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G = C13xD12order 312 = 23·3·13

Direct product of C13 and D12

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C13xD12, C39:6D4, C52:3S3, C156:5C2, C12:1C26, D6:1C26, C26.15D6, C78.20C22, C4:(S3xC13), C3:1(D4xC13), (S3xC26):4C2, C2.4(S3xC26), C6.3(C2xC26), SmallGroup(312,34)

Series: Derived Chief Lower central Upper central

C1C6 — C13xD12
C1C3C6C78S3xC26 — C13xD12
C3C6 — C13xD12
C1C26C52

Generators and relations for C13xD12
 G = < a,b,c | a13=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 68 in 32 conjugacy classes, 18 normal (14 characteristic)
Quotients: C1, C2, C22, S3, D4, D6, C13, D12, C26, C2xC26, S3xC13, D4xC13, S3xC26, C13xD12
6C2
6C2
3C22
3C22
2S3
2S3
6C26
6C26
3D4
3C2xC26
3C2xC26
2S3xC13
2S3xC13
3D4xC13

Smallest permutation representation of C13xD12
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 75 53 42 145 37 84 140 99 25 111 118)(2 76 54 43 146 38 85 141 100 26 112 119)(3 77 55 44 147 39 86 142 101 14 113 120)(4 78 56 45 148 27 87 143 102 15 114 121)(5 66 57 46 149 28 88 131 103 16 115 122)(6 67 58 47 150 29 89 132 104 17 116 123)(7 68 59 48 151 30 90 133 92 18 117 124)(8 69 60 49 152 31 91 134 93 19 105 125)(9 70 61 50 153 32 79 135 94 20 106 126)(10 71 62 51 154 33 80 136 95 21 107 127)(11 72 63 52 155 34 81 137 96 22 108 128)(12 73 64 40 156 35 82 138 97 23 109 129)(13 74 65 41 144 36 83 139 98 24 110 130)
(1 118)(2 119)(3 120)(4 121)(5 122)(6 123)(7 124)(8 125)(9 126)(10 127)(11 128)(12 129)(13 130)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 65)(25 53)(26 54)(27 87)(28 88)(29 89)(30 90)(31 91)(32 79)(33 80)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 97)(41 98)(42 99)(43 100)(44 101)(45 102)(46 103)(47 104)(48 92)(49 93)(50 94)(51 95)(52 96)(66 115)(67 116)(68 117)(69 105)(70 106)(71 107)(72 108)(73 109)(74 110)(75 111)(76 112)(77 113)(78 114)(131 149)(132 150)(133 151)(134 152)(135 153)(136 154)(137 155)(138 156)(139 144)(140 145)(141 146)(142 147)(143 148)

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), (1,75,53,42,145,37,84,140,99,25,111,118)(2,76,54,43,146,38,85,141,100,26,112,119)(3,77,55,44,147,39,86,142,101,14,113,120)(4,78,56,45,148,27,87,143,102,15,114,121)(5,66,57,46,149,28,88,131,103,16,115,122)(6,67,58,47,150,29,89,132,104,17,116,123)(7,68,59,48,151,30,90,133,92,18,117,124)(8,69,60,49,152,31,91,134,93,19,105,125)(9,70,61,50,153,32,79,135,94,20,106,126)(10,71,62,51,154,33,80,136,95,21,107,127)(11,72,63,52,155,34,81,137,96,22,108,128)(12,73,64,40,156,35,82,138,97,23,109,129)(13,74,65,41,144,36,83,139,98,24,110,130), (1,118)(2,119)(3,120)(4,121)(5,122)(6,123)(7,124)(8,125)(9,126)(10,127)(11,128)(12,129)(13,130)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,53)(26,54)(27,87)(28,88)(29,89)(30,90)(31,91)(32,79)(33,80)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,97)(41,98)(42,99)(43,100)(44,101)(45,102)(46,103)(47,104)(48,92)(49,93)(50,94)(51,95)(52,96)(66,115)(67,116)(68,117)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(131,149)(132,150)(133,151)(134,152)(135,153)(136,154)(137,155)(138,156)(139,144)(140,145)(141,146)(142,147)(143,148)>;

