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G = C3×C30order 90 = 2·32·5

Abelian group of type [3,30]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C30, SmallGroup(90,10)

Series: Derived Chief Lower central Upper central

C1 — C3×C30
C1C5C15C3×C15 — C3×C30
C1 — C3×C30
C1 — C3×C30

Generators and relations for C3×C30
 G = < a,b | a3=b30=1, ab=ba >


Smallest permutation representation of C3×C30
Regular action on 90 points
Generators in S90
(1 53 65)(2 54 66)(3 55 67)(4 56 68)(5 57 69)(6 58 70)(7 59 71)(8 60 72)(9 31 73)(10 32 74)(11 33 75)(12 34 76)(13 35 77)(14 36 78)(15 37 79)(16 38 80)(17 39 81)(18 40 82)(19 41 83)(20 42 84)(21 43 85)(22 44 86)(23 45 87)(24 46 88)(25 47 89)(26 48 90)(27 49 61)(28 50 62)(29 51 63)(30 52 64)
(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)

G:=sub<Sym(90)| (1,53,65)(2,54,66)(3,55,67)(4,56,68)(5,57,69)(6,58,70)(7,59,71)(8,60,72)(9,31,73)(10,32,74)(11,33,75)(12,34,76)(13,35,77)(14,36,78)(15,37,79)(16,38,80)(17,39,81)(18,40,82)(19,41,83)(20,42,84)(21,43,85)(22,44,86)(23,45,87)(24,46,88)(25,47,89)(26,48,90)(27,49,61)(28,50,62)(29,51,63)(30,52,64), (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)>;

G:=Group( (1,53,65)(2,54,66)(3,55,67)(4,56,68)(5,57,69)(6,58,70)(7,59,71)(8,60,72)(9,31,73)(10,32,74)(11,33,75)(12,34,76)(13,35,77)(14,36,78)(15,37,79)(16,38,80)(17,39,81)(18,40,82)(19,41,83)(20,42,84)(21,43,85)(22,44,86)(23,45,87)(24,46,88)(25,47,89)(26,48,90)(27,49,61)(28,50,62)(29,51,63)(30,52,64), (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) );

G=PermutationGroup([(1,53,65),(2,54,66),(3,55,67),(4,56,68),(5,57,69),(6,58,70),(7,59,71),(8,60,72),(9,31,73),(10,32,74),(11,33,75),(12,34,76),(13,35,77),(14,36,78),(15,37,79),(16,38,80),(17,39,81),(18,40,82),(19,41,83),(20,42,84),(21,43,85),(22,44,86),(23,45,87),(24,46,88),(25,47,89),(26,48,90),(27,49,61),(28,50,62),(29,51,63),(30,52,64)], [(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)])

C3×C30 is a maximal subgroup of   C3⋊Dic15

90 conjugacy classes

class 1  2 3A···3H5A5B5C5D6A···6H10A10B10C10D15A···15AF30A···30AF
order123···355556···61010101015···1530···30
size111···111111···111111···11···1

90 irreducible representations

dim11111111
type++
imageC1C2C3C5C6C10C15C30
kernelC3×C30C3×C15C30C3×C6C15C32C6C3
# reps1184843232

Matrix representation of C3×C30 in GL2(𝔽31) generated by

50
05
,
290
021
G:=sub<GL(2,GF(31))| [5,0,0,5],[29,0,0,21] >;

C3×C30 in GAP, Magma, Sage, TeX

C_3\times C_{30}
% in TeX

G:=Group("C3xC30");
// GroupNames label

G:=SmallGroup(90,10);
// by ID

G=gap.SmallGroup(90,10);
# by ID

G:=PCGroup([4,-2,-3,-3,-5]);
// Polycyclic

G:=Group<a,b|a^3=b^30=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C30 in TeX

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