Aliases: C4.3S4, Q8.5D6, GL2(𝔽3)⋊2C2, SL2(𝔽3)⋊2C22, C4.A4⋊1C2, C4○D4⋊2S3, C2.10(C2×S4), SmallGroup(96,193)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — C4.3S4 |
Generators and relations for C4.3S4
G = < a,b,c,d,e | a4=d3=e2=1, b2=c2=a2, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=a2b, dbd-1=a2bc, ebe=bc, dcd-1=b, ece=a2c, ede=d-1 >
6C2
12C2
12C2
4C3
3C22
3C4
6C22
6C22
12C22
12C22
4C6
8S3
8S3
3D4
3C8
3D4
3D4
3C8
6C23
6D4
4D6
4D6
4C12
3D8
3SD16
3D8
3SD16
4D12
Character table of C4.3S4
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 6 | 8A | 8B | 12A | 12B | |
| size | 1 | 1 | 6 | 12 | 12 | 8 | 2 | 6 | 8 | 12 | 12 | 8 | 8 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | -1 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 3 | 3 | -1 | 1 | 1 | 0 | 3 | -1 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from S4 |
| ρ8 | 3 | 3 | -1 | -1 | -1 | 0 | 3 | -1 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from S4 |
| ρ9 | 3 | 3 | 1 | -1 | 1 | 0 | -3 | -1 | 0 | 1 | -1 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ10 | 3 | 3 | 1 | 1 | -1 | 0 | -3 | -1 | 0 | -1 | 1 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ11 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ12 | 4 | -4 | 0 | 0 | 0 | 1 | 0 | 0 | -1 | 0 | 0 | -√3 | √3 | orthogonal faithful |
| ρ13 | 4 | -4 | 0 | 0 | 0 | 1 | 0 | 0 | -1 | 0 | 0 | √3 | -√3 | orthogonal faithful |
►On 16 points - transitive group 16T190
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 6 3 8)(2 7 4 5)(9 15 11 13)(10 16 12 14)
(1 9 3 11)(2 10 4 12)(5 16 7 14)(6 13 8 15)
(5 12 14)(6 9 15)(7 10 16)(8 11 13)
(1 4)(2 3)(5 13)(6 16)(7 15)(8 14)(9 10)(11 12)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6,3,8)(2,7,4,5)(9,15,11,13)(10,16,12,14), (1,9,3,11)(2,10,4,12)(5,16,7,14)(6,13,8,15), (5,12,14)(6,9,15)(7,10,16)(8,11,13), (1,4)(2,3)(5,13)(6,16)(7,15)(8,14)(9,10)(11,12)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6,3,8)(2,7,4,5)(9,15,11,13)(10,16,12,14), (1,9,3,11)(2,10,4,12)(5,16,7,14)(6,13,8,15), (5,12,14)(6,9,15)(7,10,16)(8,11,13), (1,4)(2,3)(5,13)(6,16)(7,15)(8,14)(9,10)(11,12) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,6,3,8),(2,7,4,5),(9,15,11,13),(10,16,12,14)], [(1,9,3,11),(2,10,4,12),(5,16,7,14),(6,13,8,15)], [(5,12,14),(6,9,15),(7,10,16),(8,11,13)], [(1,4),(2,3),(5,13),(6,16),(7,15),(8,14),(9,10),(11,12)]])
G:=TransitiveGroup(16,190);
►On 24 points - transitive group 24T127
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 12 3 10)(2 9 4 11)(5 8 7 6)(13 18 15 20)(14 19 16 17)(21 22 23 24)
(1 4 3 2)(5 24 7 22)(6 21 8 23)(9 10 11 12)(13 17 15 19)(14 18 16 20)
(1 17 7)(2 18 8)(3 19 5)(4 20 6)(9 16 24)(10 13 21)(11 14 22)(12 15 23)
(2 4)(5 19)(6 18)(7 17)(8 20)(9 12)(10 11)(13 22)(14 21)(15 24)(16 23)
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), (1,12,3,10)(2,9,4,11)(5,8,7,6)(13,18,15,20)(14,19,16,17)(21,22,23,24), (1,4,3,2)(5,24,7,22)(6,21,8,23)(9,10,11,12)(13,17,15,19)(14,18,16,20), (1,17,7)(2,18,8)(3,19,5)(4,20,6)(9,16,24)(10,13,21)(11,14,22)(12,15,23), (2,4)(5,19)(6,18)(7,17)(8,20)(9,12)(10,11)(13,22)(14,21)(15,24)(16,23)>;
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), (1,12,3,10)(2,9,4,11)(5,8,7,6)(13,18,15,20)(14,19,16,17)(21,22,23,24), (1,4,3,2)(5,24,7,22)(6,21,8,23)(9,10,11,12)(13,17,15,19)(14,18,16,20), (1,17,7)(2,18,8)(3,19,5)(4,20,6)(9,16,24)(10,13,21)(11,14,22)(12,15,23), (2,4)(5,19)(6,18)(7,17)(8,20)(9,12)(10,11)(13,22)(14,21)(15,24)(16,23) );
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)], [(1,12,3,10),(2,9,4,11),(5,8,7,6),(13,18,15,20),(14,19,16,17),(21,22,23,24)], [(1,4,3,2),(5,24,7,22),(6,21,8,23),(9,10,11,12),(13,17,15,19),(14,18,16,20)], [(1,17,7),(2,18,8),(3,19,5),(4,20,6),(9,16,24),(10,13,21),(11,14,22),(12,15,23)], [(2,4),(5,19),(6,18),(7,17),(8,20),(9,12),(10,11),(13,22),(14,21),(15,24),(16,23)]])
G:=TransitiveGroup(24,127);
C4.3S4 is a maximal subgroup of
C8.4S4 C8.3S4 Q8.5S4 D4.3S4 GL2(𝔽3)⋊C22 Q8.7S4 D4.4S4 GL2(𝔽3)⋊S3 C12.7S4 C4.3S5 GL2(𝔽3)⋊D5 C20.3S4
C4.3S4 is a maximal quotient of
SL2(𝔽3)⋊Q8 Q8⋊D12 GL2(𝔽3)⋊C4 (C2×C4).S4 SL2(𝔽3)⋊D4 C12.4S4 GL2(𝔽3)⋊S3 C12.7S4 GL2(𝔽3)⋊D5 C20.3S4
Matrix representation of C4.3S4 ►in GL4(ℚ) generated by
| 0 | 0 | 0 | -1 |
| 0 | 0 | -1 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | 0 | 1 | 0 |
| 0 | -1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 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 |
| -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())| [0,0,0,1,0,0,1,0,0,-1,0,0,-1,0,0,0],[0,0,0,1,0,0,-1,0,0,1,0,0,-1,0,0,0],[0,1,0,0,-1,0,0,0,0,0,0,-1,0,0,1,0],[-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],[-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] >;
C4.3S4 in GAP, Magma, Sage, TeX
C_4._3S_4
% in TeX
G:=Group("C4.3S4"); // GroupNames label
G:=SmallGroup(96,193);
// by ID
G=gap.SmallGroup(96,193);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,2,-2,601,295,146,579,447,117,364,286,202,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=d^3=e^2=1,b^2=c^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b*c,e*b*e=b*c,d*c*d^-1=b,e*c*e=a^2*c,e*d*e=d^-1>;
// generators/relations
Export
Subgroup lattice of C4.3S4 in TeX
Character table of C4.3S4 in TeX