metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊3S3, Q8○Dic3, C4.7D6, D12⋊4C2, C6.8C23, C12.7C22, D6.3C22, Dic3.5C22, (C4×S3)⋊3C2, C3⋊3(C4○D4), (C3×Q8)⋊3C2, C2.9(C22×S3), SmallGroup(48,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊3S3
G = < a,b,c,d | a4=c3=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >
Character table of Q8⋊3S3
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6 | 12A | 12B | 12C | |
| size | 1 | 1 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 3 | 3 | 2 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2i | -2i | -2 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2i | 2i | -2 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
►On 24 points - transitive group 24T28
(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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 21)(6 11 22)(7 12 23)(8 9 24)
(1 4)(2 3)(5 9)(6 12)(7 11)(8 10)(13 19)(14 18)(15 17)(16 20)(21 24)(22 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), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,4)(2,3)(5,9)(6,12)(7,11)(8,10)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)>;
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), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,4)(2,3)(5,9)(6,12)(7,11)(8,10)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23) );
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)], [(1,24,3,22),(2,23,4,21),(5,20,7,18),(6,19,8,17),(9,16,11,14),(10,15,12,13)], [(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,10,21),(6,11,22),(7,12,23),(8,9,24)], [(1,4),(2,3),(5,9),(6,12),(7,11),(8,10),(13,19),(14,18),(15,17),(16,20),(21,24),(22,23)]])
G:=TransitiveGroup(24,28);
Q8⋊3S3 is a maximal subgroup of
Q8⋊3D6 Q8.7D6 Q16⋊S3 D24⋊C2 Q8.15D6 S3×C4○D4 D4○D12 Q8⋊3D9 Dic3.A4 D12⋊S3 D6.6D6 C12.26D6 Q8⋊4S4 Q8.7S4 D12⋊D5 D60⋊C2 Q8⋊3D15 D12⋊D7 D84⋊C2 Q8⋊3D21 Q8.A5
Q8⋊3S3 is a maximal quotient of
C4.Dic6 C4⋊C4⋊7S3 Dic3⋊5D4 D6.D4 C12⋊D4 C4⋊C4⋊S3 Q8×Dic3 D6⋊3Q8 C12.23D4 Q8⋊3D9 D12⋊S3 D6.6D6 C12.26D6 Q8⋊4S4 D12⋊D5 D60⋊C2 Q8⋊3D15 D12⋊D7 D84⋊C2 Q8⋊3D21
Matrix representation of Q8⋊3S3 ►in GL4(𝔽5) generated by
| 3 | 0 | 3 | 3 |
| 4 | 2 | 2 | 2 |
| 2 | 2 | 3 | 0 |
| 3 | 3 | 4 | 2 |
| 3 | 3 | 2 | 0 |
| 3 | 2 | 1 | 3 |
| 3 | 0 | 3 | 3 |
| 4 | 2 | 3 | 2 |
| 3 | 3 | 0 | 3 |
| 2 | 2 | 2 | 4 |
| 0 | 3 | 1 | 2 |
| 2 | 4 | 3 | 2 |
| 0 | 0 | 2 | 1 |
| 0 | 3 | 3 | 0 |
| 0 | 4 | 2 | 0 |
| 1 | 2 | 1 | 0 |
G:=sub<GL(4,GF(5))| [3,4,2,3,0,2,2,3,3,2,3,4,3,2,0,2],[3,3,3,4,3,2,0,2,2,1,3,3,0,3,3,2],[3,2,0,2,3,2,3,4,0,2,1,3,3,4,2,2],[0,0,0,1,0,3,4,2,2,3,2,1,1,0,0,0] >;
Q8⋊3S3 in GAP, Magma, Sage, TeX
Q_8\rtimes_3S_3
% in TeX
G:=Group("Q8:3S3"); // GroupNames label
G:=SmallGroup(48,41);
// by ID
G=gap.SmallGroup(48,41);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,46,182,97,42,804]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^3=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of Q8⋊3S3 in TeX
Character table of Q8⋊3S3 in TeX