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G = C4×C12order 48 = 24·3

Abelian group of type [4,12]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C12, SmallGroup(48,20)

Series: Derived Chief Lower central Upper central

C1 — C4×C12
C1C2C22C2×C6C2×C12 — C4×C12
C1 — C4×C12
C1 — C4×C12

Generators and relations for C4×C12
 G = < a,b | a4=b12=1, ab=ba >


Smallest permutation representation of C4×C12
Regular action on 48 points
Generators in S48
(1 42 17 25)(2 43 18 26)(3 44 19 27)(4 45 20 28)(5 46 21 29)(6 47 22 30)(7 48 23 31)(8 37 24 32)(9 38 13 33)(10 39 14 34)(11 40 15 35)(12 41 16 36)
(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)

G:=sub<Sym(48)| (1,42,17,25)(2,43,18,26)(3,44,19,27)(4,45,20,28)(5,46,21,29)(6,47,22,30)(7,48,23,31)(8,37,24,32)(9,38,13,33)(10,39,14,34)(11,40,15,35)(12,41,16,36), (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)>;

G:=Group( (1,42,17,25)(2,43,18,26)(3,44,19,27)(4,45,20,28)(5,46,21,29)(6,47,22,30)(7,48,23,31)(8,37,24,32)(9,38,13,33)(10,39,14,34)(11,40,15,35)(12,41,16,36), (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) );

G=PermutationGroup([(1,42,17,25),(2,43,18,26),(3,44,19,27),(4,45,20,28),(5,46,21,29),(6,47,22,30),(7,48,23,31),(8,37,24,32),(9,38,13,33),(10,39,14,34),(11,40,15,35),(12,41,16,36)], [(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)])

C4×C12 is a maximal subgroup of
C42.S3  C12⋊C8  C424S3  C122Q8  C12.6Q8  C422S3  C4⋊D12  C427S3  C423S3  C42⋊C9

48 conjugacy classes

class 1 2A2B2C3A3B4A···4L6A···6F12A···12X
order1222334···46···612···12
size1111111···11···11···1

48 irreducible representations

dim111111
type++
imageC1C2C3C4C6C12
kernelC4×C12C2×C12C42C12C2×C4C4
# reps13212624

Matrix representation of C4×C12 in GL2(𝔽13) generated by

120
05
,
50
04
G:=sub<GL(2,GF(13))| [12,0,0,5],[5,0,0,4] >;

C4×C12 in GAP, Magma, Sage, TeX

C_4\times C_{12}
% in TeX

G:=Group("C4xC12");
// GroupNames label

G:=SmallGroup(48,20);
// by ID

G=gap.SmallGroup(48,20);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-2,60,126]);
// Polycyclic

G:=Group<a,b|a^4=b^12=1,a*b=b*a>;
// generators/relations

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Subgroup lattice of C4×C12 in TeX

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