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G = C3×D47order 282 = 2·3·47

Direct product of C3 and D47

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

Aliases: C3×D47, C47⋊C6, C1412C2, SmallGroup(282,2)

Series: Derived Chief Lower central Upper central

C1C47 — C3×D47
C1C47C141 — C3×D47
C47 — C3×D47
C1C3

Generators and relations for C3×D47
 G = < a,b,c | a3=b47=c2=1, ab=ba, ac=ca, cbc=b-1 >

47C2
47C6

Smallest permutation representation of C3×D47
On 141 points
Generators in S141
(1 124 72)(2 125 73)(3 126 74)(4 127 75)(5 128 76)(6 129 77)(7 130 78)(8 131 79)(9 132 80)(10 133 81)(11 134 82)(12 135 83)(13 136 84)(14 137 85)(15 138 86)(16 139 87)(17 140 88)(18 141 89)(19 95 90)(20 96 91)(21 97 92)(22 98 93)(23 99 94)(24 100 48)(25 101 49)(26 102 50)(27 103 51)(28 104 52)(29 105 53)(30 106 54)(31 107 55)(32 108 56)(33 109 57)(34 110 58)(35 111 59)(36 112 60)(37 113 61)(38 114 62)(39 115 63)(40 116 64)(41 117 65)(42 118 66)(43 119 67)(44 120 68)(45 121 69)(46 122 70)(47 123 71)
(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)
(1 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(49 94)(50 93)(51 92)(52 91)(53 90)(54 89)(55 88)(56 87)(57 86)(58 85)(59 84)(60 83)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(95 105)(96 104)(97 103)(98 102)(99 101)(106 141)(107 140)(108 139)(109 138)(110 137)(111 136)(112 135)(113 134)(114 133)(115 132)(116 131)(117 130)(118 129)(119 128)(120 127)(121 126)(122 125)(123 124)

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

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

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

75 conjugacy classes

class 1  2 3A3B6A6B47A···47W141A···141AT
order12336647···47141···141
size1471147472···22···2

75 irreducible representations

dim111122
type+++
imageC1C2C3C6D47C3×D47
kernelC3×D47C141D47C47C3C1
# reps11222346

Matrix representation of C3×D47 in GL3(𝔽283) generated by

23800
010
001
,
100
02571
014916
,
28200
0179184
0195104
G:=sub<GL(3,GF(283))| [238,0,0,0,1,0,0,0,1],[1,0,0,0,257,149,0,1,16],[282,0,0,0,179,195,0,184,104] >;

C3×D47 in GAP, Magma, Sage, TeX

C_3\times D_{47}
% in TeX

G:=Group("C3xD47");
// GroupNames label

G:=SmallGroup(282,2);
// by ID

G=gap.SmallGroup(282,2);
# by ID

G:=PCGroup([3,-2,-3,-47,2486]);
// Polycyclic

G:=Group<a,b,c|a^3=b^47=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D47 in TeX

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