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## G = S3×C72order 294 = 2·3·72

### Direct product of C72 and S3

Aliases: S3×C72, C213C14, C3⋊(C7×C14), (C7×C21)⋊5C2, SmallGroup(294,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C72
 Chief series C1 — C3 — C21 — C7×C21 — S3×C72
 Lower central C3 — S3×C72
 Upper central C1 — C72

Generators and relations for S3×C72
G = < a,b,c,d | a7=b7=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of S3×C72
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 37 137 23 96 8 59)(2 38 138 24 97 9 60)(3 39 139 25 98 10 61)(4 40 140 26 92 11 62)(5 41 134 27 93 12 63)(6 42 135 28 94 13 57)(7 36 136 22 95 14 58)(15 115 99 74 147 30 89)(16 116 100 75 141 31 90)(17 117 101 76 142 32 91)(18 118 102 77 143 33 85)(19 119 103 71 144 34 86)(20 113 104 72 145 35 87)(21 114 105 73 146 29 88)(43 123 82 107 131 67 51)(44 124 83 108 132 68 52)(45 125 84 109 133 69 53)(46 126 78 110 127 70 54)(47 120 79 111 128 64 55)(48 121 80 112 129 65 56)(49 122 81 106 130 66 50)
(1 55 144)(2 56 145)(3 50 146)(4 51 147)(5 52 141)(6 53 142)(7 54 143)(8 128 103)(9 129 104)(10 130 105)(11 131 99)(12 132 100)(13 133 101)(14 127 102)(15 26 82)(16 27 83)(17 28 84)(18 22 78)(19 23 79)(20 24 80)(21 25 81)(29 39 49)(30 40 43)(31 41 44)(32 42 45)(33 36 46)(34 37 47)(35 38 48)(57 69 76)(58 70 77)(59 64 71)(60 65 72)(61 66 73)(62 67 74)(63 68 75)(85 136 126)(86 137 120)(87 138 121)(88 139 122)(89 140 123)(90 134 124)(91 135 125)(92 107 115)(93 108 116)(94 109 117)(95 110 118)(96 111 119)(97 112 113)(98 106 114)
(15 82)(16 83)(17 84)(18 78)(19 79)(20 80)(21 81)(29 49)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(50 146)(51 147)(52 141)(53 142)(54 143)(55 144)(56 145)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)(85 126)(86 120)(87 121)(88 122)(89 123)(90 124)(91 125)(99 131)(100 132)(101 133)(102 127)(103 128)(104 129)(105 130)(106 114)(107 115)(108 116)(109 117)(110 118)(111 119)(112 113)

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

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

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

147 conjugacy classes

 class 1 2 3 7A ··· 7AV 14A ··· 14AV 21A ··· 21AV order 1 2 3 7 ··· 7 14 ··· 14 21 ··· 21 size 1 3 2 1 ··· 1 3 ··· 3 2 ··· 2

147 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C7 C14 S3 S3×C7 kernel S3×C72 C7×C21 S3×C7 C21 C72 C7 # reps 1 1 48 48 1 48

Matrix representation of S3×C72 in GL3(𝔽43) generated by

 21 0 0 0 35 0 0 0 35
,
 4 0 0 0 4 0 0 0 4
,
 1 0 0 0 0 42 0 1 42
,
 42 0 0 0 1 42 0 0 42
G:=sub<GL(3,GF(43))| [21,0,0,0,35,0,0,0,35],[4,0,0,0,4,0,0,0,4],[1,0,0,0,0,1,0,42,42],[42,0,0,0,1,0,0,42,42] >;

S3×C72 in GAP, Magma, Sage, TeX

S_3\times C_7^2
% in TeX

G:=Group("S3xC7^2");
// GroupNames label

G:=SmallGroup(294,20);
// by ID

G=gap.SmallGroup(294,20);
# by ID

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

G:=Group<a,b,c,d|a^7=b^7=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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