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G = C6×D39order 468 = 22·32·13

Direct product of C6 and D39

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C6×D39, C785C6, C782S3, C397D6, C325D26, C6⋊(C3×D13), C135(S3×C6), C263(C3×S3), C397(C2×C6), (C3×C78)⋊2C2, (C3×C6)⋊1D13, C32(C6×D13), (C3×C39)⋊7C22, SmallGroup(468,52)

Series: Derived Chief Lower central Upper central

C1C39 — C6×D39
C1C13C39C3×C39C3×D39 — C6×D39
C39 — C6×D39
C1C6

Generators and relations for C6×D39
 G = < a,b,c | a6=b39=c2=1, ab=ba, ac=ca, cbc=b-1 >

39C2
39C2
2C3
39C22
2C6
13S3
13S3
39C6
39C6
3D13
3D13
2C39
13D6
39C2×C6
13C3×S3
13C3×S3
3D26
2C78
3C3×D13
3C3×D13
13S3×C6
3C6×D13

Smallest permutation representation of C6×D39
On 156 points
Generators in S156
(1 41 27 67 14 54)(2 42 28 68 15 55)(3 43 29 69 16 56)(4 44 30 70 17 57)(5 45 31 71 18 58)(6 46 32 72 19 59)(7 47 33 73 20 60)(8 48 34 74 21 61)(9 49 35 75 22 62)(10 50 36 76 23 63)(11 51 37 77 24 64)(12 52 38 78 25 65)(13 53 39 40 26 66)(79 136 92 149 105 123)(80 137 93 150 106 124)(81 138 94 151 107 125)(82 139 95 152 108 126)(83 140 96 153 109 127)(84 141 97 154 110 128)(85 142 98 155 111 129)(86 143 99 156 112 130)(87 144 100 118 113 131)(88 145 101 119 114 132)(89 146 102 120 115 133)(90 147 103 121 116 134)(91 148 104 122 117 135)
(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)
(1 148)(2 147)(3 146)(4 145)(5 144)(6 143)(7 142)(8 141)(9 140)(10 139)(11 138)(12 137)(13 136)(14 135)(15 134)(16 133)(17 132)(18 131)(19 130)(20 129)(21 128)(22 127)(23 126)(24 125)(25 124)(26 123)(27 122)(28 121)(29 120)(30 119)(31 118)(32 156)(33 155)(34 154)(35 153)(36 152)(37 151)(38 150)(39 149)(40 105)(41 104)(42 103)(43 102)(44 101)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)(61 84)(62 83)(63 82)(64 81)(65 80)(66 79)(67 117)(68 116)(69 115)(70 114)(71 113)(72 112)(73 111)(74 110)(75 109)(76 108)(77 107)(78 106)

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

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

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

126 conjugacy classes

class 1 2A2B2C3A3B3C3D3E6A6B6C6D6E6F6G6H6I13A···13F26A···26F39A···39AV78A···78AV
order12223333366666666613···1326···2639···3978···78
size1139391122211222393939392···22···22···22···2

126 irreducible representations

dim111111222222222222
type+++++++++
imageC1C2C2C3C6C6S3D6C3×S3D13S3×C6D26C3×D13D39C6×D13D78C3×D39C6×D39
kernelC6×D39C3×D39C3×C78D78D39C78C78C39C26C3×C6C13C32C6C6C3C3C2C1
# reps121242112626121212122424

Matrix representation of C6×D39 in GL3(𝔽79) generated by

2400
0230
0023
,
100
07223
0045
,
100
06753
04512
G:=sub<GL(3,GF(79))| [24,0,0,0,23,0,0,0,23],[1,0,0,0,72,0,0,23,45],[1,0,0,0,67,45,0,53,12] >;

C6×D39 in GAP, Magma, Sage, TeX

C_6\times D_{39}
% in TeX

G:=Group("C6xD39");
// GroupNames label

G:=SmallGroup(468,52);
// by ID

G=gap.SmallGroup(468,52);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-13,483,10804]);
// Polycyclic

G:=Group<a,b,c|a^6=b^39=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C6×D39 in TeX

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