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## G = C3×A4⋊C4order 144 = 24·32

### Direct product of C3 and A4⋊C4

Aliases: C3×A4⋊C4, A4⋊C12, C6.10S4, (C2×A4).C6, (C3×A4)⋊1C4, C2.1(C3×S4), C23.(C3×S3), (C6×A4).1C2, (C2×C6)⋊1Dic3, C22⋊(C3×Dic3), (C22×C6).1S3, SmallGroup(144,123)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C3×A4⋊C4
 Chief series C1 — C22 — A4 — C2×A4 — C6×A4 — C3×A4⋊C4
 Lower central A4 — C3×A4⋊C4
 Upper central C1 — C6

Generators and relations for C3×A4⋊C4
G = < a,b,c,d,e | a3=b2=c2=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >

Character table of C3×A4⋊C4

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 3 3 1 1 8 8 8 6 6 6 6 1 1 3 3 3 3 8 8 8 6 6 6 6 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 ζ3 ζ32 ζ3 1 ζ32 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 3 ρ4 1 1 1 1 ζ32 ζ3 ζ32 1 ζ3 -1 -1 -1 -1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 ζ3 ζ32 ζ65 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 ζ6 linear of order 6 ρ5 1 1 1 1 ζ3 ζ32 ζ3 1 ζ32 -1 -1 -1 -1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 ζ32 ζ3 ζ6 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 ζ65 linear of order 6 ρ6 1 1 1 1 ζ32 ζ3 ζ32 1 ζ3 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 3 ρ7 1 -1 -1 1 1 1 1 1 1 i -i -i i -1 -1 -1 -1 1 1 -1 -1 -1 i -i i i -i -i i -i linear of order 4 ρ8 1 -1 -1 1 1 1 1 1 1 -i i i -i -1 -1 -1 -1 1 1 -1 -1 -1 -i i -i -i i i -i i linear of order 4 ρ9 1 -1 -1 1 ζ32 ζ3 ζ32 1 ζ3 -i i i -i ζ65 ζ6 ζ6 ζ65 ζ3 ζ32 -1 ζ65 ζ6 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ43ζ32 ζ4ζ3 ζ4ζ3 ζ43ζ3 ζ4ζ32 linear of order 12 ρ10 1 -1 -1 1 ζ3 ζ32 ζ3 1 ζ32 i -i -i i ζ6 ζ65 ζ65 ζ6 ζ32 ζ3 -1 ζ6 ζ65 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ4ζ3 ζ43ζ32 ζ43ζ32 ζ4ζ32 ζ43ζ3 linear of order 12 ρ11 1 -1 -1 1 ζ32 ζ3 ζ32 1 ζ3 i -i -i i ζ65 ζ6 ζ6 ζ65 ζ3 ζ32 -1 ζ65 ζ6 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ4ζ32 ζ43ζ3 ζ43ζ3 ζ4ζ3 ζ43ζ32 linear of order 12 ρ12 1 -1 -1 1 ζ3 ζ32 ζ3 1 ζ32 -i i i -i ζ6 ζ65 ζ65 ζ6 ζ32 ζ3 -1 ζ6 ζ65 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ43ζ3 ζ4ζ32 ζ4ζ32 ζ43ζ32 ζ4ζ3 linear of order 12 ρ13 2 2 2 2 2 2 -1 -1 -1 0 0 0 0 2 2 2 2 2 2 -1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from S3 ρ14 2 -2 -2 2 2 2 -1 -1 -1 0 0 0 0 -2 -2 -2 -2 2 2 1 1 1 0 0 0 0 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ15 2 -2 -2 2 -1-√-3 -1+√-3 ζ6 -1 ζ65 0 0 0 0 1-√-3 1+√-3 1+√-3 1-√-3 -1+√-3 -1-√-3 1 ζ3 ζ32 0 0 0 0 0 0 0 0 complex lifted from C3×Dic3 ρ16 2 2 2 2 -1+√-3 -1-√-3 ζ65 -1 ζ6 0 0 0 0 -1-√-3 -1+√-3 -1+√-3 -1-√-3 -1-√-3 -1+√-3 -1 ζ6 ζ65 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ17 2 -2 -2 2 -1+√-3 -1-√-3 ζ65 -1 ζ6 0 0 0 0 1+√-3 1-√-3 1-√-3 1+√-3 -1-√-3 -1+√-3 1 ζ32 ζ3 0 0 0 0 0 0 0 0 complex lifted from C3×Dic3 ρ18 2 2 2 2 -1-√-3 -1+√-3 ζ6 -1 ζ65 0 0 0 0 -1+√-3 -1-√-3 -1-√-3 -1+√-3 -1+√-3 -1-√-3 -1 ζ65 ζ6 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ19 3 3 -1 -1 3 3 0 0 0 1 -1 1 -1 3 3 -1 -1 -1 -1 0 0 0 1 1 -1 1 -1 1 -1 -1 orthogonal lifted from S4 ρ20 3 3 -1 -1 3 3 0 0 0 -1 1 -1 1 3 3 -1 -1 -1 -1 0 0 0 -1 -1 1 -1 1 -1 1 1 orthogonal lifted from S4 ρ21 3 3 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 -1 1 -1 1 -3-3√-3/2 -3+3√-3/2 ζ65 ζ6 ζ6 ζ65 0 0 0 ζ6 ζ65 ζ3 ζ65 ζ32 ζ6 ζ32 ζ3 complex lifted from C3×S4 ρ22 3 3 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 -1 1 -1 1 -3+3√-3/2 -3-3√-3/2 ζ6 ζ65 ζ65 ζ6 0 0 0 ζ65 ζ6 ζ32 ζ6 ζ3 ζ65 ζ3 ζ32 complex lifted from C3×S4 ρ23 3 3 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 1 -1 1 -1 -3-3√-3/2 -3+3√-3/2 ζ65 ζ6 ζ6 ζ65 0 0 0 ζ32 ζ3 ζ65 ζ3 ζ6 ζ32 ζ6 ζ65 complex lifted from C3×S4 ρ24 3 3 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 1 -1 1 -1 -3+3√-3/2 -3-3√-3/2 ζ6 ζ65 ζ65 ζ6 0 0 0 ζ3 ζ32 ζ6 ζ32 ζ65 ζ3 ζ65 ζ6 complex lifted from C3×S4 ρ25 3 -3 1 -1 3 3 0 0 0 -i -i i i -3 -3 1 1 -1 -1 0 0 0 -i i i -i -i i i -i complex lifted from A4⋊C4 ρ26 3 -3 1 -1 3 3 0 0 0 i i -i -i -3 -3 1 1 -1 -1 0 0 0 i -i -i i i -i -i i complex lifted from A4⋊C4 ρ27 3 -3 1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 -i -i i i 3+3√-3/2 3-3√-3/2 ζ3 ζ32 ζ6 ζ65 0 0 0 ζ43ζ32 ζ4ζ3 ζ4ζ3 ζ43ζ3 ζ43ζ32 ζ4ζ32 ζ4ζ32 ζ43ζ3 complex faithful ρ28 3 -3 1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 -i -i i i 3-3√-3/2 3+3√-3/2 ζ32 ζ3 ζ65 ζ6 0 0 0 ζ43ζ3 ζ4ζ32 ζ4ζ32 ζ43ζ32 ζ43ζ3 ζ4ζ3 ζ4ζ3 ζ43ζ32 complex faithful ρ29 3 -3 1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 i i -i -i 3+3√-3/2 3-3√-3/2 ζ3 ζ32 ζ6 ζ65 0 0 0 ζ4ζ32 ζ43ζ3 ζ43ζ3 ζ4ζ3 ζ4ζ32 ζ43ζ32 ζ43ζ32 ζ4ζ3 complex faithful ρ30 3 -3 1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 i i -i -i 3-3√-3/2 3+3√-3/2 ζ32 ζ3 ζ65 ζ6 0 0 0 ζ4ζ3 ζ43ζ32 ζ43ζ32 ζ4ζ32 ζ4ζ3 ζ43ζ3 ζ43ζ3 ζ4ζ32 complex faithful

