Copied to
clipboard

## G = A4×C3⋊S3order 216 = 23·33

### Direct product of A4 and C3⋊S3

Aliases: A4×C3⋊S3, C626C6, C3⋊(S3×A4), (C3×A4)⋊5S3, C324(C2×A4), (C32×A4)⋊5C2, (C2×C6)⋊3(C3×S3), C222(C3×C3⋊S3), (C22×C3⋊S3)⋊3C3, SmallGroup(216,167)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — A4×C3⋊S3
 Chief series C1 — C3 — C32 — C62 — C32×A4 — A4×C3⋊S3
 Lower central C62 — A4×C3⋊S3
 Upper central C1

Generators and relations for A4×C3⋊S3
G = < a,b,c,d,e,f | a2=b2=c3=d3=e3=f2=1, cac-1=ab=ba, ad=da, ae=ea, af=fa, cbc-1=a, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 492 in 88 conjugacy classes, 21 normal (9 characteristic)
C1, C2, C3, C3, C22, C22, S3, C6, C23, C32, C32, A4, A4, D6, C2×C6, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C2×A4, C22×S3, C33, C3×A4, C3×A4, C2×C3⋊S3, C62, C3×C3⋊S3, S3×A4, C22×C3⋊S3, C32×A4, A4×C3⋊S3
Quotients: C1, C2, C3, S3, C6, A4, C3×S3, C3⋊S3, C2×A4, C3×C3⋊S3, S3×A4, A4×C3⋊S3

Character table of A4×C3⋊S3

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M 3N 6A 6B 6C 6D 6E 6F size 1 3 9 27 2 2 2 2 4 4 8 8 8 8 8 8 8 8 6 6 6 6 36 36 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 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 1 ζ32 ζ3 linear of order 3 ρ4 1 1 -1 -1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 1 ζ65 ζ6 linear of order 6 ρ5 1 1 -1 -1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 1 ζ6 ζ65 linear of order 6 ρ6 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 1 ζ3 ζ32 linear of order 3 ρ7 2 2 0 0 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 2 -1 2 -1 -1 -1 2 0 0 orthogonal lifted from S3 ρ8 2 2 0 0 -1 2 -1 -1 2 2 -1 2 2 -1 -1 -1 -1 -1 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ9 2 2 0 0 2 -1 -1 -1 2 2 -1 -1 -1 -1 2 -1 2 -1 -1 -1 2 -1 0 0 orthogonal lifted from S3 ρ10 2 2 0 0 -1 -1 2 -1 2 2 2 -1 -1 2 -1 -1 -1 -1 -1 2 -1 -1 0 0 orthogonal lifted from S3 ρ11 2 2 0 0 -1 -1 -1 2 -1-√-3 -1+√-3 ζ65 ζ65 ζ6 ζ6 ζ6 -1-√-3 ζ65 -1+√-3 -1 -1 -1 2 0 0 complex lifted from C3×S3 ρ12 2 2 0 0 -1 2 -1 -1 -1+√-3 -1-√-3 ζ6 -1-√-3 -1+√-3 ζ65 ζ65 ζ65 ζ6 ζ6 2 -1 -1 -1 0 0 complex lifted from C3×S3 ρ13 2 2 0 0 -1 -1 2 -1 -1-√-3 -1+√-3 -1+√-3 ζ65 ζ6 -1-√-3 ζ6 ζ6 ζ65 ζ65 -1 2 -1 -1 0 0 complex lifted from C3×S3 ρ14 2 2 0 0 2 -1 -1 -1 -1-√-3 -1+√-3 ζ65 ζ65 ζ6 ζ6 -1-√-3 ζ6 -1+√-3 ζ65 -1 -1 2 -1 0 0 complex lifted from C3×S3 ρ15 2 2 0 0 -1 2 -1 -1 -1-√-3 -1+√-3 ζ65 -1+√-3 -1-√-3 ζ6 ζ6 ζ6 ζ65 ζ65 2 -1 -1 -1 0 0 complex lifted from C3×S3 ρ16 2 2 0 0 -1 -1 -1 2 -1+√-3 -1-√-3 ζ6 ζ6 ζ65 ζ65 ζ65 -1+√-3 ζ6 -1-√-3 -1 -1 -1 2 0 0 complex lifted from C3×S3 ρ17 2 2 0 0 -1 -1 2 -1 -1+√-3 -1-√-3 -1-√-3 ζ6 ζ65 -1+√-3 ζ65 ζ65 ζ6 ζ6 -1 2 -1 -1 0 0 complex lifted from C3×S3 ρ18 2 2 0 0 2 -1 -1 -1 -1+√-3 -1-√-3 ζ6 ζ6 ζ65 ζ65 -1+√-3 ζ65 -1-√-3 ζ6 -1 -1 2 -1 0 0 complex lifted from C3×S3 ρ19 3 -1 3 -1 3 3 3 3 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 orthogonal lifted from A4 ρ20 3 -1 -3 1 3 3 3 3 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 orthogonal lifted from C2×A4 ρ21 6 -2 0 0 6 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 1 1 -2 1 0 0 orthogonal lifted from S3×A4 ρ22 6 -2 0 0 -3 -3 -3 6 0 0 0 0 0 0 0 0 0 0 1 1 1 -2 0 0 orthogonal lifted from S3×A4 ρ23 6 -2 0 0 -3 6 -3 -3 0 0 0 0 0 0 0 0 0 0 -2 1 1 1 0 0 orthogonal lifted from S3×A4 ρ24 6 -2 0 0 -3 -3 6 -3 0 0 0 0 0 0 0 0 0 0 1 -2 1 1 0 0 orthogonal lifted from S3×A4

