non-abelian, soluble, monomial, rational
Aliases: Q8:2S4, C23.6S4, 2+ 1+4:5S3, C23:A4:3C2, C2.7(C22:S4), Hol(Q8), SmallGroup(192,1494)
Series: Derived ►Chief ►Lower central ►Upper central
| C23:A4 — Q8:2S4 |
Generators and relations for Q8:2S4
G = < a,b,c,d,e,f | a4=c2=d2=e3=f2=1, b2=a2, bab-1=cac=dad=fbf=a-1, eae-1=a-1b, faf=dbd=a2b, bc=cb, ebe-1=a, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >
Subgroups: 469 in 78 conjugacy classes, 8 normal (6 characteristic)
C1, C2, C2, C3, C4, C22, S3, C6, C8, C2xC4, D4, Q8, C23, C23, A4, D6, C42, M4(2), D8, SD16, C2xD4, C4oD4, SL2(F3), S4, C2xA4, C4.D4, C4wrC2, C4:1D4, C8:C22, 2+ 1+4, GL2(F3), C2xS4, D4:4D4, C23:A4, Q8:2S4
Quotients: C1, C2, S3, S4, C22:S4, Q8:2S4
Character table of Q8:2S4
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 6 | 8A | 8B | |
| size | 1 | 1 | 6 | 12 | 24 | 32 | 6 | 6 | 12 | 12 | 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 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ4 | 3 | 3 | 3 | -1 | -1 | 0 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | orthogonal lifted from S4 |
| ρ5 | 3 | 3 | -1 | -1 | 1 | 0 | 3 | -1 | -1 | -1 | 0 | -1 | 1 | orthogonal lifted from S4 |
| ρ6 | 3 | 3 | -1 | -1 | -1 | 0 | -1 | 3 | 1 | 1 | 0 | -1 | 1 | orthogonal lifted from S4 |
| ρ7 | 3 | 3 | -1 | -1 | 1 | 0 | -1 | 3 | -1 | -1 | 0 | 1 | -1 | orthogonal lifted from S4 |
| ρ8 | 3 | 3 | -1 | -1 | -1 | 0 | 3 | -1 | 1 | 1 | 0 | 1 | -1 | orthogonal lifted from S4 |
| ρ9 | 3 | 3 | 3 | -1 | 1 | 0 | -1 | -1 | 1 | 1 | 0 | -1 | -1 | orthogonal lifted from S4 |
| ρ10 | 4 | -4 | 0 | 0 | 0 | 1 | 0 | 0 | -2 | 2 | -1 | 0 | 0 | orthogonal faithful |
| ρ11 | 4 | -4 | 0 | 0 | 0 | 1 | 0 | 0 | 2 | -2 | -1 | 0 | 0 | orthogonal faithful |
| ρ12 | 6 | 6 | -2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C22:S4 |
| ρ13 | 8 | -8 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | orthogonal faithful |
►On 8 points - transitive group 8T40
Generators in S8
(1 2 3 4)(5 6 7 8)
(1 6 3 8)(2 5 4 7)
(2 4)(5 7)
(2 4)(6 8)
(2 6 7)(4 8 5)
(1 3)(2 6)(4 8)
G:=sub<Sym(8)| (1,2,3,4)(5,6,7,8), (1,6,3,8)(2,5,4,7), (2,4)(5,7), (2,4)(6,8), (2,6,7)(4,8,5), (1,3)(2,6)(4,8)>;
G:=Group( (1,2,3,4)(5,6,7,8), (1,6,3,8)(2,5,4,7), (2,4)(5,7), (2,4)(6,8), (2,6,7)(4,8,5), (1,3)(2,6)(4,8) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8)], [(1,6,3,8),(2,5,4,7)], [(2,4),(5,7)], [(2,4),(6,8)], [(2,6,7),(4,8,5)], [(1,3),(2,6),(4,8)]])
G:=TransitiveGroup(8,40);
►On 16 points - transitive group 16T444
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)
(1 4)(2 3)(5 13)(6 16)(7 15)(8 14)(9 10)(11 12)
(1 11)(2 10)(3 9)(4 12)(5 6)(7 8)(13 16)(14 15)
(2 9 10)(4 11 12)(5 6 13)(7 8 15)
(1 14)(2 5)(3 16)(4 7)(6 10)(8 12)(9 13)(11 15)
