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G = C3×C42order 126 = 2·32·7

Abelian group of type [3,42]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C42, SmallGroup(126,16)

Series: Derived Chief Lower central Upper central

C1 — C3×C42
C1C7C21C3×C21 — C3×C42
C1 — C3×C42
C1 — C3×C42

Generators and relations for C3×C42
 G = < a,b | a3=b42=1, ab=ba >


Smallest permutation representation of C3×C42
Regular action on 126 points
Generators in S126
(1 109 82)(2 110 83)(3 111 84)(4 112 43)(5 113 44)(6 114 45)(7 115 46)(8 116 47)(9 117 48)(10 118 49)(11 119 50)(12 120 51)(13 121 52)(14 122 53)(15 123 54)(16 124 55)(17 125 56)(18 126 57)(19 85 58)(20 86 59)(21 87 60)(22 88 61)(23 89 62)(24 90 63)(25 91 64)(26 92 65)(27 93 66)(28 94 67)(29 95 68)(30 96 69)(31 97 70)(32 98 71)(33 99 72)(34 100 73)(35 101 74)(36 102 75)(37 103 76)(38 104 77)(39 105 78)(40 106 79)(41 107 80)(42 108 81)
(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)

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

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

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

C3×C42 is a maximal subgroup of   C3⋊Dic21

126 conjugacy classes

class 1  2 3A···3H6A···6H7A···7F14A···14F21A···21AV42A···42AV
order123···36···67···714···1421···2142···42
size111···11···11···11···11···11···1

126 irreducible representations

dim11111111
type++
imageC1C2C3C6C7C14C21C42
kernelC3×C42C3×C21C42C21C3×C6C32C6C3
# reps1188664848

Matrix representation of C3×C42 in GL2(𝔽43) generated by

10
06
,
200
036
G:=sub<GL(2,GF(43))| [1,0,0,6],[20,0,0,36] >;

C3×C42 in GAP, Magma, Sage, TeX

C_3\times C_{42}
% in TeX

G:=Group("C3xC42");
// GroupNames label

G:=SmallGroup(126,16);
// by ID

G=gap.SmallGroup(126,16);
# by ID

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

G:=Group<a,b|a^3=b^42=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C42 in TeX

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