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G = Q8×S4order 192 = 26·3

Direct product of Q8 and S4

direct product, non-abelian, soluble, monomial, rational

Aliases: Q8×S4, C22⋊(S3×Q8), A4⋊Q84C2, A42(C2×Q8), (Q8×A4)⋊2C2, (C4×S4).1C2, C4.10(C2×S4), (C22×Q8)⋊5S3, (C22×C4).6D6, A4⋊C4.4C22, (C4×A4).6C22, (C2×A4).7C23, (C2×S4).5C22, C2.11(C22×S4), C23.7(C22×S3), SmallGroup(192,1477)

Series: Derived Chief Lower central Upper central

C1C22C2×A4 — Q8×S4
C1C22A4C2×A4C2×S4C4×S4 — Q8×S4
A4C2×A4 — Q8×S4
C1C2Q8

Generators and relations for Q8×S4
 G = < a,b,c,d,e,f | a4=c2=d2=e3=f2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 506 in 156 conjugacy classes, 31 normal (11 characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C4 [×11], C22, C22 [×6], S3 [×2], C6, C2×C4 [×19], D4 [×4], Q8, Q8 [×10], C23, C23, Dic3 [×3], C12 [×3], A4, D6, C42 [×3], C22⋊C4 [×6], C4⋊C4 [×12], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×Q8 [×11], Dic6 [×3], C4×S3 [×3], C3×Q8, S4 [×2], C2×A4, C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C4⋊Q8 [×3], C22×Q8, C22×Q8, A4⋊C4 [×3], C4×A4 [×3], S3×Q8, C2×S4, D4×Q8, A4⋊Q8 [×3], C4×S4 [×3], Q8×A4, Q8×S4
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D6 [×3], C2×Q8, S4, C22×S3, S3×Q8, C2×S4 [×3], C22×S4, Q8×S4

Character table of Q8×S4

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H4I4J4K4L4M4N612A12B12C
 size 1133668222666661212121212128161616
ρ11111111111111111111111111    trivial
ρ21111111-1-11-11-111-11-1-1-111-11-1    linear of order 2
ρ31111-1-11-1-11-11-1-1-11-1111-11-11-1    linear of order 2
ρ41111-1-11111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ511111111-1-1-1-11111-1-1-11-111-1-1    linear of order 2
ρ61111111-11-11-1-111-1-111-1-11-1-11    linear of order 2
ρ71111-1-11-11-11-1-1-1-111-1-1111-1-11    linear of order 2
ρ81111-1-111-1-1-1-11-1-1-1111-1111-1-1    linear of order 2
ρ9222200-12-2-2-2-2200000000-1-111    orthogonal lifted from D6
ρ10222200-1-2-22-22-200000000-11-11    orthogonal lifted from D6
ρ11222200-122222200000000-1-1-1-1    orthogonal lifted from S3
ρ12222200-1-22-22-2-200000000-111-1    orthogonal lifted from D6
ρ132-22-22-22000000-22000000-2000    symplectic lifted from Q8, Schur index 2
ρ142-22-2-2220000002-2000000-2000    symplectic lifted from Q8, Schur index 2
ρ1533-1-1110333-1-1-1-1-1-1-11-1110000    orthogonal lifted from S4
ρ1633-1-1-1-10-33-3-11111-1-1-11110000    orthogonal lifted from C2×S4
ρ1733-1-11103-3-311-1-1-1-11-111-10000    orthogonal lifted from C2×S4
ρ1833-1-1-1-10-3-331-1111-111-11-10000    orthogonal lifted from C2×S4
ρ1933-1-1-1-10333-1-1-11111-11-1-10000    orthogonal lifted from S4
ρ2033-1-1110-33-3-111-1-1111-1-1-10000    orthogonal lifted from C2×S4
ρ2133-1-1-1-103-3-311-1111-11-1-110000    orthogonal lifted from C2×S4
ρ2233-1-1110-3-331-11-1-11-1-11-110000    orthogonal lifted from C2×S4
ρ234-44-400-2000000000000002000    symplectic lifted from S3×Q8, Schur index 2
ρ246-6-222-200000002-20000000000    symplectic faithful, Schur index 2
ρ256-6-22-220000000-220000000000    symplectic faithful, Schur index 2

Permutation representations of Q8×S4
On 24 points - transitive group 24T321
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 23 3 21)(2 22 4 24)(5 16 7 14)(6 15 8 13)(9 19 11 17)(10 18 12 20)
(1 3)(2 4)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 11 16)(2 12 13)(3 9 14)(4 10 15)(5 21 19)(6 22 20)(7 23 17)(8 24 18)
(1 3)(2 4)(5 17)(6 18)(7 19)(8 20)(9 16)(10 13)(11 14)(12 15)(21 23)(22 24)

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), (1,23,3,21)(2,22,4,24)(5,16,7,14)(6,15,8,13)(9,19,11,17)(10,18,12,20), (1,3)(2,4)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24), (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,11,16)(2,12,13)(3,9,14)(4,10,15)(5,21,19)(6,22,20)(7,23,17)(8,24,18), (1,3)(2,4)(5,17)(6,18)(7,19)(8,20)(9,16)(10,13)(11,14)(12,15)(21,23)(22,24)>;

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), (1,23,3,21)(2,22,4,24)(5,16,7,14)(6,15,8,13)(9,19,11,17)(10,18,12,20), (1,3)(2,4)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24), (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,11,16)(2,12,13)(3,9,14)(4,10,15)(5,21,19)(6,22,20)(7,23,17)(8,24,18), (1,3)(2,4)(5,17)(6,18)(7,19)(8,20)(9,16)(10,13)(11,14)(12,15)(21,23)(22,24) );

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)], [(1,23,3,21),(2,22,4,24),(5,16,7,14),(6,15,8,13),(9,19,11,17),(10,18,12,20)], [(1,3),(2,4),(9,11),(10,12),(17,19),(18,20),(21,23),(22,24)], [(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20)], [(1,11,16),(2,12,13),(3,9,14),(4,10,15),(5,21,19),(6,22,20),(7,23,17),(8,24,18)], [(1,3),(2,4),(5,17),(6,18),(7,19),(8,20),(9,16),(10,13),(11,14),(12,15),(21,23),(22,24)])

G:=TransitiveGroup(24,321);

Matrix representation of Q8×S4 in GL5(𝔽13)

05000
50000
001200
000120
000012
,
012000
10000
00100
00010
00001
,
10000
01000
000121
000120
001120
,
10000
01000
000112
001012
000012
,
10000
01000
001210
001200
001201
,
120000
012000
00010
00100
00001

G:=sub<GL(5,GF(13))| [0,5,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[0,1,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,12,12,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,12,12,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

Q8×S4 in GAP, Magma, Sage, TeX

Q_8\times S_4
% in TeX

G:=Group("Q8xS4");
// GroupNames label

G:=SmallGroup(192,1477);
// by ID

G=gap.SmallGroup(192,1477);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,64,135,58,1124,4037,285,2358,475]);
// 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=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,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

Export

Character table of Q8×S4 in TeX

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