Aliases: C4.2S4, Q8.3D6, CSU2(𝔽3)⋊2C2, SL2(𝔽3).3C22, C2.8(C2×S4), C4○D4.2S3, C4.A4.1C2, SmallGroup(96,191)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — C4.S4 |
Generators and relations for C4.S4
G = < a,b,c,d,e | a4=d3=1, b2=c2=e2=a2, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc-1=a2b, dbd-1=a2bc, ebe-1=bc, dcd-1=b, ece-1=a2c, ede-1=d-1 >
Character table of C4.S4
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 6 | 8A | 8B | 12A | 12B | |
| size | 1 | 1 | 6 | 8 | 2 | 6 | 12 | 12 | 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 | -1 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 3 | 3 | 1 | 0 | -3 | -1 | 1 | -1 | 0 | 1 | -1 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ8 | 3 | 3 | 1 | 0 | -3 | -1 | -1 | 1 | 0 | -1 | 1 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ9 | 3 | 3 | -1 | 0 | 3 | -1 | -1 | -1 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from S4 |
| ρ10 | 3 | 3 | -1 | 0 | 3 | -1 | 1 | 1 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from S4 |
| ρ11 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ12 | 4 | -4 | 0 | 1 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | √3 | -√3 | symplectic faithful, Schur index 2 |
| ρ13 | 4 | -4 | 0 | 1 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -√3 | √3 | symplectic faithful, Schur index 2 |
►On 32 points
Generators in S32
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 8 3 6)(2 5 4 7)(9 28 11 26)(10 25 12 27)(13 18 15 20)(14 19 16 17)(21 30 23 32)(22 31 24 29)
(1 18 3 20)(2 19 4 17)(5 14 7 16)(6 15 8 13)(9 24 11 22)(10 21 12 23)(25 32 27 30)(26 29 28 31)
(5 19 16)(6 20 13)(7 17 14)(8 18 15)(9 28 31)(10 25 32)(11 26 29)(12 27 30)
(1 24 3 22)(2 23 4 21)(5 27 7 25)(6 26 8 28)(9 20 11 18)(10 19 12 17)(13 29 15 31)(14 32 16 30)
G:=sub<Sym(32)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,8,3,6)(2,5,4,7)(9,28,11,26)(10,25,12,27)(13,18,15,20)(14,19,16,17)(21,30,23,32)(22,31,24,29), (1,18,3,20)(2,19,4,17)(5,14,7,16)(6,15,8,13)(9,24,11,22)(10,21,12,23)(25,32,27,30)(26,29,28,31), (5,19,16)(6,20,13)(7,17,14)(8,18,15)(9,28,31)(10,25,32)(11,26,29)(12,27,30), (1,24,3,22)(2,23,4,21)(5,27,7,25)(6,26,8,28)(9,20,11,18)(10,19,12,17)(13,29,15,31)(14,32,16,30)>;
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)(25,26,27,28)(29,30,31,32), (1,8,3,6)(2,5,4,7)(9,28,11,26)(10,25,12,27)(13,18,15,20)(14,19,16,17)(21,30,23,32)(22,31,24,29), (1,18,3,20)(2,19,4,17)(5,14,7,16)(6,15,8,13)(9,24,11,22)(10,21,12,23)(25,32,27,30)(26,29,28,31), (5,19,16)(6,20,13)(7,17,14)(8,18,15)(9,28,31)(10,25,32)(11,26,29)(12,27,30), (1,24,3,22)(2,23,4,21)(5,27,7,25)(6,26,8,28)(9,20,11,18)(10,19,12,17)(13,29,15,31)(14,32,16,30) );
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),(25,26,27,28),(29,30,31,32)], [(1,8,3,6),(2,5,4,7),(9,28,11,26),(10,25,12,27),(13,18,15,20),(14,19,16,17),(21,30,23,32),(22,31,24,29)], [(1,18,3,20),(2,19,4,17),(5,14,7,16),(6,15,8,13),(9,24,11,22),(10,21,12,23),(25,32,27,30),(26,29,28,31)], [(5,19,16),(6,20,13),(7,17,14),(8,18,15),(9,28,31),(10,25,32),(11,26,29),(12,27,30)], [(1,24,3,22),(2,23,4,21),(5,27,7,25),(6,26,8,28),(9,20,11,18),(10,19,12,17),(13,29,15,31),(14,32,16,30)]])
C4.S4 is a maximal subgroup of
C8.S4 C8.4S4 Q8.4S4 D4.S4 GL2(𝔽3)⋊C22 Q8.6S4 D4.5S4 CSU2(𝔽3)⋊S3 C12.6S4 C4.S5 CSU2(𝔽3)⋊D5 C20.2S4
C4.S4 is a maximal quotient of
Q8⋊Dic6 CSU2(𝔽3)⋊C4 Q8.D12 SL2(𝔽3).D4 (C2×C4).S4 C12.3S4 CSU2(𝔽3)⋊S3 C12.6S4 CSU2(𝔽3)⋊D5 C20.2S4
Matrix representation of C4.S4 ►in GL4(𝔽3) generated by
| 0 | 2 | 1 | 1 |
| 1 | 1 | 2 | 0 |
| 1 | 1 | 0 | 1 |
| 2 | 0 | 2 | 2 |
| 2 | 1 | 1 | 1 |
| 0 | 2 | 2 | 1 |
| 2 | 2 | 0 | 1 |
| 2 | 0 | 2 | 2 |
| 2 | 2 | 2 | 2 |
| 0 | 1 | 0 | 1 |
| 2 | 2 | 1 | 1 |
| 0 | 1 | 0 | 2 |
| 2 | 2 | 2 | 2 |
| 1 | 2 | 1 | 2 |
| 2 | 2 | 0 | 1 |
| 1 | 2 | 2 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(3))| [0,1,1,2,2,1,1,0,1,2,0,2,1,0,1,2],[2,0,2,2,1,2,2,0,1,2,0,2,1,1,1,2],[2,0,2,0,2,1,2,1,2,0,1,0,2,1,1,2],[2,1,2,1,2,2,2,2,2,1,0,2,2,2,1,0],[0,0,1,0,0,0,0,1,2,0,0,0,0,2,0,0] >;
C4.S4 in GAP, Magma, Sage, TeX
C_4.S_4
% in TeX
G:=Group("C4.S4"); // GroupNames label
G:=SmallGroup(96,191);
// by ID
G=gap.SmallGroup(96,191);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,2,-2,288,601,295,146,579,447,117,364,286,202,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=d^3=1,b^2=c^2=e^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b*c,e*b*e^-1=b*c,d*c*d^-1=b,e*c*e^-1=a^2*c,e*d*e^-1=d^-1>;
// generators/relations
Export
Subgroup lattice of C4.S4 in TeX
Character table of C4.S4 in TeX