Aliases: C4.6S4, Q8.4D6, C4○GL2(𝔽3), C4○CSU2(𝔽3), GL2(𝔽3)⋊3C2, CSU2(𝔽3)⋊3C2, SL2(𝔽3).4C22, C4.A4⋊2C2, C4○D4⋊1S3, C2.9(C2×S4), SmallGroup(96,192)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — C4.6S4 |
Generators and relations for C4.6S4
G = < a,b,c,d,e | a4=d3=e2=1, b2=c2=a2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=a2b, dbd-1=a2bc, ebe=bc, dcd-1=b, ece=a2c, ede=d-1 >
Character table of C4.6S4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 8A | 8B | 8C | 8D | 12A | 12B | |
| size | 1 | 1 | 6 | 12 | 8 | 1 | 1 | 6 | 12 | 8 | 6 | 6 | 6 | 6 | 8 | 8 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -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 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -2 | 0 | -1 | -2 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | -2 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 1 | -√2 | -√-2 | √2 | √-2 | -i | i | complex faithful |
| ρ8 | 2 | -2 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 1 | -√2 | √-2 | √2 | -√-2 | i | -i | complex faithful |
| ρ9 | 2 | -2 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 1 | √2 | √-2 | -√2 | -√-2 | -i | i | complex faithful |
| ρ10 | 2 | -2 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 1 | √2 | -√-2 | -√2 | √-2 | i | -i | complex faithful |
| ρ11 | 3 | 3 | 1 | -1 | 0 | -3 | -3 | -1 | 1 | 0 | -1 | 1 | -1 | 1 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ12 | 3 | 3 | -1 | 1 | 0 | 3 | 3 | -1 | 1 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S4 |
| ρ13 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | orthogonal lifted from S4 |
| ρ14 | 3 | 3 | 1 | 1 | 0 | -3 | -3 | -1 | -1 | 0 | 1 | -1 | 1 | -1 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ15 | 4 | -4 | 0 | 0 | 1 | -4i | 4i | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -i | i | complex faithful |
| ρ16 | 4 | -4 | 0 | 0 | 1 | 4i | -4i | 0 | 0 | -1 | 0 | 0 | 0 | 0 | i | -i | complex faithful |
►On 16 points - transitive group 16T189
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 8 3 6)(2 5 4 7)(9 15 11 13)(10 16 12 14)
(1 9 3 11)(2 10 4 12)(5 14 7 16)(6 15 8 13)
(5 10 16)(6 11 13)(7 12 14)(8 9 15)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8,3,6)(2,5,4,7)(9,15,11,13)(10,16,12,14), (1,9,3,11)(2,10,4,12)(5,14,7,16)(6,15,8,13), (5,10,16)(6,11,13)(7,12,14)(8,9,15), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8,3,6)(2,5,4,7)(9,15,11,13)(10,16,12,14), (1,9,3,11)(2,10,4,12)(5,14,7,16)(6,15,8,13), (5,10,16)(6,11,13)(7,12,14)(8,9,15), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,8,3,6),(2,5,4,7),(9,15,11,13),(10,16,12,14)], [(1,9,3,11),(2,10,4,12),(5,14,7,16),(6,15,8,13)], [(5,10,16),(6,11,13),(7,12,14),(8,9,15)], [(1,3),(2,4),(5,16),(6,13),(7,14),(8,15)]])
G:=TransitiveGroup(16,189);
C4.6S4 is a maximal subgroup of
CU2(𝔽3) C8.5S4 GL2(𝔽3)⋊C22 Q8.6S4 Q8.7S4 D4.4S4 D4.5S4 Dic3.4S4 Dic3.5S4 C12.14S4 C4.6S5 Dic5.6S4 Dic5.7S4 C20.6S4
C4.6S4 is a maximal quotient of
C4×CSU2(𝔽3) Q8.Dic6 C4×GL2(𝔽3) Q8.2D12 C4.A4⋊C4 SL2(𝔽3).D4 SL2(𝔽3)⋊D4 C12.11S4 Dic3.4S4 Dic3.5S4 C12.14S4 Dic5.6S4 Dic5.7S4 C20.6S4
Matrix representation of C4.6S4 ►in GL2(𝔽17) generated by
| 4 | 0 |
| 0 | 4 |
| 9 | 9 |
| 6 | 8 |
| 6 | 11 |
| 9 | 11 |
| 1 | 10 |
| 15 | 15 |
| 1 | 10 |
| 0 | 16 |
G:=sub<GL(2,GF(17))| [4,0,0,4],[9,6,9,8],[6,9,11,11],[1,15,10,15],[1,0,10,16] >;
C4.6S4 in GAP, Magma, Sage, TeX
C_4._6S_4
% in TeX
G:=Group("C4.6S4"); // GroupNames label
G:=SmallGroup(96,192);
// by ID
G=gap.SmallGroup(96,192);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,2,-2,295,146,579,447,117,364,286,202,88]);
// 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,a*e=e*a,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 C4.6S4 in TeX
Character table of C4.6S4 in TeX