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## G = Q8×S4order 192 = 26·3

### Direct product of Q8 and S4

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

 Derived series C1 — C22 — C2×A4 — Q8×S4
 Chief series C1 — C22 — A4 — C2×A4 — C2×S4 — C4×S4 — Q8×S4
 Lower central A4 — C2×A4 — Q8×S4
 Upper central C1 — C2 — Q8

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 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 6 12A 12B 12C size 1 1 3 3 6 6 8 2 2 2 6 6 6 6 6 12 12 12 12 12 12 8 16 16 16 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 2 0 0 -1 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 -1 -1 1 1 orthogonal lifted from D6 ρ10 2 2 2 2 0 0 -1 -2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 -1 1 -1 1 orthogonal lifted from D6 ρ11 2 2 2 2 0 0 -1 2 2 2 2 2 2 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 2 0 0 -1 -2 2 -2 2 -2 -2 0 0 0 0 0 0 0 0 -1 1 1 -1 orthogonal lifted from D6 ρ13 2 -2 2 -2 2 -2 2 0 0 0 0 0 0 -2 2 0 0 0 0 0 0 -2 0 0 0 symplectic lifted from Q8, Schur index 2 ρ14 2 -2 2 -2 -2 2 2 0 0 0 0 0 0 2 -2 0 0 0 0 0 0 -2 0 0 0 symplectic lifted from Q8, Schur index 2 ρ15 3 3 -1 -1 1 1 0 3 3 3 -1 -1 -1 -1 -1 -1 -1 1 -1 1 1 0 0 0 0 orthogonal lifted from S4 ρ16 3 3 -1 -1 -1 -1 0 -3 3 -3 -1 1 1 1 1 -1 -1 -1 1 1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ17 3 3 -1 -1 1 1 0 3 -3 -3 1 1 -1 -1 -1 -1 1 -1 1 1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ18 3 3 -1 -1 -1 -1 0 -3 -3 3 1 -1 1 1 1 -1 1 1 -1 1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ19 3 3 -1 -1 -1 -1 0 3 3 3 -1 -1 -1 1 1 1 1 -1 1 -1 -1 0 0 0 0 orthogonal lifted from S4 ρ20 3 3 -1 -1 1 1 0 -3 3 -3 -1 1 1 -1 -1 1 1 1 -1 -1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ21 3 3 -1 -1 -1 -1 0 3 -3 -3 1 1 -1 1 1 1 -1 1 -1 -1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ22 3 3 -1 -1 1 1 0 -3 -3 3 1 -1 1 -1 -1 1 -1 -1 1 -1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ23 4 -4 4 -4 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 symplectic lifted from S3×Q8, Schur index 2 ρ24 6 -6 -2 2 2 -2 0 0 0 0 0 0 0 2 -2 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ25 6 -6 -2 2 -2 2 0 0 0 0 0 0 0 -2 2 0 0 0 0 0 0 0 0 0 0 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)

 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 12 0 0 0 1 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 12 1 0 0 0 12 0 0 0 1 12 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 1 0 12 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 1 0 0 0 12 0 0 0 0 12 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

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

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