direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C3⋊S3, C62⋊6C6, C3⋊(S3×A4), (C3×A4)⋊5S3, C32⋊4(C2×A4), (C32×A4)⋊5C2, (C2×C6)⋊3(C3×S3), C22⋊2(C3×C3⋊S3), (C22×C3⋊S3)⋊3C3, SmallGroup(216,167)
Series: Derived ►Chief ►Lower central ►Upper central
| C62 — A4×C3⋊S3 |
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 |
►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 C62⋊10D6 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
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