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G = Q8xS4  order 192 = 26·3

Direct product of Q8 and S4

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

Aliases: Q8xS4, C22:(S3xQ8), A4:Q8:4C2, A4:2(C2xQ8), (Q8xA4):2C2, (C4xS4).1C2, C4.10(C2xS4), (C22xQ8):5S3, (C22xC4).6D6, A4:C4.4C22, (C4xA4).6C22, (C2xA4).7C23, (C2xS4).5C22, C2.11(C22xS4), C23.7(C22xS3), SmallGroup(192,1477)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C2xA4 — Q8xS4
Chief series
C1 — C22 — A4 — C2xA4 — C2xS4 — C4xS4 — Q8xS4
Lower central
A4 — C2xA4 — Q8xS4
Upper central
C1 — C2 — Q8

Generators and relations for Q8xS4
 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, C3, C4, C4, C22, C22, S3, C6, C2xC4, D4, Q8, Q8, C23, C23, Dic3, C12, A4, D6, C42, C22:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xQ8, Dic6, C4xS3, C3xQ8, S4, C2xA4, C4xD4, C4xQ8, C22:Q8, C4:Q8, C22xQ8, C22xQ8, A4:C4, C4xA4, S3xQ8, C2xS4, D4xQ8, A4:Q8, C4xS4, Q8xA4, Q8xS4
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2xQ8, S4, C22xS3, S3xQ8, C2xS4, C22xS4, Q8xS4

Character table of Q8xS4

 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 C2xS4
ρ1733-1-11103-3-311-1-1-1-11-111-10000    orthogonal lifted from C2xS4
ρ1833-1-1-1-10-3-331-1111-111-11-10000    orthogonal lifted from C2xS4
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 C2xS4
ρ2133-1-1-1-103-3-311-1111-11-1-110000    orthogonal lifted from C2xS4
ρ2233-1-1110-3-331-11-1-11-1-11-110000    orthogonal lifted from C2xS4
ρ234-44-400-2000000000000002000    symplectic lifted from S3xQ8, 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 Q8xS4
►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 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 3)(2 4)(5 17)(6 18)(7 19)(8 20)(9 14)(10 15)(11 16)(12 13)(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)(9,17,11,19)(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), (1,3)(2,4)(5,17)(6,18)(7,19)(8,20)(9,14)(10,15)(11,16)(12,13)(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)(9,17,11,19)(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), (1,3)(2,4)(5,17)(6,18)(7,19)(8,20)(9,14)(10,15)(11,16)(12,13)(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),(9,17,11,19),(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)], [(1,3),(2,4),(5,17),(6,18),(7,19),(8,20),(9,14),(10,15),(11,16),(12,13),(21,23),(22,24)]])
 
G:=TransitiveGroup(24,321);
 

Matrix representation of Q8xS4 ►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] >;
 

Q8xS4 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 Q8xS4 in TeX

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