direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C13×C3⋊D4, C39⋊9D4, D6⋊2C26, Dic3⋊C26, C26.17D6, C78.22C22, C3⋊2(D4×C13), (C2×C26)⋊3S3, (C2×C6)⋊2C26, (C2×C78)⋊6C2, (S3×C26)⋊5C2, C2.5(S3×C26), C6.5(C2×C26), C22⋊2(S3×C13), (Dic3×C13)⋊4C2, SmallGroup(312,36)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C3⋊D4
G = < a,b,c,d | a13=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Smallest permutation representation of C13×C3⋊D4
►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 63 28)(2 64 29)(3 65 30)(4 53 31)(5 54 32)(6 55 33)(7 56 34)(8 57 35)(9 58 36)(10 59 37)(11 60 38)(12 61 39)(13 62 27)(14 147 120)(15 148 121)(16 149 122)(17 150 123)(18 151 124)(19 152 125)(20 153 126)(21 154 127)(22 155 128)(23 156 129)(24 144 130)(25 145 118)(26 146 119)(40 75 132)(41 76 133)(42 77 134)(43 78 135)(44 66 136)(45 67 137)(46 68 138)(47 69 139)(48 70 140)(49 71 141)(50 72 142)(51 73 143)(52 74 131)(79 107 96)(80 108 97)(81 109 98)(82 110 99)(83 111 100)(84 112 101)(85 113 102)(86 114 103)(87 115 104)(88 116 92)(89 117 93)(90 105 94)(91 106 95)
(1 146 131 80)(2 147 132 81)(3 148 133 82)(4 149 134 83)(5 150 135 84)(6 151 136 85)(7 152 137 86)(8 153 138 87)(9 154 139 88)(10 155 140 89)(11 156 141 90)(12 144 142 91)(13 145 143 79)(14 40 98 64)(15 41 99 65)(16 42 100 53)(17 43 101 54)(18 44 102 55)(19 45 103 56)(20 46 104 57)(21 47 92 58)(22 48 93 59)(23 49 94 60)(24 50 95 61)(25 51 96 62)(26 52 97 63)(27 118 73 107)(28 119 74 108)(29 120 75 109)(30 121 76 110)(31 122 77 111)(32 123 78 112)(33 124 66 113)(34 125 67 114)(35 126 68 115)(36 127 69 116)(37 128 70 117)(38 129 71 105)(39 130 72 106)
(14 109)(15 110)(16 111)(17 112)(18 113)(19 114)(20 115)(21 116)(22 117)(23 105)(24 106)(25 107)(26 108)(27 62)(28 63)(29 64)(30 65)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 61)(40 75)(41 76)(42 77)(43 78)(44 66)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(79 145)(80 146)(81 147)(82 148)(83 149)(84 150)(85 151)(86 152)(87 153)(88 154)(89 155)(90 156)(91 144)(92 127)(93 128)(94 129)(95 130)(96 118)(97 119)(98 120)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)
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), (1,63,28)(2,64,29)(3,65,30)(4,53,31)(5,54,32)(6,55,33)(7,56,34)(8,57,35)(9,58,36)(10,59,37)(11,60,38)(12,61,39)(13,62,27)(14,147,120)(15,148,121)(16,149,122)(17,150,123)(18,151,124)(19,152,125)(20,153,126)(21,154,127)(22,155,128)(23,156,129)(24,144,130)(25,145,118)(26,146,119)(40,75,132)(41,76,133)(42,77,134)(43,78,135)(44,66,136)(45,67,137)(46,68,138)(47,69,139)(48,70,140)(49,71,141)(50,72,142)(51,73,143)(52,74,131)(79,107,96)(80,108,97)(81,109,98)(82,110,99)(83,111,100)(84,112,101)(85,113,102)(86,114,103)(87,115,104)(88,116,92)(89,117,93)(90,105,94)(91,106,95), (1,146,131,80)(2,147,132,81)(3,148,133,82)(4,149,134,83)(5,150,135,84)(6,151,136,85)(7,152,137,86)(8,153,138,87)(9,154,139,88)(10,155,140,89)(11,156,141,90)(12,144,142,91)(13,145,143,79)(14,40,98,64)(15,41,99,65)(16,42,100,53)(17,43,101,54)(18,44,102,55)(19,45,103,56)(20,46,104,57)(21,47,92,58)(22,48,93,59)(23,49,94,60)(24,50,95,61)(25,51,96,62)(26,52,97,63)(27,118,73,107)(28,119,74,108)(29,120,75,109)(30,121,76,110)(31,122,77,111)(32,123,78,112)(33,124,66,113)(34,125,67,114)(35,126,68,115)(36,127,69,116)(37,128,70,117)(38,129,71,105)(39,130,72,106), (14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,105)(24,106)(25,107)(26,108)(27,62)(28,63)(29,64)(30,65)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,75)(41,76)(42,77)(43,78)(44,66)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(79,145)(80,146)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156)(91,144)(92,127)(93,128)(94,129)(95,130)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)>;
