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## G = Q8⋊4S4order 192 = 26·3

### The semidirect product of Q8 and S4 acting through Inn(Q8)

Aliases: Q84S4, C4⋊S44C2, (C4×S4)⋊3C2, (Q8×A4)⋊3C2, C4.11(C2×S4), A43(C4○D4), (C22×Q8)⋊6S3, (C22×C4).7D6, A4⋊C4.5C22, (C2×S4).3C22, (C4×A4).7C22, (C2×A4).8C23, C2.12(C22×S4), C22⋊(Q83S3), C23.8(C22×S3), SmallGroup(192,1478)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×A4 — Q8⋊4S4
 Chief series C1 — C22 — A4 — C2×A4 — C2×S4 — C4×S4 — Q8⋊4S4
 Lower central A4 — C2×A4 — Q8⋊4S4
 Upper central C1 — C2 — Q8

Generators and relations for Q84S4
G = < a,b,c,d,e,f | a4=c2=d2=e3=f2=1, b2=a2, bab-1=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 598 in 163 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2 [×5], C3, C4 [×3], C4 [×8], C22, C22 [×11], S3 [×3], C6, C2×C4 [×17], D4 [×12], Q8, Q8 [×4], C23, C23 [×3], Dic3, C12 [×3], A4, D6 [×3], C42 [×3], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×3], C2×D4 [×6], C2×Q8 [×4], C4○D4 [×4], C4×S3 [×3], D12 [×3], C3×Q8, S4 [×3], C2×A4, C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, A4⋊C4, C4×A4 [×3], Q83S3, C2×S4 [×3], Q85D4, C4×S4 [×3], C4⋊S4 [×3], Q8×A4, Q84S4
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], C4○D4, S4, C22×S3, Q83S3, C2×S4 [×3], C22×S4, Q84S4

Character table of Q84S4

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 6 12A 12B 12C size 1 1 3 3 12 12 12 8 2 2 2 6 6 6 6 6 6 6 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 0 -1 2 2 2 2 2 2 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 2 0 0 0 -1 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 -1 -1 1 1 orthogonal lifted from D6 ρ11 2 2 2 2 0 0 0 -1 -2 -2 2 2 -2 -2 0 0 0 0 0 0 0 -1 1 1 -1 orthogonal lifted from D6 ρ12 2 2 2 2 0 0 0 -1 -2 2 -2 -2 2 -2 0 0 0 0 0 0 0 -1 1 -1 1 orthogonal lifted from D6 ρ13 2 -2 2 -2 0 0 0 2 0 0 0 0 0 0 2i 2i -2i -2i 0 0 0 -2 0 0 0 complex lifted from C4○D4 ρ14 2 -2 2 -2 0 0 0 2 0 0 0 0 0 0 -2i -2i 2i 2i 0 0 0 -2 0 0 0 complex lifted from C4○D4 ρ15 3 3 -1 -1 -1 1 -1 0 -3 -3 3 -1 1 1 1 -1 1 -1 1 1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ16 3 3 -1 -1 1 1 -1 0 3 -3 -3 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 1 0 3 3 3 -1 -1 -1 1 -1 1 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S4 ρ18 3 3 -1 -1 -1 1 1 0 -3 3 -3 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 -1 0 3 3 3 -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 -1 0 -3 3 -3 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 1 0 -3 -3 3 -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 1 0 3 -3 -3 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 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2 ρ24 6 -6 -2 2 0 0 0 0 0 0 0 0 0 0 2i -2i -2i 2i 0 0 0 0 0 0 0 complex faithful ρ25 6 -6 -2 2 0 0 0 0 0 0 0 0 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 0 complex faithful

Permutation representations of Q84S4
On 24 points - transitive group 24T320
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 17 11 19)(10 20 12 18)
(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 9 16)(2 10 13)(3 11 14)(4 12 15)(5 21 19)(6 22 20)(7 23 17)(8 24 18)
(1 4)(2 3)(5 20)(6 19)(7 18)(8 17)(9 15)(10 14)(11 13)(12 16)(21 22)(23 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)(9,17,11,19)(10,20,12,18), (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), (1,9,16)(2,10,13)(3,11,14)(4,12,15)(5,21,19)(6,22,20)(7,23,17)(8,24,18), (1,4)(2,3)(5,20)(6,19)(7,18)(8,17)(9,15)(10,14)(11,13)(12,16)(21,22)(23,24)>;

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

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

G:=TransitiveGroup(24,320);

Matrix representation of Q84S4 in GL5(𝔽13)

 8 7 0 0 0 0 5 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 8 0 0 0 0 4 5 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 6 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(5,GF(13))| [8,0,0,0,0,7,5,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[8,4,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[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],[12,6,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

Q84S4 in GAP, Magma, Sage, TeX

Q_8\rtimes_4S_4
% in TeX

G:=Group("Q8:4S4");
// GroupNames label

G:=SmallGroup(192,1478);
// by ID

G=gap.SmallGroup(192,1478);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,64,254,135,58,1124,4037,285,2358,475]);
// 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=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*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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