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), (1,4)(2,3)(5,13)(6,16)(7,15)(8,14)(9,10)(11,12), (1,11)(2,10)(3,9)(4,12)(5,6)(7,8)(13,16)(14,15), (2,9,10)(4,11,12)(5,6,13)(7,8,15), (1,14)(2,5)(3,16)(4,7)(6,10)(8,12)(9,13)(11,15)>;
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), (1,4)(2,3)(5,13)(6,16)(7,15)(8,14)(9,10)(11,12), (1,11)(2,10)(3,9)(4,12)(5,6)(7,8)(13,16)(14,15), (2,9,10)(4,11,12)(5,6,13)(7,8,15), (1,14)(2,5)(3,16)(4,7)(6,10)(8,12)(9,13)(11,15) );
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)], [(1,4),(2,3),(5,13),(6,16),(7,15),(8,14),(9,10),(11,12)], [(1,11),(2,10),(3,9),(4,12),(5,6),(7,8),(13,16),(14,15)], [(2,9,10),(4,11,12),(5,6,13),(7,8,15)], [(1,14),(2,5),(3,16),(4,7),(6,10),(8,12),(9,13),(11,15)]])
G:=TransitiveGroup(16,444);
►On 16 points - transitive group 16T445
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 13)(2 6)(3 15)(4 8)(5 12)(7 10)(9 16)(11 14)
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), (1,13)(2,6)(3,15)(4,8)(5,12)(7,10)(9,16)(11,14)>;
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), (1,13)(2,6)(3,15)(4,8)(5,12)(7,10)(9,16)(11,14) );
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)], [(1,13),(2,6),(3,15),(4,8),(5,12),(7,10),(9,16),(11,14)]])
G:=TransitiveGroup(16,445);
►On 24 points - transitive group 24T332
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 10 3 12)(2 9 4 11)(5 23 7 21)(6 22 8 24)(13 17 15 19)(14 20 16 18)
(2 4)(6 8)(9 11)(14 16)(18 20)(22 24)
(2 4)(5 7)(10 12)(13 15)(18 20)(22 24)
(1 17 23)(2 15 8)(3 19 21)(4 13 6)(5 9 20)(7 11 18)(10 16 24)(12 14 22)
(1 3)(2 10)(4 12)(5 20)(6 14)(7 18)(8 16)(13 22)(15 24)(17 21)(19 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), (1,10,3,12)(2,9,4,11)(5,23,7,21)(6,22,8,24)(13,17,15,19)(14,20,16,18), (2,4)(6,8)(9,11)(14,16)(18,20)(22,24), (2,4)(5,7)(10,12)(13,15)(18,20)(22,24), (1,17,23)(2,15,8)(3,19,21)(4,13,6)(5,9,20)(7,11,18)(10,16,24)(12,14,22), (1,3)(2,10)(4,12)(5,20)(6,14)(7,18)(8,16)(13,22)(15,24)(17,21)(19,23)>;
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), (1,10,3,12)(2,9,4,11)(5,23,7,21)(6,22,8,24)(13,17,15,19)(14,20,16,18), (2,4)(6,8)(9,11)(14,16)(18,20)(22,24), (2,4)(5,7)(10,12)(13,15)(18,20)(22,24), (1,17,23)(2,15,8)(3,19,21)(4,13,6)(5,9,20)(7,11,18)(10,16,24)(12,14,22), (1,3)(2,10)(4,12)(5,20)(6,14)(7,18)(8,16)(13,22)(15,24)(17,21)(19,23) );
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)], [(1,10,3,12),(2,9,4,11),(5,23,7,21),(6,22,8,24),(13,17,15,19),(14,20,16,18)], [(2,4),(6,8),(9,11),(14,16),(18,20),(22,24)], [(2,4),(5,7),(10,12),(13,15),(18,20),(22,24)], [(1,17,23),(2,15,8),(3,19,21),(4,13,6),(5,9,20),(7,11,18),(10,16,24),(12,14,22)], [(1,3),(2,10),(4,12),(5,20),(6,14),(7,18),(8,16),(13,22),(15,24),(17,21),(19,23)]])
G:=TransitiveGroup(24,332);
►On 24 points - transitive group 24T430
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 7 3 5)(2 6 4 8)(9 14 11 16)(10 13 12 15)(17 23 19 21)(18 22 20 24)
