non-abelian, soluble, monomial
Aliases: S32⋊Q8, C4.13S3≀C2, (C3×C12).8D4, C3⋊Dic3.26D4, Dic3.D6⋊6C2, C32⋊1(C22⋊Q8), C6.D6.1C22, S32⋊C4.C2, (C4×S32).3C2, C2.4(C2×S3≀C2), (C3×C6).1(C2×D4), C3⋊S3.Q8⋊1C2, C3⋊S3.1(C2×Q8), C4⋊(C32⋊C4)⋊3C2, (C2×S32).5C22, C3⋊S3.1(C4○D4), (C2×C3⋊S3).1C23, (C4×C3⋊S3).27C22, (C2×C32⋊C4).4C22, SmallGroup(288,868)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S32⋊Q8
G = < a,b,c,d,e,f | a3=b2=c3=d2=e4=1, f2=e2, bab=fcf-1=a-1, ac=ca, ad=da, ae=ea, faf-1=dcd=c-1, bc=cb, bd=db, be=eb, fbf-1=d, ce=ec, de=ed, fdf-1=b, fef-1=e-1 >
Subgroups: 600 in 120 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, Q8, C23, C32, Dic3, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C3×S3, C3⋊S3, C3×C6, Dic6, C4×S3, C2×Dic3, C2×C12, C3×Q8, C22×S3, C22⋊Q8, C3×Dic3, C3⋊Dic3, C3×C12, C32⋊C4, S32, S32, S3×C6, C2×C3⋊S3, S3×C2×C4, S3×Q8, S3×Dic3, C6.D6, C6.D6, C32⋊2Q8, C3×Dic6, S3×C12, C4×C3⋊S3, C2×C32⋊C4, C2×S32, S32⋊C4, C3⋊S3.Q8, C4⋊(C32⋊C4), Dic3.D6, C4×S32, S32⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C22⋊Q8, S3≀C2, C2×S3≀C2, S32⋊Q8
Character table of S32⋊Q8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | |
| size | 1 | 1 | 6 | 6 | 9 | 9 | 4 | 4 | 2 | 6 | 6 | 12 | 12 | 18 | 36 | 36 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 12 | 12 | 24 | 24 | |
| ρ1 | 1 | 1 | 1 | 1 | 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 | -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 | -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 | 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 | -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 | 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 | 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 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ12 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ13 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | 2 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | 2 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ15 | 4 | 4 | -2 | -2 | 0 | 0 | -2 | 1 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | 1 | 1 | -2 | 1 | 1 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ16 | 4 | 4 | 2 | 2 | 0 | 0 | -2 | 1 | 4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | -1 | -1 | 1 | 1 | -2 | -1 | -1 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ17 | 4 | 4 | 0 | 0 | 0 | 0 | 1 | -2 | -4 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | -1 | 1 | orthogonal lifted from C2×S3≀C2 |
| ρ18 | 4 | 4 | 0 | 0 | 0 | 0 | 1 | -2 | 4 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | -1 | -1 | orthogonal lifted from S3≀C2 |
| ρ19 | 4 | 4 | 0 | 0 | 0 | 0 | 1 | -2 | 4 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 1 | 1 | orthogonal lifted from S3≀C2 |
| ρ20 | 4 | 4 | 2 | 2 | 0 | 0 | -2 | 1 | -4 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | -1 | -1 | -1 | -1 | 2 | 1 | 1 | 0 | 0 | orthogonal lifted from C2×S3≀C2 |
| ρ21 | 4 | 4 | -2 | -2 | 0 | 0 | -2 | 1 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | orthogonal lifted from C2×S3≀C2 |
| ρ22 | 4 | 4 | 0 | 0 | 0 | 0 | 1 | -2 | -4 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 1 | -1 | orthogonal lifted from C2×S3≀C2 |
| ρ23 | 4 | -4 | -2 | 2 | 0 | 0 | -2 | 1 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | 1 | -3i | 3i | 0 | -i | i | 0 | 0 | complex faithful |
| ρ24 | 4 | -4 | 2 | -2 | 0 | 0 | -2 | 1 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 2 | -1 | 1 | -1 | -3i | 3i | 0 | i | -i | 0 | 0 | complex faithful |
| ρ25 | 4 | -4 | -2 | 2 | 0 | 0 | -2 | 1 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | 1 | 3i | -3i | 0 | i | -i | 0 | 0 | complex faithful |
