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G = S3×C69order 414 = 2·32·23

Direct product of C69 and S3

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

Aliases: S3×C69, C3⋊C138, C693C6, C321C46, (C3×C69)⋊4C2, SmallGroup(414,6)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C69
C1C3C69C3×C69 — S3×C69
C3 — S3×C69
C1C69

Generators and relations for S3×C69
 G = < a,b,c | a69=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
2C3
3C6
3C46
2C69
3C138

Smallest permutation representation of S3×C69
On 138 points
Generators in S138
(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)
(1 47 24)(2 48 25)(3 49 26)(4 50 27)(5 51 28)(6 52 29)(7 53 30)(8 54 31)(9 55 32)(10 56 33)(11 57 34)(12 58 35)(13 59 36)(14 60 37)(15 61 38)(16 62 39)(17 63 40)(18 64 41)(19 65 42)(20 66 43)(21 67 44)(22 68 45)(23 69 46)(70 93 116)(71 94 117)(72 95 118)(73 96 119)(74 97 120)(75 98 121)(76 99 122)(77 100 123)(78 101 124)(79 102 125)(80 103 126)(81 104 127)(82 105 128)(83 106 129)(84 107 130)(85 108 131)(86 109 132)(87 110 133)(88 111 134)(89 112 135)(90 113 136)(91 114 137)(92 115 138)
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 97)(14 98)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 113)(30 114)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 121)(38 122)(39 123)(40 124)(41 125)(42 126)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(49 133)(50 134)(51 135)(52 136)(53 137)(54 138)(55 70)(56 71)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)

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

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

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

207 conjugacy classes

class 1  2 3A3B3C3D3E6A6B23A···23V46A···46V69A···69AR69AS···69DF138A···138AR
order12333336623···2346···4669···6969···69138···138
size1311222331···13···31···12···23···3

207 irreducible representations

dim111111112222
type+++
imageC1C2C3C6C23C46C69C138S3C3×S3S3×C23S3×C69
kernelS3×C69C3×C69S3×C23C69C3×S3C32S3C3C69C23C3C1
# reps112222224444122244

Matrix representation of S3×C69 in GL2(𝔽139) generated by

540
054
,
9659
042
,
115111
9524
G:=sub<GL(2,GF(139))| [54,0,0,54],[96,0,59,42],[115,95,111,24] >;

S3×C69 in GAP, Magma, Sage, TeX

S_3\times C_{69}
% in TeX

G:=Group("S3xC69");
// GroupNames label

G:=SmallGroup(414,6);
// by ID

G=gap.SmallGroup(414,6);
# by ID

G:=PCGroup([4,-2,-3,-23,-3,4419]);
// Polycyclic

G:=Group<a,b,c|a^69=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of S3×C69 in TeX

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