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G = S3×C49order 294 = 2·3·72

Direct product of C49 and S3

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

Aliases: S3×C49, C3⋊C98, C1473C2, C21.C14, C7.(S3×C7), (S3×C7).C7, SmallGroup(294,3)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C49
C1C3C21C147 — S3×C49
C3 — S3×C49
C1C49

Generators and relations for S3×C49
 G = < a,b,c | a49=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C14
3C98

Smallest permutation representation of S3×C49
On 147 points
Generators in S147
(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)
(1 128 91)(2 129 92)(3 130 93)(4 131 94)(5 132 95)(6 133 96)(7 134 97)(8 135 98)(9 136 50)(10 137 51)(11 138 52)(12 139 53)(13 140 54)(14 141 55)(15 142 56)(16 143 57)(17 144 58)(18 145 59)(19 146 60)(20 147 61)(21 99 62)(22 100 63)(23 101 64)(24 102 65)(25 103 66)(26 104 67)(27 105 68)(28 106 69)(29 107 70)(30 108 71)(31 109 72)(32 110 73)(33 111 74)(34 112 75)(35 113 76)(36 114 77)(37 115 78)(38 116 79)(39 117 80)(40 118 81)(41 119 82)(42 120 83)(43 121 84)(44 122 85)(45 123 86)(46 124 87)(47 125 88)(48 126 89)(49 127 90)
(50 136)(51 137)(52 138)(53 139)(54 140)(55 141)(56 142)(57 143)(58 144)(59 145)(60 146)(61 147)(62 99)(63 100)(64 101)(65 102)(66 103)(67 104)(68 105)(69 106)(70 107)(71 108)(72 109)(73 110)(74 111)(75 112)(76 113)(77 114)(78 115)(79 116)(80 117)(81 118)(82 119)(83 120)(84 121)(85 122)(86 123)(87 124)(88 125)(89 126)(90 127)(91 128)(92 129)(93 130)(94 131)(95 132)(96 133)(97 134)(98 135)

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

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

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

147 conjugacy classes

class 1  2  3 7A···7F14A···14F21A···21F49A···49AP98A···98AP147A···147AP
order1237···714···1421···2149···4998···98147···147
size1321···13···32···21···13···32···2

147 irreducible representations

dim111111222
type+++
imageC1C2C7C14C49C98S3S3×C7S3×C49
kernelS3×C49C147S3×C7C21S3C3C49C7C1
# reps116642421642

Matrix representation of S3×C49 in GL2(𝔽883) generated by

520
052
,
882882
10
,
01
10
G:=sub<GL(2,GF(883))| [52,0,0,52],[882,1,882,0],[0,1,1,0] >;

S3×C49 in GAP, Magma, Sage, TeX

S_3\times C_{49}
% in TeX

G:=Group("S3xC49");
// GroupNames label

G:=SmallGroup(294,3);
// by ID

G=gap.SmallGroup(294,3);
# by ID

G:=PCGroup([4,-2,-7,-7,-3,61,3139]);
// Polycyclic

G:=Group<a,b,c|a^49=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×C49 in TeX

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