| ρ26 | 4 | -4 | 2 | -2 | 0 | 0 | -2 | 1 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 2 | -1 | 1 | -1 | 3i | -3i | 0 | -i | i | 0 | 0 | complex faithful |
| ρ27 | 8 | -8 | 0 | 0 | 0 | 0 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 24 points - transitive group 24T641
Generators in S24
(5 21 10)(6 22 11)(7 23 12)(8 24 9)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 20)(14 17)(15 18)(16 19)(21 23)(22 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)
G:=sub<Sym(24)| (5,21,10)(6,22,11)(7,23,12)(8,24,9), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,19,14)(2,20,15)(3,17,16)(4,18,13), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24), (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)>;
G:=Group( (5,21,10)(6,22,11)(7,23,12)(8,24,9), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,19,14)(2,20,15)(3,17,16)(4,18,13), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24), (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) );
G=PermutationGroup([[(5,21,10),(6,22,11),(7,23,12),(8,24,9)], [(1,3),(2,4),(5,12),(6,9),(7,10),(8,11),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,19,14),(2,20,15),(3,17,16),(4,18,13)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,20),(14,17),(15,18),(16,19),(21,23),(22,24)], [(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)]])
G:=TransitiveGroup(24,641);
►On 24 points - transitive group 24T648
Generators in S24
(5 21 10)(6 22 11)(7 23 12)(8 24 9)
(1 3)(2 4)(5 10)(6 11)(7 12)(8 9)(13 15)(14 16)(17 19)(18 20)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)
(5 7)(6 8)(9 11)(10 12)(13 18)(14 19)(15 20)(16 17)(21 23)(22 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)
G:=sub<Sym(24)| (5,21,10)(6,22,11)(7,23,12)(8,24,9), (1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,15)(14,16)(17,19)(18,20), (1,19,14)(2,20,15)(3,17,16)(4,18,13), (5,7)(6,8)(9,11)(10,12)(13,18)(14,19)(15,20)(16,17)(21,23)(22,24), (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)>;
G:=Group( (5,21,10)(6,22,11)(7,23,12)(8,24,9), (1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,15)(14,16)(17,19)(18,20), (1,19,14)(2,20,15)(3,17,16)(4,18,13), (5,7)(6,8)(9,11)(10,12)(13,18)(14,19)(15,20)(16,17)(21,23)(22,24), (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) );
G=PermutationGroup([[(5,21,10),(6,22,11),(7,23,12),(8,24,9)], [(1,3),(2,4),(5,10),(6,11),(7,12),(8,9),(13,15),(14,16),(17,19),(18,20)], [(1,19,14),(2,20,15),(3,17,16),(4,18,13)], [(5,7),(6,8),(9,11),(10,12),(13,18),(14,19),(15,20),(16,17),(21,23),(22,24)], [(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)]])
G:=TransitiveGroup(24,648);
Matrix representation of S32⋊Q8 ►in GL4(𝔽5) generated by
| 0 | 0 | 2 | 0 |
| 0 | 1 | 0 | 0 |
| 2 | 0 | 4 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 3 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 1 | 0 |
| 0 | 3 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 3 | 0 | 4 |
| 3 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 3 | 0 |
| 0 | 3 | 0 | 0 |
| 3 | 0 | 0 | 0 |
G:=sub<GL(4,GF(5))| [0,0,2,0,0,1,0,0,2,0,4,0,0,0,0,1],[4,0,0,0,0,4,0,0,3,0,1,0,0,0,0,4],[1,0,0,0,0,0,0,3,0,0,1,0,0,3,0,4],[4,0,0,0,0,1,0,3,0,0,4,0,0,0,0,4],[3,0,0,0,0,2,0,0,0,0,3,0,0,0,0,2],[0,0,0,3,0,0,3,0,0,3,0,0,3,0,0,0] >;
S32⋊Q8 in GAP, Magma, Sage, TeX
S_3^2\rtimes Q_8
% in TeX
G:=Group("S3^2:Q8"); // GroupNames label
G:=SmallGroup(288,868);
// by ID
G=gap.SmallGroup(288,868);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,120,219,100,2693,2028,362,797,1203]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^3=d^2=e^4=1,f^2=e^2,b*a*b=f*c*f^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=d*c*d=c^-1,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=d,c*e=e*c,d*e=e*d,f*d*f^-1=b,f*e*f^-1=e^-1>;
// generators/relations
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