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G = C3×C12order 36 = 22·32

Abelian group of type [3,12]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C12, SmallGroup(36,8)

Series: Derived Chief Lower central Upper central

C1 — C3×C12
C1C2C6C3×C6 — C3×C12
C1 — C3×C12
C1 — C3×C12

Generators and relations for C3×C12
 G = < a,b | a3=b12=1, ab=ba >


Smallest permutation representation of C3×C12
Regular action on 36 points
Generators in S36
(1 23 33)(2 24 34)(3 13 35)(4 14 36)(5 15 25)(6 16 26)(7 17 27)(8 18 28)(9 19 29)(10 20 30)(11 21 31)(12 22 32)
(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)

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

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

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

36 conjugacy classes

class 1  2 3A···3H4A4B6A···6H12A···12P
order123···3446···612···12
size111···1111···11···1

36 irreducible representations

dim111111
type++
imageC1C2C3C4C6C12
kernelC3×C12C3×C6C12C32C6C3
# reps1182816

Matrix representation of C3×C12 in GL2(𝔽13) generated by

90
09
,
70
02
G:=sub<GL(2,GF(13))| [9,0,0,9],[7,0,0,2] >;

C3×C12 in GAP, Magma, Sage, TeX

C_3\times C_{12}
% in TeX

G:=Group("C3xC12");
// GroupNames label

G:=SmallGroup(36,8);
// by ID

G=gap.SmallGroup(36,8);
# by ID

G:=PCGroup([4,-2,-3,-3,-2,72]);
// Polycyclic

G:=Group<a,b|a^3=b^12=1,a*b=b*a>;
// generators/relations

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