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G = C3×C39order 117 = 32·13

Abelian group of type [3,39]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C39, SmallGroup(117,4)

Series: Derived Chief Lower central Upper central

C1 — C3×C39
C1C13C39 — C3×C39
C1 — C3×C39
C1 — C3×C39

Generators and relations for C3×C39
 G = < a,b | a3=b39=1, ab=ba >


Smallest permutation representation of C3×C39
Regular action on 117 points
Generators in S117
(1 92 49)(2 93 50)(3 94 51)(4 95 52)(5 96 53)(6 97 54)(7 98 55)(8 99 56)(9 100 57)(10 101 58)(11 102 59)(12 103 60)(13 104 61)(14 105 62)(15 106 63)(16 107 64)(17 108 65)(18 109 66)(19 110 67)(20 111 68)(21 112 69)(22 113 70)(23 114 71)(24 115 72)(25 116 73)(26 117 74)(27 79 75)(28 80 76)(29 81 77)(30 82 78)(31 83 40)(32 84 41)(33 85 42)(34 86 43)(35 87 44)(36 88 45)(37 89 46)(38 90 47)(39 91 48)
(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)

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

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

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

C3×C39 is a maximal subgroup of   C3⋊D39  C39.C32  C13⋊He3

117 conjugacy classes

class 1 3A···3H13A···13L39A···39CR
order13···313···1339···39
size11···11···11···1

117 irreducible representations

dim1111
type+
imageC1C3C13C39
kernelC3×C39C39C32C3
# reps181296

Matrix representation of C3×C39 in GL2(𝔽79) generated by

10
055
,
20
064
G:=sub<GL(2,GF(79))| [1,0,0,55],[2,0,0,64] >;

C3×C39 in GAP, Magma, Sage, TeX

C_3\times C_{39}
% in TeX

G:=Group("C3xC39");
// GroupNames label

G:=SmallGroup(117,4);
// by ID

G=gap.SmallGroup(117,4);
# by ID

G:=PCGroup([3,-3,-3,-13]);
// Polycyclic

G:=Group<a,b|a^3=b^39=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C39 in TeX

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