Copied to
clipboard

## G = Dic3×A4order 144 = 24·32

### Direct product of Dic3 and A4

Aliases: Dic3×A4, C3⋊(C4×A4), (C2×C6)⋊C12, (C3×A4)⋊3C4, C6.3(C2×A4), C2.1(S3×A4), (C22×C6).C6, (C2×A4).2S3, (C6×A4).3C2, C23.2(C3×S3), (C22×Dic3)⋊C3, C222(C3×Dic3), SmallGroup(144,129)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Dic3×A4
 Chief series C1 — C3 — C2×C6 — C22×C6 — C6×A4 — Dic3×A4
 Lower central C2×C6 — Dic3×A4
 Upper central C1 — C2

Generators and relations for Dic3×A4
G = < a,b,c,d,e | a6=c2=d2=e3=1, b2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Character table of Dic3×A4

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D size 1 1 3 3 2 4 4 8 8 3 3 9 9 2 4 4 6 6 8 8 12 12 12 12 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 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 linear of order 2 ρ3 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 -1 -1 -1 -1 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ4 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 -1 -1 -1 -1 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ5 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ6 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 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 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 -i i -i i linear of order 4 ρ9 1 -1 1 -1 1 ζ3 ζ32 ζ3 ζ32 i -i i -i -1 ζ6 ζ65 -1 1 ζ65 ζ6 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 linear of order 12 ρ10 1 -1 1 -1 1 ζ3 ζ32 ζ3 ζ32 -i i -i i -1 ζ6 ζ65 -1 1 ζ65 ζ6 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 linear of order 12 ρ11 1 -1 1 -1 1 ζ32 ζ3 ζ32 ζ3 i -i i -i -1 ζ65 ζ6 -1 1 ζ6 ζ65 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 linear of order 12 ρ12 1 -1 1 -1 1 ζ32 ζ3 ζ32 ζ3 -i i -i i -1 ζ65 ζ6 -1 1 ζ6 ζ65 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 linear of order 12 ρ13 2 2 2 2 -1 2 2 -1 -1 0 0 0 0 -1 2 2 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ14 2 -2 2 -2 -1 2 2 -1 -1 0 0 0 0 1 -2 -2 1 -1 1 1 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ15 2 2 2 2 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 0 0 -1 -1+√-3 -1-√-3 -1 -1 ζ6 ζ65 0 0 0 0 complex lifted from C3×S3 ρ16 2 -2 2 -2 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 0 0 1 1-√-3 1+√-3 1 -1 ζ32 ζ3 0 0 0 0 complex lifted from C3×Dic3 ρ17 2 -2 2 -2 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 0 0 1 1+√-3 1-√-3 1 -1 ζ3 ζ32 0 0 0 0 complex lifted from C3×Dic3 ρ18 2 2 2 2 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 0 0 -1 -1-√-3 -1+√-3 -1 -1 ζ65 ζ6 0 0 0 0 complex lifted from C3×S3 ρ19 3 3 -1 -1 3 0 0 0 0 -3 -3 1 1 3 0 0 -1 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ20 3 3 -1 -1 3 0 0 0 0 3 3 -1 -1 3 0 0 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ21 3 -3 -1 1 3 0 0 0 0 3i -3i -i i -3 0 0 1 -1 0 0 0 0 0 0 complex lifted from C4×A4 ρ22 3 -3 -1 1 3 0 0 0 0 -3i 3i i -i -3 0 0 1 -1 0 0 0 0 0 0 complex lifted from C4×A4 ρ23 6 6 -2 -2 -3 0 0 0 0 0 0 0 0 -3 0 0 1 1 0 0 0 0 0 0 orthogonal lifted from S3×A4 ρ24 6 -6 -2 2 -3 0 0 0 0 0 0 0 0 3 0 0 -1 1 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of Dic3×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)
(1 22 4 19)(2 21 5 24)(3 20 6 23)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 32 16 35)(14 31 17 34)(15 36 18 33)
(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(31 34)(32 35)(33 36)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)
(1 17 11)(2 18 12)(3 13 7)(4 14 8)(5 15 9)(6 16 10)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)

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

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

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

Dic3×A4 is a maximal subgroup of   Dic3.S4  Dic32S4  Dic3⋊S4  C4×S3×A4  Dic9⋊A4  C624C12
Dic3×A4 is a maximal quotient of   SL2(𝔽3).Dic3  Dic9⋊A4  C624C12

Matrix representation of Dic3×A4 in GL5(𝔽13)

 1 12 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 5 8 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 8
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0

G:=sub<GL(5,GF(13))| [1,1,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[5,0,0,0,0,8,8,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;

Dic3×A4 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times A_4
% in TeX

G:=Group("Dic3xA4");
// GroupNames label

G:=SmallGroup(144,129);
// by ID

G=gap.SmallGroup(144,129);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,2,-3,36,441,190,3461]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=c^2=d^2=e^3=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

Export

׿
×
𝔽