Aliases: C6.6S4, C3⋊GL2(𝔽3), SL2(𝔽3)⋊S3, Q8⋊(C3⋊S3), (C3×Q8)⋊1S3, C2.3(C3⋊S4), (C3×SL2(𝔽3))⋊1C2, SmallGroup(144,125)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C3×SL2(𝔽3) — C6.6S4 |
| C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C6.6S4 |
| C3×SL2(𝔽3) — C6.6S4 |
Generators and relations for C6.6S4
G = < a,b,c,d,e | a6=d3=e2=1, b2=c2=a3, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=a3b, dbd-1=a3bc, ebe=bc, dcd-1=b, ece=a3c, ede=d-1 >
36C2
4C3
4C3
4C3
3C4
18C22
4C6
4C6
4C6
12S3
12S3
12S3
12S3
12S3
12S3
12S3
4C32
9C8
9D4
3C12
6D6
12D6
12D6
12D6
9SD16
3D12
Character table of C6.6S4
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4 | 6A | 6B | 6C | 6D | 8A | 8B | 12 | |
| size | 1 | 1 | 36 | 2 | 8 | 8 | 8 | 6 | 2 | 8 | 8 | 8 | 18 | 18 | 12 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ4 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ5 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | orthogonal lifted from S3 |
| ρ7 | 2 | -2 | 0 | 2 | -1 | -1 | -1 | 0 | -2 | 1 | 1 | 1 | -√-2 | √-2 | 0 | complex lifted from GL2(𝔽3) |
| ρ8 | 2 | -2 | 0 | 2 | -1 | -1 | -1 | 0 | -2 | 1 | 1 | 1 | √-2 | -√-2 | 0 | complex lifted from GL2(𝔽3) |
| ρ9 | 3 | 3 | 1 | 3 | 0 | 0 | 0 | -1 | 3 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ10 | 3 | 3 | -1 | 3 | 0 | 0 | 0 | -1 | 3 | 0 | 0 | 0 | 1 | 1 | -1 | orthogonal lifted from S4 |
| ρ11 | 4 | -4 | 0 | -2 | -2 | 1 | 1 | 0 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | orthogonal faithful |
| ρ12 | 4 | -4 | 0 | 4 | 1 | 1 | 1 | 0 | -4 | -1 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from GL2(𝔽3) |
| ρ13 | 4 | -4 | 0 | -2 | 1 | 1 | -2 | 0 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | orthogonal faithful |
| ρ14 | 4 | -4 | 0 | -2 | 1 | -2 | 1 | 0 | 2 | -1 | -1 | 2 | 0 | 0 | 0 | orthogonal faithful |
| ρ15 | 6 | 6 | 0 | -3 | 0 | 0 | 0 | -2 | -3 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from C3⋊S4 |
►On 24 points - transitive group 24T252
Generators in S24
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 22 4 19)(2 23 5 20)(3 24 6 21)(7 17 10 14)(8 18 11 15)(9 13 12 16)
(1 11 4 8)(2 12 5 9)(3 7 6 10)(13 20 16 23)(14 21 17 24)(15 22 18 19)
(7 17 24)(8 18 19)(9 13 20)(10 14 21)(11 15 22)(12 16 23)
(1 4)(2 3)(5 6)(7 9)(10 12)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)
G:=sub<Sym(24)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,22,4,19)(2,23,5,20)(3,24,6,21)(7,17,10,14)(8,18,11,15)(9,13,12,16), (1,11,4,8)(2,12,5,9)(3,7,6,10)(13,20,16,23)(14,21,17,24)(15,22,18,19), (7,17,24)(8,18,19)(9,13,20)(10,14,21)(11,15,22)(12,16,23), (1,4)(2,3)(5,6)(7,9)(10,12)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)>;
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), (1,22,4,19)(2,23,5,20)(3,24,6,21)(7,17,10,14)(8,18,11,15)(9,13,12,16), (1,11,4,8)(2,12,5,9)(3,7,6,10)(13,20,16,23)(14,21,17,24)(15,22,18,19), (7,17,24)(8,18,19)(9,13,20)(10,14,21)(11,15,22)(12,16,23), (1,4)(2,3)(5,6)(7,9)(10,12)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19) );
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)], [(1,22,4,19),(2,23,5,20),(3,24,6,21),(7,17,10,14),(8,18,11,15),(9,13,12,16)], [(1,11,4,8),(2,12,5,9),(3,7,6,10),(13,20,16,23),(14,21,17,24),(15,22,18,19)], [(7,17,24),(8,18,19),(9,13,20),(10,14,21),(11,15,22),(12,16,23)], [(1,4),(2,3),(5,6),(7,9),(10,12),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19)]])
G:=TransitiveGroup(24,252);
C6.6S4 is a maximal subgroup of
Dic3.5S4 GL2(𝔽3)⋊S3 D6.2S4 S3×GL2(𝔽3) SL2(𝔽3).D6 C12.14S4 C12.7S4 C32⋊2GL2(𝔽3) C18.6S4 C32⋊5GL2(𝔽3)
C6.6S4 is a maximal quotient of
C6.GL2(𝔽3) C18.6S4 C32.3GL2(𝔽3) C32⋊3GL2(𝔽3) C32⋊5GL2(𝔽3)
Matrix representation of C6.6S4 ►in GL4(ℚ) generated by
| 1/2 | 1/2 | -1/2 | 1/2 |
| -1/2 | 1/2 | 1/2 | 1/2 |
| 1/2 | -1/2 | 1/2 | 1/2 |
| -1/2 | -1/2 | -1/2 | 1/2 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| -1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | 0 | 1 | 0 |
| 0 | -1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| -1/2 | -1/2 | 1/2 | -1/2 |
| -1/2 | 1/2 | -1/2 | -1/2 |
| 1/2 | -1/2 | -1/2 | -1/2 |
| -1/2 | -1/2 | -1/2 | 1/2 |
G:=sub<GL(4,Rationals())| [1/2,-1/2,1/2,-1/2,1/2,1/2,-1/2,-1/2,-1/2,1/2,1/2,-1/2,1/2,1/2,1/2,1/2],[0,0,-1,0,0,0,0,-1,1,0,0,0,0,1,0,0],[0,0,0,1,0,0,-1,0,0,1,0,0,-1,0,0,0],[1,0,0,0,0,0,-1,0,0,0,0,-1,0,1,0,0],[-1/2,-1/2,1/2,-1/2,-1/2,1/2,-1/2,-1/2,1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,1/2] >;
C6.6S4 in GAP, Magma, Sage, TeX
C_6._6S_4
% in TeX
G:=Group("C6.6S4"); // GroupNames label
G:=SmallGroup(144,125);
// by ID
G=gap.SmallGroup(144,125);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,2,-2,49,218,867,1305,117,544,820,202,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=d^3=e^2=1,b^2=c^2=a^3,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,c*b*c^-1=a^3*b,d*b*d^-1=a^3*b*c,e*b*e=b*c,d*c*d^-1=b,e*c*e=a^3*c,e*d*e=d^-1>;
// generators/relations
Export
Subgroup lattice of C6.6S4 in TeX
Character table of C6.6S4 in TeX