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G = S3×C78order 468 = 22·32·13

Direct product of C78 and S3

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

Aliases: S3×C78, C6⋊C78, C787C6, C3⋊(C2×C78), (C3×C6)⋊1C26, (C3×C78)⋊4C2, C399(C2×C6), (C3×C39)⋊9C22, C322(C2×C26), SmallGroup(468,51)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C78
C1C3C39C3×C39S3×C39 — S3×C78
C3 — S3×C78
C1C78

Generators and relations for S3×C78
 G = < a,b,c | a78=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
2C3
3C22
2C6
3C6
3C6
3C26
3C26
2C39
3C2×C6
3C2×C26
2C78
3C78
3C78
3C2×C78

Smallest permutation representation of S3×C78
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 27 53)(2 28 54)(3 29 55)(4 30 56)(5 31 57)(6 32 58)(7 33 59)(8 34 60)(9 35 61)(10 36 62)(11 37 63)(12 38 64)(13 39 65)(14 40 66)(15 41 67)(16 42 68)(17 43 69)(18 44 70)(19 45 71)(20 46 72)(21 47 73)(22 48 74)(23 49 75)(24 50 76)(25 51 77)(26 52 78)(79 131 105)(80 132 106)(81 133 107)(82 134 108)(83 135 109)(84 136 110)(85 137 111)(86 138 112)(87 139 113)(88 140 114)(89 141 115)(90 142 116)(91 143 117)(92 144 118)(93 145 119)(94 146 120)(95 147 121)(96 148 122)(97 149 123)(98 150 124)(99 151 125)(100 152 126)(101 153 127)(102 154 128)(103 155 129)(104 156 130)
(1 135)(2 136)(3 137)(4 138)(5 139)(6 140)(7 141)(8 142)(9 143)(10 144)(11 145)(12 146)(13 147)(14 148)(15 149)(16 150)(17 151)(18 152)(19 153)(20 154)(21 155)(22 156)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(61 117)(62 118)(63 119)(64 120)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 127)(72 128)(73 129)(74 130)(75 131)(76 132)(77 133)(78 134)

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

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

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

234 conjugacy classes

class 1 2A2B2C3A3B3C3D3E6A6B6C6D6E6F6G6H6I13A···13L26A···26L26M···26AJ39A···39X39Y···39BH78A···78X78Y···78BH78BI···78DD
order12223333366666666613···1326···2626···2639···3939···3978···7878···7878···78
size1133112221122233331···11···13···31···12···21···12···23···3

234 irreducible representations

dim11111111111122222222
type+++++
imageC1C2C2C3C6C6C13C26C26C39C78C78S3D6C3×S3S3×C6S3×C13S3×C26S3×C39S3×C78
kernelS3×C78S3×C39C3×C78S3×C26S3×C13C78S3×C6C3×S3C3×C6D6S3C6C78C39C26C13C6C3C2C1
# reps121242122412244824112212122424

Matrix representation of S3×C78 in GL3(𝔽79) generated by

5600
0110
0011
,
100
0550
02923
,
7800
06422
07615
G:=sub<GL(3,GF(79))| [56,0,0,0,11,0,0,0,11],[1,0,0,0,55,29,0,0,23],[78,0,0,0,64,76,0,22,15] >;

S3×C78 in GAP, Magma, Sage, TeX

S_3\times C_{78}
% in TeX

G:=Group("S3xC78");
// GroupNames label

G:=SmallGroup(468,51);
// by ID

G=gap.SmallGroup(468,51);
# by ID

G:=PCGroup([5,-2,-2,-3,-13,-3,7804]);
// Polycyclic

G:=Group<a,b,c|a^78=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×C78 in TeX

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