Smallest permutation representation of A4×C3⋊S3
On 36 points
Generators in S36
(1 18)(2 22)(3 11)(4 29)(5 31)(6 25)(7 20)(8 13)(9 34)(10 12)(14 15)(16 17)(19 21)(23 24)(26 27)(28 30)(32 33)(35 36)
(1 16)(2 23)(3 12)(4 30)(5 32)(6 26)(7 21)(8 14)(9 35)(10 11)(13 15)(17 18)(19 20)(22 24)(25 27)(28 29)(31 33)(34 36)
(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 3 2)(4 6 5)(7 9 8)(10 24 17)(11 22 18)(12 23 16)(13 20 34)(14 21 35)(15 19 36)(25 31 29)(26 32 30)(27 33 28)
(1 4 7)(2 5 8)(3 6 9)(10 27 36)(11 25 34)(12 26 35)(13 22 31)(14 23 32)(15 24 33)(16 30 21)(17 28 19)(18 29 20)
(2 3)(4 7)(5 9)(6 8)(10 24)(11 22)(12 23)(13 25)(14 26)(15 27)(19 28)(20 29)(21 30)(31 34)(32 35)(33 36)

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

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

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

A4×C3⋊S3 is a maximal subgroup of   C62⋊Dic3  C6210D6  S32×A4
A4×C3⋊S3 is a maximal quotient of   C3⋊Dic3.2A4

Matrix representation of A4×C3⋊S3 in GL7(𝔽7)

 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 1 0 0 0 0 6 1 0
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 6 0 0 0 0 0 1 6 0
,
 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0
,
 6 1 0 0 0 0 0 6 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 0 6 0 0 0 0 0 1 6 0 0 0 0 0 0 0 6 6 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 6 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1

G:=sub<GL(7,GF(7))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,6,6,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,6,6,6,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[6,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,6,6,0,0,0,0,0,0,0,6,1,0,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1] >;

A4×C3⋊S3 in GAP, Magma, Sage, TeX

A_4\times C_3\rtimes S_3
% in TeX

G:=Group("A4xC3:S3");
// GroupNames label

G:=SmallGroup(216,167);
// by ID

G=gap.SmallGroup(216,167);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-3,-3,170,81,1444,5189]);
// Polycyclic

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

Export

׿
×
𝔽