Aliases: D4.4S4, 2- (1+4)⋊2S3, GL2(𝔽3)⋊5C22, CSU2(𝔽3)⋊5C22, SL2(𝔽3).8C23, C4○D4⋊D6, (C2×Q8)⋊D6, C4.14(C2×S4), D4.A4⋊2C2, C4.3S4⋊5C2, C4.6S4⋊3C2, C4.A4⋊2C22, C22.5(C2×S4), C2.19(C22×S4), Q8.D6⋊1C2, Q8.9(C22×S3), (C2×GL2(𝔽3))⋊3C2, (C2×SL2(𝔽3))⋊C22, SmallGroup(192,1485)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — D4.4S4 |
Subgroups: 615 in 152 conjugacy classes, 27 normal (15 characteristic)
C1, C2, C2 [×6], C3, C4, C4 [×4], C22 [×2], C22 [×7], S3 [×4], C6 [×3], C8 [×4], C2×C4 [×7], D4, D4 [×12], Q8, Q8 [×4], C23 [×3], Dic3, C12, D6 [×7], C2×C6 [×2], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×6], C2×Q8 [×2], C2×Q8, C4○D4, C4○D4 [×7], SL2(𝔽3), C4×S3, D12, C3⋊D4 [×2], C3×D4, C22×S3 [×2], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ (1+4), 2- (1+4), CSU2(𝔽3), GL2(𝔽3), GL2(𝔽3) [×2], C2×SL2(𝔽3) [×2], C4.A4, S3×D4, D4○SD16, C2×GL2(𝔽3) [×2], Q8.D6 [×2], C4.6S4, C4.3S4, D4.A4, D4.4S4
Quotients:
C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], S4, C22×S3, C2×S4 [×3], C22×S4, D4.4S4
Generators and relations
G = < a,b,c,d,e,f | a4=b2=e3=f2=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=a2c, ece-1=a2cd, fcf=cd, ede-1=c, fdf=a2d, fef=e-1 >
►On 16 points - transitive group 16T422
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(2 4)(5 7)(9 11)(14 16)
(1 6 3 8)(2 7 4 5)(9 16 11 14)(10 13 12 15)
(1 10 3 12)(2 11 4 9)(5 14 7 16)(6 15 8 13)
(5 9 16)(6 10 13)(7 11 14)(8 12 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), (2,4)(5,7)(9,11)(14,16), (1,6,3,8)(2,7,4,5)(9,16,11,14)(10,13,12,15), (1,10,3,12)(2,11,4,9)(5,14,7,16)(6,15,8,13), (5,9,16)(6,10,13)(7,11,14)(8,12,15), (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), (2,4)(5,7)(9,11)(14,16), (1,6,3,8)(2,7,4,5)(9,16,11,14)(10,13,12,15), (1,10,3,12)(2,11,4,9)(5,14,7,16)(6,15,8,13), (5,9,16)(6,10,13)(7,11,14)(8,12,15), (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)], [(2,4),(5,7),(9,11),(14,16)], [(1,6,3,8),(2,7,4,5),(9,16,11,14),(10,13,12,15)], [(1,10,3,12),(2,11,4,9),(5,14,7,16),(6,15,8,13)], [(5,9,16),(6,10,13),(7,11,14),(8,12,15)], [(1,3),(2,4),(5,16),(6,13),(7,14),(8,15)])
G:=TransitiveGroup(16,422);
Matrix representation ►G ⊆ GL4(𝔽3) generated by
| 0 | 0 | 2 | 0 |
| 0 | 0 | 2 | 2 |
| 1 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 1 | 0 | 0 |
| 1 | 2 | 0 | 0 |
| 0 | 0 | 2 | 1 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 2 |
| 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 |
| 2 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 2 | 1 |
G:=sub<GL(4,GF(3))| [0,0,1,2,0,0,0,1,2,2,0,0,0,2,0,0],[2,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1],[1,1,0,0,1,2,0,0,0,0,2,1,0,0,1,1],[0,2,0,0,1,0,0,0,0,0,1,1,0,0,1,2],[1,1,0,0,0,1,0,0,0,0,1,1,0,0,0,1],[2,0,0,0,0,1,0,0,0,0,2,2,0,0,0,1] >;
Character table of D4.4S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 12 | |
| size | 1 | 1 | 2 | 2 | 6 | 12 | 12 | 12 | 8 | 2 | 6 | 6 | 6 | 12 | 8 | 16 | 16 | 6 | 6 | 12 | 12 | 12 | 16 | |
| ρ1 | 1 | 1 | 1 | 1 | 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 | -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 | -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 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | -2 | 2 | 2 | -2 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | 2 | -2 | 2 | -2 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | 2 | 0 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ13 | 3 | 3 | -3 | 3 | 1 | -1 | 1 | 1 | 0 | -3 | 1 | -1 | -1 | -1 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | 0 | orthogonal lifted from C2×S4 |
| ρ14 | 3 | 3 | -3 | -3 | -1 | 1 | -1 | 1 | 0 | 3 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | 0 | orthogonal lifted from C2×S4 |
| ρ15 | 3 | 3 | 3 | -3 | 1 | 1 | 1 | -1 | 0 | -3 | -1 | -1 | 1 | -1 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 0 | orthogonal lifted from C2×S4 |
| ρ16 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | orthogonal lifted from S4 |
| ρ17 | 3 | 3 | 3 | -3 | 1 | -1 | -1 | 1 | 0 | -3 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | 0 | orthogonal lifted from C2×S4 |
| ρ18 | 3 | 3 | 3 | 3 | -1 | 1 | 1 | 1 | 0 | 3 | -1 | -1 | -1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 0 | orthogonal lifted from S4 |
| ρ19 | 3 | 3 | -3 | 3 | 1 | 1 | -1 | -1 | 0 | -3 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | 0 | orthogonal lifted from C2×S4 |
| ρ20 | 3 | 3 | -3 | -3 | -1 | -1 | 1 | -1 | 0 | 3 | 1 | -1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | 0 | orthogonal lifted from C2×S4 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2√-2 | 2√-2 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2√-2 | 2√-2 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
In GAP, Magma, Sage, TeX
D_4._4S_4
% in TeX
G:=Group("D4.4S4"); // GroupNames label
G:=SmallGroup(192,1485);
// by ID
G=gap.SmallGroup(192,1485);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,1059,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=b^2=e^3=f^2=1,c^2=d^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=a^2*c,e*c*e^-1=a^2*c*d,f*c*f=c*d,e*d*e^-1=c,f*d*f=a^2*d,f*e*f=e^-1>;
// generators/relations