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G = C3×D49order 294 = 2·3·72

Direct product of C3 and D49

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

Aliases: C3×D49, C493C6, C1472C2, C21.2D7, C7.(C3×D7), SmallGroup(294,4)

Series: Derived Chief Lower central Upper central

C1C49 — C3×D49
C1C7C49C147 — C3×D49
C49 — C3×D49
C1C3

Generators and relations for C3×D49
 G = < a,b,c | a3=b49=c2=1, ab=ba, ac=ca, cbc=b-1 >

49C2
49C6
7D7
7C3×D7

Smallest permutation representation of C3×D49
On 147 points
Generators in S147
(1 131 88)(2 132 89)(3 133 90)(4 134 91)(5 135 92)(6 136 93)(7 137 94)(8 138 95)(9 139 96)(10 140 97)(11 141 98)(12 142 50)(13 143 51)(14 144 52)(15 145 53)(16 146 54)(17 147 55)(18 99 56)(19 100 57)(20 101 58)(21 102 59)(22 103 60)(23 104 61)(24 105 62)(25 106 63)(26 107 64)(27 108 65)(28 109 66)(29 110 67)(30 111 68)(31 112 69)(32 113 70)(33 114 71)(34 115 72)(35 116 73)(36 117 74)(37 118 75)(38 119 76)(39 120 77)(40 121 78)(41 122 79)(42 123 80)(43 124 81)(44 125 82)(45 126 83)(46 127 84)(47 128 85)(48 129 86)(49 130 87)
(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 49)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)(50 76)(51 75)(52 74)(53 73)(54 72)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(77 98)(78 97)(79 96)(80 95)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)(99 113)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)(114 147)(115 146)(116 145)(117 144)(118 143)(119 142)(120 141)(121 140)(122 139)(123 138)(124 137)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)

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

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

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

78 conjugacy classes

class 1  2 3A3B6A6B7A7B7C21A···21F49A···49U147A···147AP
order12336677721···2149···49147···147
size1491149492222···22···22···2

78 irreducible representations

dim11112222
type++++
imageC1C2C3C6D7C3×D7D49C3×D49
kernelC3×D49C147D49C49C21C7C3C1
# reps1122362142

Matrix representation of C3×D49 in GL3(𝔽883) generated by

33700
010
001
,
100
0784406
071351
,
88200
0130337
0618753
G:=sub<GL(3,GF(883))| [337,0,0,0,1,0,0,0,1],[1,0,0,0,784,71,0,406,351],[882,0,0,0,130,618,0,337,753] >;

C3×D49 in GAP, Magma, Sage, TeX

C_3\times D_{49}
% in TeX

G:=Group("C3xD49");
// GroupNames label

G:=SmallGroup(294,4);
// by ID

G=gap.SmallGroup(294,4);
# by ID

G:=PCGroup([4,-2,-3,-7,-7,938,514,4035]);
// Polycyclic

G:=Group<a,b,c|a^3=b^49=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D49 in TeX

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