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## G = C12×A4order 144 = 24·32

### Direct product of C12 and A4

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

 Derived series C1 — C22 — C12×A4
 Chief series C1 — C22 — C23 — C22×C6 — C6×A4 — C12×A4
 Lower central C22 — C12×A4
 Upper central C1 — C12

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 >

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

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G ··· 6L 12A 12B 12C 12D 12E 12F 12G 12H 12I ··· 12T order 1 2 2 2 3 3 3 ··· 3 4 4 4 4 6 6 6 6 6 6 6 ··· 6 12 12 12 12 12 12 12 12 12 ··· 12 size 1 1 3 3 1 1 4 ··· 4 1 1 3 3 1 1 3 3 3 3 4 ··· 4 1 1 1 1 3 3 3 3 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C4 C6 C6 C12 C12 A4 C2×A4 C3×A4 C4×A4 C6×A4 C12×A4 kernel C12×A4 C6×A4 C4×A4 C22×C12 C3×A4 C2×A4 C22×C6 A4 C2×C6 C12 C6 C4 C3 C2 C1 # reps 1 1 6 2 2 6 2 12 4 1 1 2 2 2 4

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

 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 4 0 0 0 4 9 0 0
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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