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## G = C3×C12order 36 = 22·32

### Abelian group of type [3,12]

Aliases: C3×C12, SmallGroup(36,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12
 Chief series C1 — C2 — C6 — C3×C6 — C3×C12
 Lower central C1 — C3×C12
 Upper central 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 16 27)(2 17 28)(3 18 29)(4 19 30)(5 20 31)(6 21 32)(7 22 33)(8 23 34)(9 24 35)(10 13 36)(11 14 25)(12 15 26)
(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)| (1,16,27)(2,17,28)(3,18,29)(4,19,30)(5,20,31)(6,21,32)(7,22,33)(8,23,34)(9,24,35)(10,13,36)(11,14,25)(12,15,26), (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( (1,16,27)(2,17,28)(3,18,29)(4,19,30)(5,20,31)(6,21,32)(7,22,33)(8,23,34)(9,24,35)(10,13,36)(11,14,25)(12,15,26), (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([[(1,16,27),(2,17,28),(3,18,29),(4,19,30),(5,20,31),(6,21,32),(7,22,33),(8,23,34),(9,24,35),(10,13,36),(11,14,25),(12,15,26)], [(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)]])

C3×C12 is a maximal subgroup of   C324C8  C324Q8  C12⋊S3

36 conjugacy classes

 class 1 2 3A ··· 3H 4A 4B 6A ··· 6H 12A ··· 12P order 1 2 3 ··· 3 4 4 6 ··· 6 12 ··· 12 size 1 1 1 ··· 1 1 1 1 ··· 1 1 ··· 1

36 irreducible representations

 dim 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C6 C12 kernel C3×C12 C3×C6 C12 C32 C6 C3 # reps 1 1 8 2 8 16

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

 9 0 0 9
,
 7 0 0 2
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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