non-abelian, soluble, monomial
Aliases: Q8.2S4, C23.5S4, 2+ (1+4).3S3, C2.5(C22⋊S4), C23⋊A4.3C2, SmallGroup(192,1492)
Series: Derived ►Chief ►Lower central ►Upper central
| C23⋊A4 — Q8.S4 |
Subgroups: 333 in 66 conjugacy classes, 8 normal (6 characteristic)
C1, C2, C2 [×2], C3, C4 [×4], C22 [×5], C6, C8 [×2], C2×C4 [×4], D4 [×4], Q8 [×2], Q8 [×2], C23, C23 [×2], Dic3, A4 [×2], C42, C22⋊C4 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×2], SL2(𝔽3) [×2], C2×A4 [×2], C4.D4, C4≀C2 [×2], C4.4D4, C8.C22 [×2], 2+ (1+4), CSU2(𝔽3) [×2], A4⋊C4, D4.9D4, C23⋊A4, Q8.S4
Quotients:
C1, C2, S3, S4 [×3], C22⋊S4, Q8.S4
Generators and relations
G = < a,b,c,d,e,f | a4=c2=d2=e3=1, b2=f2=a2, bab-1=cac=dad=fbf-1=a-1, eae-1=a-1b, faf-1=dbd=a2b, bc=cb, ebe-1=a, ece-1=fcf-1=cd=dc, ede-1=c, df=fd, fef-1=e-1 >
►On 16 points - transitive group 16T443
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 11 3 9)(2 10 4 12)(5 16 7 14)(6 15 8 13)
(1 4)(2 3)(5 15)(6 14)(7 13)(8 16)(9 10)(11 12)
(1 9)(2 12)(3 11)(4 10)(5 6)(7 8)(13 16)(14 15)
(2 11 12)(4 9 10)(5 6 15)(7 8 13)
(1 14 3 16)(2 7 4 5)(6 12 8 10)(9 15 11 13)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,11,3,9)(2,10,4,12)(5,16,7,14)(6,15,8,13), (1,4)(2,3)(5,15)(6,14)(7,13)(8,16)(9,10)(11,12), (1,9)(2,12)(3,11)(4,10)(5,6)(7,8)(13,16)(14,15), (2,11,12)(4,9,10)(5,6,15)(7,8,13), (1,14,3,16)(2,7,4,5)(6,12,8,10)(9,15,11,13)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,11,3,9)(2,10,4,12)(5,16,7,14)(6,15,8,13), (1,4)(2,3)(5,15)(6,14)(7,13)(8,16)(9,10)(11,12), (1,9)(2,12)(3,11)(4,10)(5,6)(7,8)(13,16)(14,15), (2,11,12)(4,9,10)(5,6,15)(7,8,13), (1,14,3,16)(2,7,4,5)(6,12,8,10)(9,15,11,13) );
G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,11,3,9),(2,10,4,12),(5,16,7,14),(6,15,8,13)], [(1,4),(2,3),(5,15),(6,14),(7,13),(8,16),(9,10),(11,12)], [(1,9),(2,12),(3,11),(4,10),(5,6),(7,8),(13,16),(14,15)], [(2,11,12),(4,9,10),(5,6,15),(7,8,13)], [(1,14,3,16),(2,7,4,5),(6,12,8,10),(9,15,11,13)])
G:=TransitiveGroup(16,443);
►On 16 points - transitive group 16T446
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 9 3 11)(2 12 4 10)(5 14 7 16)(6 13 8 15)
(2 4)(5 7)(10 12)(14 16)
(2 4)(6 8)(9 11)(14 16)
(2 9 10)(4 11 12)(5 14 8)(6 7 16)
(1 15 3 13)(2 8 4 6)(5 12 7 10)(9 14 11 16)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,9,3,11)(2,12,4,10)(5,14,7,16)(6,13,8,15), (2,4)(5,7)(10,12)(14,16), (2,4)(6,8)(9,11)(14,16), (2,9,10)(4,11,12)(5,14,8)(6,7,16), (1,15,3,13)(2,8,4,6)(5,12,7,10)(9,14,11,16)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,9,3,11)(2,12,4,10)(5,14,7,16)(6,13,8,15), (2,4)(5,7)(10,12)(14,16), (2,4)(6,8)(9,11)(14,16), (2,9,10)(4,11,12)(5,14,8)(6,7,16), (1,15,3,13)(2,8,4,6)(5,12,7,10)(9,14,11,16) );
G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,9,3,11),(2,12,4,10),(5,14,7,16),(6,13,8,15)], [(2,4),(5,7),(10,12),(14,16)], [(2,4),(6,8),(9,11),(14,16)], [(2,9,10),(4,11,12),(5,14,8),(6,7,16)], [(1,15,3,13),(2,8,4,6),(5,12,7,10),(9,14,11,16)])
G:=TransitiveGroup(16,446);
Matrix representation ►G ⊆ GL4(𝔽5) generated by
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 3 | 0 |
| 0 | 3 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(5))| [0,0,4,0,0,0,0,1,1,0,0,0,0,4,0,0],[0,0,0,2,0,0,3,0,0,3,0,0,2,0,0,0],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,0,3,0,0,0,0,3,0,4,0,0],[2,0,0,0,0,3,0,0,0,0,0,4,0,0,1,0] >;
Character table of Q8.S4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6 | 8A | 8B | |
| size | 1 | 1 | 6 | 12 | 32 | 6 | 6 | 12 | 12 | 24 | 32 | 24 | 24 | |
| 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 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ4 | 3 | 3 | 3 | -1 | 0 | -1 | -1 | 1 | 1 | 1 | 0 | -1 | -1 | orthogonal lifted from S4 |
| ρ5 | 3 | 3 | -1 | -1 | 0 | 3 | -1 | 1 | 1 | -1 | 0 | -1 | 1 | orthogonal lifted from S4 |
| ρ6 | 3 | 3 | 3 | -1 | 0 | -1 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | orthogonal lifted from S4 |
| ρ7 | 3 | 3 | -1 | -1 | 0 | 3 | -1 | -1 | -1 | 1 | 0 | 1 | -1 | orthogonal lifted from S4 |
| ρ8 | 3 | 3 | -1 | -1 | 0 | -1 | 3 | -1 | -1 | 1 | 0 | -1 | 1 | orthogonal lifted from S4 |
| ρ9 | 3 | 3 | -1 | -1 | 0 | -1 | 3 | 1 | 1 | -1 | 0 | 1 | -1 | orthogonal lifted from S4 |
| ρ10 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 2i | 2i | 0 | -1 | 0 | 0 | complex faithful |
| ρ11 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 2i | 2i | 0 | -1 | 0 | 0 | complex faithful |
| ρ12 | 6 | 6 | -2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C22⋊S4 |
| ρ13 | 8 | -8 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | symplectic faithful, Schur index 2 |
In GAP, Magma, Sage, TeX
Q_8.S_4
% in TeX
G:=Group("Q8.S4"); // GroupNames label
G:=SmallGroup(192,1492);
// by ID
G=gap.SmallGroup(192,1492);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,57,254,135,171,262,1684,1271,718,172,1013,2532,530,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=c^2=d^2=e^3=1,b^2=f^2=a^2,b*a*b^-1=c*a*c=d*a*d=f*b*f^-1=a^-1,e*a*e^-1=a^-1*b,f*a*f^-1=d*b*d=a^2*b,b*c=c*b,e*b*e^-1=a,e*c*e^-1=f*c*f^-1=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations
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ℤ
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