Smallest permutation representation of C3×A4⋊C4
On 36 points
Generators in S36
(1 26 14)(2 27 15)(3 28 16)(4 25 13)(5 29 17)(6 30 18)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)
(1 4)(2 3)(5 11)(6 8)(7 9)(10 12)(13 14)(15 16)(17 23)(18 20)(19 21)(22 24)(25 26)(27 28)(29 35)(30 32)(31 33)(34 36)
(1 3)(2 4)(5 9)(6 10)(7 11)(8 12)(13 15)(14 16)(17 21)(18 22)(19 23)(20 24)(25 27)(26 28)(29 33)(30 34)(31 35)(32 36)
(1 5 10)(2 11 6)(3 7 12)(4 9 8)(13 21 20)(14 17 22)(15 23 18)(16 19 24)(25 33 32)(26 29 34)(27 35 30)(28 31 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)

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

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

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

C3×A4⋊C4 is a maximal subgroup of   Dic3.S4  Dic32S4  A4⋊D12  C12×S4  C626Dic3
C3×A4⋊C4 is a maximal quotient of   C62.Dic3  C625Dic3

Matrix representation of C3×A4⋊C4 in GL3(𝔽13) generated by

 9 0 0 0 9 0 0 0 9
,
 12 0 0 0 1 0 0 0 12
,
 1 0 0 0 12 0 0 0 12
,
 0 2 0 0 0 1 7 0 0
,
 5 0 0 0 0 5 0 5 0
G:=sub<GL(3,GF(13))| [9,0,0,0,9,0,0,0,9],[12,0,0,0,1,0,0,0,12],[1,0,0,0,12,0,0,0,12],[0,0,7,2,0,0,0,1,0],[5,0,0,0,0,5,0,5,0] >;

C3×A4⋊C4 in GAP, Magma, Sage, TeX

C_3\times A_4\rtimes C_4
% in TeX

G:=Group("C3xA4:C4");
// GroupNames label

G:=SmallGroup(144,123);
// by ID

G=gap.SmallGroup(144,123);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-2,2,36,579,2164,202,1301,347]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^3=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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