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G = C3×C15order 45 = 32·5

Abelian group of type [3,15]

Aliases: C3×C15, SmallGroup(45,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15
 Chief series C1 — C5 — C15 — C3×C15
 Lower central C1 — C3×C15
 Upper central 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 16)(2 39 17)(3 40 18)(4 41 19)(5 42 20)(6 43 21)(7 44 22)(8 45 23)(9 31 24)(10 32 25)(11 33 26)(12 34 27)(13 35 28)(14 36 29)(15 37 30)
(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)| (1,38,16)(2,39,17)(3,40,18)(4,41,19)(5,42,20)(6,43,21)(7,44,22)(8,45,23)(9,31,24)(10,32,25)(11,33,26)(12,34,27)(13,35,28)(14,36,29)(15,37,30), (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( (1,38,16)(2,39,17)(3,40,18)(4,41,19)(5,42,20)(6,43,21)(7,44,22)(8,45,23)(9,31,24)(10,32,25)(11,33,26)(12,34,27)(13,35,28)(14,36,29)(15,37,30), (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([[(1,38,16),(2,39,17),(3,40,18),(4,41,19),(5,42,20),(6,43,21),(7,44,22),(8,45,23),(9,31,24),(10,32,25),(11,33,26),(12,34,27),(13,35,28),(14,36,29),(15,37,30)], [(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)]])

C3×C15 is a maximal subgroup of   C3⋊D15

45 conjugacy classes

 class 1 3A ··· 3H 5A 5B 5C 5D 15A ··· 15AF order 1 3 ··· 3 5 5 5 5 15 ··· 15 size 1 1 ··· 1 1 1 1 1 1 ··· 1

45 irreducible representations

 dim 1 1 1 1 type + image C1 C3 C5 C15 kernel C3×C15 C15 C32 C3 # reps 1 8 4 32

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

 5 0 0 5
,
 10 0 0 7
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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