0 | 56 | 0 |
| 0 | 0 | 0 | 56 |
| 1 | 0 | 0 | 0 |
| 0 | 3 | 55 | 35 |
| 0 | 1 | 0 | 39 |
| 0 | 0 | 1 | 66 |
| 1 | 0 | 0 | 0 |
| 0 | 17 | 3 | 2 |
| 0 | 70 | 8 | 52 |
| 0 | 4 | 73 | 54 |
G:=sub<GL(4,GF(79))| [78,0,0,0,0,78,0,0,0,0,78,0,0,0,0,78],[23,0,0,0,0,56,0,0,0,0,56,0,0,0,0,56],[1,0,0,0,0,3,1,0,0,55,0,1,0,35,39,66],[1,0,0,0,0,17,70,4,0,3,8,73,0,2,52,54] >;
C2×C6×C13⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_6\times C_{13}\rtimes C_3 % in TeX
G:=Group("C2xC6xC13:C3"); // GroupNames label
G:=SmallGroup(468,47);
// by ID
G=gap.SmallGroup(468,47);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-13,689]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^13=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations