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G = C12×A4order 144 = 24·32

Direct product of C12 and A4

direct product, metabelian, soluble, monomial, A-group

Aliases: C12×A4, (C2×C6)⋊2C12, (C22×C12)⋊C3, C22⋊(C3×C12), C6.8(C2×A4), C2.1(C6×A4), C23.(C3×C6), (C22×C4)⋊C32, (C6×A4).4C2, (C2×A4).2C6, (C22×C6).4C6, SmallGroup(144,155)

Series: Derived Chief Lower central Upper central

C1C22 — C12×A4
C1C22C23C22×C6C6×A4 — C12×A4
C22 — C12×A4
C1C12

Generators and relations for C12×A4
 G = < a,b,c,d | a12=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

3C2
3C2
4C3
4C3
4C3
3C22
3C22
3C4
3C6
3C6
4C6
4C6
4C6
4C32
3C2×C4
3C2×C4
3C12
3C2×C6
3C2×C6
4C12
4C12
4C12
4C3×C6
3C2×C12
3C2×C12
4C3×C12

Smallest permutation representation of C12×A4
On 36 points
Generators in S36
(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)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 19 30)(2 20 31)(3 21 32)(4 22 33)(5 23 34)(6 24 35)(7 13 36)(8 14 25)(9 15 26)(10 16 27)(11 17 28)(12 18 29)

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

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

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

C12×A4 is a maximal subgroup of   C12.12S4  A4⋊Dic6  C12⋊S4

48 conjugacy classes

class 1 2A2B2C3A3B3C···3H4A4B4C4D6A6B6C6D6E6F6G···6L12A12B12C12D12E12F12G12H12I···12T
order1222333···344446666666···6121212121212121212···12
size1133114···411331133334···4111133334···4

48 irreducible representations

dim111111111333333
type++++
imageC1C2C3C3C4C6C6C12C12A4C2×A4C3×A4C4×A4C6×A4C12×A4
kernelC12×A4C6×A4C4×A4C22×C12C3×A4C2×A4C22×C6A4C2×C6C12C6C4C3C2C1
# reps1162262124112224

Matrix representation of C12×A4 in GL3(𝔽13) generated by

600
060
006
,
1200
0120
001
,
1200
010
0012
,
040
004
900
G:=sub<GL(3,GF(13))| [6,0,0,0,6,0,0,0,6],[12,0,0,0,12,0,0,0,1],[12,0,0,0,1,0,0,0,12],[0,0,9,4,0,0,0,4,0] >;

C12×A4 in GAP, Magma, Sage, TeX

C_{12}\times A_4
% in TeX

G:=Group("C12xA4");
// GroupNames label

G:=SmallGroup(144,155);
// by ID

G=gap.SmallGroup(144,155);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-2,2,108,1090,1955]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

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Subgroup lattice of C12×A4 in TeX

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