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G = C3×C15order 45 = 32·5

Abelian group of type [3,15]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C15, SmallGroup(45,2)

Series: Derived Chief Lower central Upper central

C1 — C3×C15
C1C5C15 — C3×C15
C1 — C3×C15
C1 — C3×C15

Generators and relations for C3×C15
 G = < a,b | a3=b15=1, ab=ba >


Smallest permutation representation of C3×C15
Regular action on 45 points
Generators in S45
(1 38 29)(2 39 30)(3 40 16)(4 41 17)(5 42 18)(6 43 19)(7 44 20)(8 45 21)(9 31 22)(10 32 23)(11 33 24)(12 34 25)(13 35 26)(14 36 27)(15 37 28)
(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)

G:=sub<Sym(45)| (1,38,29)(2,39,30)(3,40,16)(4,41,17)(5,42,18)(6,43,19)(7,44,20)(8,45,21)(9,31,22)(10,32,23)(11,33,24)(12,34,25)(13,35,26)(14,36,27)(15,37,28), (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)>;

G:=Group( (1,38,29)(2,39,30)(3,40,16)(4,41,17)(5,42,18)(6,43,19)(7,44,20)(8,45,21)(9,31,22)(10,32,23)(11,33,24)(12,34,25)(13,35,26)(14,36,27)(15,37,28), (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) );

G=PermutationGroup([(1,38,29),(2,39,30),(3,40,16),(4,41,17),(5,42,18),(6,43,19),(7,44,20),(8,45,21),(9,31,22),(10,32,23),(11,33,24),(12,34,25),(13,35,26),(14,36,27),(15,37,28)], [(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)])

45 conjugacy classes

class 1 3A···3H5A5B5C5D15A···15AF
order13···3555515···15
size11···111111···1

45 irreducible representations

dim1111
type+
imageC1C3C5C15
kernelC3×C15C15C32C3
# reps18432

Matrix representation of C3×C15 in GL2(𝔽31) generated by

50
05
,
100
07
G:=sub<GL(2,GF(31))| [5,0,0,5],[10,0,0,7] >;

C3×C15 in GAP, Magma, Sage, TeX

C_3\times C_{15}
% in TeX

G:=Group("C3xC15");
// GroupNames label

G:=SmallGroup(45,2);
// by ID

G=gap.SmallGroup(45,2);
# by ID

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

G:=Group<a,b|a^3=b^15=1,a*b=b*a>;
// generators/relations

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