(1 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 16)(14 15)(18 20)(22 24)
(1 3)(6 8)(9 12)(10 11)(13 14)(15 16)(17 21)(18 24)(19 23)(20 22)
(1 15 20)(2 10 21)(3 13 18)(4 12 23)(5 9 19)(6 14 22)(7 11 17)(8 16 24)
(2 5)(4 7)(6 8)(9 21)(10 19)(11 23)(12 17)(13 18)(14 24)(15 20)(16 22)
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), (1,7,3,5)(2,6,4,8)(9,14,11,16)(10,13,12,15)(17,23,19,21)(18,22,20,24), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,16)(14,15)(18,20)(22,24), (1,3)(6,8)(9,12)(10,11)(13,14)(15,16)(17,21)(18,24)(19,23)(20,22), (1,15,20)(2,10,21)(3,13,18)(4,12,23)(5,9,19)(6,14,22)(7,11,17)(8,16,24), (2,5)(4,7)(6,8)(9,21)(10,19)(11,23)(12,17)(13,18)(14,24)(15,20)(16,22)>;
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), (1,7,3,5)(2,6,4,8)(9,14,11,16)(10,13,12,15)(17,23,19,21)(18,22,20,24), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,16)(14,15)(18,20)(22,24), (1,3)(6,8)(9,12)(10,11)(13,14)(15,16)(17,21)(18,24)(19,23)(20,22), (1,15,20)(2,10,21)(3,13,18)(4,12,23)(5,9,19)(6,14,22)(7,11,17)(8,16,24), (2,5)(4,7)(6,8)(9,21)(10,19)(11,23)(12,17)(13,18)(14,24)(15,20)(16,22) );
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)], [(1,7,3,5),(2,6,4,8),(9,14,11,16),(10,13,12,15),(17,23,19,21),(18,22,20,24)], [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,16),(14,15),(18,20),(22,24)], [(1,3),(6,8),(9,12),(10,11),(13,14),(15,16),(17,21),(18,24),(19,23),(20,22)], [(1,15,20),(2,10,21),(3,13,18),(4,12,23),(5,9,19),(6,14,22),(7,11,17),(8,16,24)], [(2,5),(4,7),(6,8),(9,21),(10,19),(11,23),(12,17),(13,18),(14,24),(15,20),(16,22)]])
G:=TransitiveGroup(24,430);
Polynomial with Galois group Q8:2S4 over Q
| action | f(x) | Disc(f) |
|---|---|---|
| 8T40 | x8+4x7-2x6-20x5-10x4+18x3+10x2-3x-1 | 33·174·593 |
Matrix representation of Q8:2S4 ►in GL4(Z) generated by
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | -1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | -1 | 0 | 0 |
| -1 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,Integers())| [0,0,1,0,0,0,0,-1,-1,0,0,0,0,1,0,0],[0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1],[1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,1],[1,0,0,0,0,0,0,-1,0,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,1] >;
Q8:2S4 in GAP, Magma, Sage, TeX
Q_8\rtimes_2S_4
% in TeX
G:=Group("Q8:2S4"); // GroupNames label
G:=SmallGroup(192,1494);
// by ID
G=gap.SmallGroup(192,1494);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,57,254,135,171,262,1684,1271,718,172,1013,2532,530,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=c^2=d^2=e^3=f^2=1,b^2=a^2,b*a*b^-1=c*a*c=d*a*d=f*b*f=a^-1,e*a*e^-1=a^-1*b,f*a*f=d*b*d=a^2*b,b*c=c*b,e*b*e^-1=a,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations
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