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), (1,63,28)(2,64,29)(3,65,30)(4,53,31)(5,54,32)(6,55,33)(7,56,34)(8,57,35)(9,58,36)(10,59,37)(11,60,38)(12,61,39)(13,62,27)(14,147,120)(15,148,121)(16,149,122)(17,150,123)(18,151,124)(19,152,125)(20,153,126)(21,154,127)(22,155,128)(23,156,129)(24,144,130)(25,145,118)(26,146,119)(40,75,132)(41,76,133)(42,77,134)(43,78,135)(44,66,136)(45,67,137)(46,68,138)(47,69,139)(48,70,140)(49,71,141)(50,72,142)(51,73,143)(52,74,131)(79,107,96)(80,108,97)(81,109,98)(82,110,99)(83,111,100)(84,112,101)(85,113,102)(86,114,103)(87,115,104)(88,116,92)(89,117,93)(90,105,94)(91,106,95), (1,146,131,80)(2,147,132,81)(3,148,133,82)(4,149,134,83)(5,150,135,84)(6,151,136,85)(7,152,137,86)(8,153,138,87)(9,154,139,88)(10,155,140,89)(11,156,141,90)(12,144,142,91)(13,145,143,79)(14,40,98,64)(15,41,99,65)(16,42,100,53)(17,43,101,54)(18,44,102,55)(19,45,103,56)(20,46,104,57)(21,47,92,58)(22,48,93,59)(23,49,94,60)(24,50,95,61)(25,51,96,62)(26,52,97,63)(27,118,73,107)(28,119,74,108)(29,120,75,109)(30,121,76,110)(31,122,77,111)(32,123,78,112)(33,124,66,113)(34,125,67,114)(35,126,68,115)(36,127,69,116)(37,128,70,117)(38,129,71,105)(39,130,72,106), (14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,105)(24,106)(25,107)(26,108)(27,62)(28,63)(29,64)(30,65)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,75)(41,76)(42,77)(43,78)(44,66)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(79,145)(80,146)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156)(91,144)(92,127)(93,128)(94,129)(95,130)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126) );
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)], [(1,63,28),(2,64,29),(3,65,30),(4,53,31),(5,54,32),(6,55,33),(7,56,34),(8,57,35),(9,58,36),(10,59,37),(11,60,38),(12,61,39),(13,62,27),(14,147,120),(15,148,121),(16,149,122),(17,150,123),(18,151,124),(19,152,125),(20,153,126),(21,154,127),(22,155,128),(23,156,129),(24,144,130),(25,145,118),(26,146,119),(40,75,132),(41,76,133),(42,77,134),(43,78,135),(44,66,136),(45,67,137),(46,68,138),(47,69,139),(48,70,140),(49,71,141),(50,72,142),(51,73,143),(52,74,131),(79,107,96),(80,108,97),(81,109,98),(82,110,99),(83,111,100),(84,112,101),(85,113,102),(86,114,103),(87,115,104),(88,116,92),(89,117,93),(90,105,94),(91,106,95)], [(1,146,131,80),(2,147,132,81),(3,148,133,82),(4,149,134,83),(5,150,135,84),(6,151,136,85),(7,152,137,86),(8,153,138,87),(9,154,139,88),(10,155,140,89),(11,156,141,90),(12,144,142,91),(13,145,143,79),(14,40,98,64),(15,41,99,65),(16,42,100,53),(17,43,101,54),(18,44,102,55),(19,45,103,56),(20,46,104,57),(21,47,92,58),(22,48,93,59),(23,49,94,60),(24,50,95,61),(25,51,96,62),(26,52,97,63),(27,118,73,107),(28,119,74,108),(29,120,75,109),(30,121,76,110),(31,122,77,111),(32,123,78,112),(33,124,66,113),(34,125,67,114),(35,126,68,115),(36,127,69,116),(37,128,70,117),(38,129,71,105),(39,130,72,106)], [(14,109),(15,110),(16,111),(17,112),(18,113),(19,114),(20,115),(21,116),(22,117),(23,105),(24,106),(25,107),(26,108),(27,62),(28,63),(29,64),(30,65),(31,53),(32,54),(33,55),(34,56),(35,57),(36,58),(37,59),(38,60),(39,61),(40,75),(41,76),(42,77),(43,78),(44,66),(45,67),(46,68),(47,69),(48,70),(49,71),(50,72),(51,73),(52,74),(79,145),(80,146),(81,147),(82,148),(83,149),(84,150),(85,151),(86,152),(87,153),(88,154),(89,155),(90,156),(91,144),(92,127),(93,128),(94,129),(95,130),(96,118),(97,119),(98,120),(99,121),(100,122),(101,123),(102,124),(103,125),(104,126)]])
117 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26X | 26Y | ··· | 26AJ | 39A | ··· | 39L | 52A | ··· | 52L | 78A | ··· | 78AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 |
| size | 1 | 1 | 2 | 6 | 2 | 6 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
117 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C13 | C26 | C26 | C26 | S3 | D4 | D6 | C3⋊D4 | S3×C13 | D4×C13 | S3×C26 | C13×C3⋊D4 |
| kernel | C13×C3⋊D4 | Dic3×C13 | S3×C26 | C2×C78 | C3⋊D4 | Dic3 | D6 | C2×C6 | C2×C26 | C39 | C26 | C13 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 1 | 1 | 1 | 2 | 12 | 12 | 12 | 24 |
Matrix representation of C13×C3⋊D4 ►in GL2(𝔽157) generated by
| 130 | 0 |
| 0 | 130 |
| 156 | 156 |
| 1 | 0 |
| 44 | 88 |
| 44 | 113 |
| 1 | 0 |
| 156 | 156 |
G:=sub<GL(2,GF(157))| [130,0,0,130],[156,1,156,0],[44,44,88,113],[1,156,0,156] >;
C13×C3⋊D4 in GAP, Magma, Sage, TeX
C_{13}\times C_3\rtimes D_4 % in TeX
G:=Group("C13xC3:D4"); // GroupNames label
G:=SmallGroup(312,36);
// by ID
G=gap.SmallGroup(312,36);
# by ID
G:=PCGroup([5,-2,-2,-13,-2,-3,541,5204]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
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