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## G = S3×C49order 294 = 2·3·72

### Direct product of C49 and S3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C49
 Chief series C1 — C3 — C21 — C147 — S3×C49
 Lower central C3 — S3×C49
 Upper central C1 — C49

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

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 ··· 7F 14A ··· 14F 21A ··· 21F 49A ··· 49AP 98A ··· 98AP 147A ··· 147AP order 1 2 3 7 ··· 7 14 ··· 14 21 ··· 21 49 ··· 49 98 ··· 98 147 ··· 147 size 1 3 2 1 ··· 1 3 ··· 3 2 ··· 2 1 ··· 1 3 ··· 3 2 ··· 2

147 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 type + + + image C1 C2 C7 C14 C49 C98 S3 S3×C7 S3×C49 kernel S3×C49 C147 S3×C7 C21 S3 C3 C49 C7 C1 # reps 1 1 6 6 42 42 1 6 42

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

 52 0 0 52
,
 882 882 1 0
,
 0 1 1 0
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

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