non-abelian, soluble, monomial
Aliases: C32⋊2S4, C62⋊5S3, (C3×A4)⋊1S3, C3.3(C3⋊S4), C32⋊A4⋊2C2, C22⋊(He3⋊C2), (C2×C6).1(C3⋊S3), SmallGroup(216,95)
Series: Derived ►Chief ►Lower central ►Upper central
| C32⋊A4 — C32⋊S4 |
Generators and relations for C32⋊S4
G = < a,b,c,d | a6=b6=c3=d2=1, ab=ba, cac-1=a4b, dad=a2b3, cbc-1=a3b, dbd=a3b4, dcd=c-1 >
Subgroups: 310 in 57 conjugacy classes, 11 normal (8 characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, C6, D4, C32, C32, Dic3, C12, A4, D6, C2×C6, C2×C6, C3×S3, C3×C6, C3⋊D4, C3×D4, S4, He3, C3×Dic3, C3×A4, S3×C6, C62, He3⋊C2, C3×C3⋊D4, C3×S4, C32⋊A4, C32⋊S4
Quotients: C1, C2, S3, C3⋊S3, S4, He3⋊C2, C3⋊S4, C32⋊S4
Character table of C32⋊S4
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 3F | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | |
| size | 1 | 3 | 18 | 1 | 1 | 6 | 24 | 24 | 24 | 18 | 3 | 3 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 2 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ4 | 2 | 2 | 0 | 2 | 2 | -1 | -1 | 2 | -1 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ5 | 2 | 2 | 0 | 2 | 2 | -1 | 2 | -1 | -1 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ7 | 3 | -1 | -1 | 3 | 3 | 3 | 0 | 0 | 0 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from S4 |
| ρ8 | 3 | -1 | 1 | 3 | 3 | 3 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | orthogonal lifted from S4 |
| ρ9 | 3 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | ζ32 | ζ3 | ζ32 | ζ3 | complex lifted from He3⋊C2 |
| ρ10 | 3 | -1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | -1 | ζ65 | ζ6 | 2 | -1-√-3 | -1+√-3 | ζ3 | ζ32 | ζ65 | ζ6 | complex faithful |
| ρ11 | 3 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | ζ65 | ζ6 | ζ65 | ζ6 | complex lifted from He3⋊C2 |
| ρ12 | 3 | -1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | -1 | ζ6 | ζ65 | 2 | -1+√-3 | -1-√-3 | ζ32 | ζ3 | ζ6 | ζ65 | complex faithful |
| ρ13 | 3 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | ζ3 | ζ32 | ζ3 | ζ32 | complex lifted from He3⋊C2 |
| ρ14 | 3 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | ζ6 | ζ65 | ζ6 | ζ65 | complex lifted from He3⋊C2 |
| ρ15 | 3 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 1 | ζ6 | ζ65 | 2 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | ζ32 | ζ3 | complex faithful |
| ρ16 | 3 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 1 | ζ65 | ζ6 | 2 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | ζ3 | ζ32 | complex faithful |
| ρ17 | 6 | -2 | 0 | 6 | 6 | -3 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C3⋊S4 |
| ρ18 | 6 | -2 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | -2 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ19 | 6 | -2 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | -2 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | complex faithful |
►On 18 points - transitive group 18T107
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)
(1 4 5 2 3 6)(7 12 8 10 9 11)(13 17 15)(14 18 16)
(1 16 11)(2 13 8)(3 14 10)(4 17 7)(5 18 12)(6 15 9)
(1 2)(3 4)(5 6)(7 14)(8 16)(9 18)(10 17)(11 13)(12 15)
G:=sub<Sym(18)| (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,12,8,10,9,11)(13,17,15)(14,18,16), (1,16,11)(2,13,8)(3,14,10)(4,17,7)(5,18,12)(6,15,9), (1,2)(3,4)(5,6)(7,14)(8,16)(9,18)(10,17)(11,13)(12,15)>;
G:=Group( (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,12,8,10,9,11)(13,17,15)(14,18,16), (1,16,11)(2,13,8)(3,14,10)(4,17,7)(5,18,12)(6,15,9), (1,2)(3,4)(5,6)(7,14)(8,16)(9,18)(10,17)(11,13)(12,15) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18)], [(1,4,5,2,3,6),(7,12,8,10,9,11),(13,17,15),(14,18,16)], [(1,16,11),(2,13,8),(3,14,10),(4,17,7),(5,18,12),(6,15,9)], [(1,2),(3,4),(5,6),(7,14),(8,16),(9,18),(10,17),(11,13),(12,15)]])
G:=TransitiveGroup(18,107);
►On 18 points - transitive group 18T108
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)
(1 4 5 2 3 6)(7 12 8 10 9 11)(13 17 15)(14 18 16)
(1 14 11)(2 17 8)(3 18 10)(4 15 7)(5 16 12)(6 13 9)
(7 15)(8 17)(9 13)(10 18)(11 14)(12 16)
G:=sub<Sym(18)| (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,12,8,10,9,11)(13,17,15)(14,18,16), (1,14,11)(2,17,8)(3,18,10)(4,15,7)(5,16,12)(6,13,9), (7,15)(8,17)(9,13)(10,18)(11,14)(12,16)>;
G:=Group( (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,12,8,10,9,11)(13,17,15)(14,18,16), (1,14,11)(2,17,8)(3,18,10)(4,15,7)(5,16,12)(6,13,9), (7,15)(8,17)(9,13)(10,18)(11,14)(12,16) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18)], [(1,4,5,2,3,6),(7,12,8,10,9,11),(13,17,15),(14,18,16)], [(1,14,11),(2,17,8),(3,18,10),(4,15,7),(5,16,12),(6,13,9)], [(7,15),(8,17),(9,13),(10,18),(11,14),(12,16)]])
G:=TransitiveGroup(18,108);
C32⋊S4 is a maximal subgroup of
C62⋊5D6
C32⋊S4 is a maximal quotient of C32⋊2CSU2(𝔽3) C32⋊3GL2(𝔽3) C62⋊6Dic3
Matrix representation of C32⋊S4 ►in GL3(𝔽7) generated by
| 0 | 0 | 6 |
| 0 | 3 | 1 |
| 5 | 2 | 5 |
| 6 | 1 | 1 |
| 1 | 6 | 1 |
| 2 | 2 | 0 |
| 1 | 3 | 2 |
| 0 | 6 | 2 |
| 0 | 3 | 0 |
| 6 | 0 | 0 |
| 0 | 0 | 2 |
| 0 | 4 | 0 |
G:=sub<GL(3,GF(7))| [0,0,5,0,3,2,6,1,5],[6,1,2,1,6,2,1,1,0],[1,0,0,3,6,3,2,2,0],[6,0,0,0,0,4,0,2,0] >;
C32⋊S4 in GAP, Magma, Sage, TeX
C_3^2\rtimes S_4
% in TeX
G:=Group("C3^2:S4"); // GroupNames label
G:=SmallGroup(216,95);
// by ID
G=gap.SmallGroup(216,95);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-2,2,49,218,224,3244,1630,1949,2927]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^3=d^2=1,a*b=b*a,c*a*c^-1=a^4*b,d*a*d=a^2*b^3,c*b*c^-1=a^3*b,d*b*d=a^3*b^4,d*c*d=c^-1>;
// generators/relations
Export