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), (1,75,53,42,145,37,84,140,99,25,111,118)(2,76,54,43,146,38,85,141,100,26,112,119)(3,77,55,44,147,39,86,142,101,14,113,120)(4,78,56,45,148,27,87,143,102,15,114,121)(5,66,57,46,149,28,88,131,103,16,115,122)(6,67,58,47,150,29,89,132,104,17,116,123)(7,68,59,48,151,30,90,133,92,18,117,124)(8,69,60,49,152,31,91,134,93,19,105,125)(9,70,61,50,153,32,79,135,94,20,106,126)(10,71,62,51,154,33,80,136,95,21,107,127)(11,72,63,52,155,34,81,137,96,22,108,128)(12,73,64,40,156,35,82,138,97,23,109,129)(13,74,65,41,144,36,83,139,98,24,110,130), (1,118)(2,119)(3,120)(4,121)(5,122)(6,123)(7,124)(8,125)(9,126)(10,127)(11,128)(12,129)(13,130)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,53)(26,54)(27,87)(28,88)(29,89)(30,90)(31,91)(32,79)(33,80)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,97)(41,98)(42,99)(43,100)(44,101)(45,102)(46,103)(47,104)(48,92)(49,93)(50,94)(51,95)(52,96)(66,115)(67,116)(68,117)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(131,149)(132,150)(133,151)(134,152)(135,153)(136,154)(137,155)(138,156)(139,144)(140,145)(141,146)(142,147)(143,148) );

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)], [(1,75,53,42,145,37,84,140,99,25,111,118),(2,76,54,43,146,38,85,141,100,26,112,119),(3,77,55,44,147,39,86,142,101,14,113,120),(4,78,56,45,148,27,87,143,102,15,114,121),(5,66,57,46,149,28,88,131,103,16,115,122),(6,67,58,47,150,29,89,132,104,17,116,123),(7,68,59,48,151,30,90,133,92,18,117,124),(8,69,60,49,152,31,91,134,93,19,105,125),(9,70,61,50,153,32,79,135,94,20,106,126),(10,71,62,51,154,33,80,136,95,21,107,127),(11,72,63,52,155,34,81,137,96,22,108,128),(12,73,64,40,156,35,82,138,97,23,109,129),(13,74,65,41,144,36,83,139,98,24,110,130)], [(1,118),(2,119),(3,120),(4,121),(5,122),(6,123),(7,124),(8,125),(9,126),(10,127),(11,128),(12,129),(13,130),(14,55),(15,56),(16,57),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,65),(25,53),(26,54),(27,87),(28,88),(29,89),(30,90),(31,91),(32,79),(33,80),(34,81),(35,82),(36,83),(37,84),(38,85),(39,86),(40,97),(41,98),(42,99),(43,100),(44,101),(45,102),(46,103),(47,104),(48,92),(49,93),(50,94),(51,95),(52,96),(66,115),(67,116),(68,117),(69,105),(70,106),(71,107),(72,108),(73,109),(74,110),(75,111),(76,112),(77,113),(78,114),(131,149),(132,150),(133,151),(134,152),(135,153),(136,154),(137,155),(138,156),(139,144),(140,145),(141,146),(142,147),(143,148)]])

117 conjugacy classes

class 1 2A2B2C 3  4  6 12A12B13A···13L26A···26L26M···26AJ39A···39L52A···52L78A···78L156A···156X
order1222346121213···1326···2626···2639···3952···5278···78156···156
size1166222221···11···16···62···22···22···22···2

117 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C13C26C26S3D4D6D12S3xC13D4xC13S3xC26C13xD12
kernelC13xD12C156S3xC26D12C12D6C52C39C26C13C4C3C2C1
# reps112121224111212121224

Matrix representation of C13xD12 in GL4(F157) generated by

39000
03900
0010
0001
,
15615600
1000
0024133
002448
,
15615600
0100
0024133
00109133
G:=sub<GL(4,GF(157))| [39,0,0,0,0,39,0,0,0,0,1,0,0,0,0,1],[156,1,0,0,156,0,0,0,0,0,24,24,0,0,133,48],[156,0,0,0,156,1,0,0,0,0,24,109,0,0,133,133] >;

C13xD12 in GAP, Magma, Sage, TeX

C_{13}\times D_{12}
% in TeX

G:=Group("C13xD12");
// GroupNames label

G:=SmallGroup(312,34);
// by ID

G=gap.SmallGroup(312,34);
# by ID

G:=PCGroup([5,-2,-2,-13,-2,-3,541,266,5204]);
// Polycyclic

G:=Group<a,b,c|a^13=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C13xD12 in TeX

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