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G = Q8⋊4S4  order 192 = 26·3

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

non-abelian, soluble, monomial

Aliases: Q8⋊4S4, C4⋊S4⋊4C2, (C4×S4)⋊3C2, (Q8×A4)⋊3C2, C4.11(C2×S4), A4⋊3(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⋊(Q8⋊3S3), 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 Q8⋊4S4
 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, C3, C4, C4, C22, C22, S3, C6, C2×C4, D4, Q8, Q8, C23, C23, Dic3, C12, A4, D6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4×S3, D12, C3×Q8, S4, C2×A4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, A4⋊C4, C4×A4, Q8⋊3S3, C2×S4, Q8⋊5D4, C4×S4, C4⋊S4, Q8×A4, Q8⋊4S4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, S4, C22×S3, Q8⋊3S3, C2×S4, C22×S4, Q8⋊4S4

Character table of Q8⋊4S4

 class 12A2B2C2D2E2F34A4B4C4D4E4F4G4H4I4J4K4L4M612A12B12C
 size 1133121212822266666661212128161616
ρ11111111111111111111111111    trivial
ρ21111-1-1111-1-1-1-111111-11-111-1-1    linear of order 2
ρ31111-1111-11-1-11-1-1-1-1-1-1111-11-1    linear of order 2
ρ411111-111-1-111-1-1-1-1-1-111-11-1-11    linear of order 2
ρ511111-1-11-11-1-11-111111-1-11-11-1    linear of order 2
ρ61111-11-11-1-111-1-11111-1-111-1-11    linear of order 2
ρ71111-1-1-11111111-1-1-1-1-1-1-11111    linear of order 2
ρ8111111-111-1-1-1-11-1-1-1-11-1111-1-1    linear of order 2
ρ92222000-12222220000000-1-1-1-1    orthogonal lifted from S3
ρ102222000-12-2-2-2-220000000-1-111    orthogonal lifted from D6
ρ112222000-1-2-222-2-20000000-111-1    orthogonal lifted from D6
ρ122222000-1-22-2-22-20000000-11-11    orthogonal lifted from D6
ρ132-22-200020000002i2i-2i-2i000-2000    complex lifted from C4○D4
ρ142-22-20002000000-2i-2i2i2i000-2000    complex lifted from C4○D4
ρ1533-1-1-11-10-3-33-1111-11-111-10000    orthogonal lifted from C2×S4
ρ1633-1-111-103-3-311-1-11-11-11-10000    orthogonal lifted from C2×S4
ρ1733-1-11110333-1-1-11-11-1-1-1-10000    orthogonal lifted from S4
ρ1833-1-1-1110-33-31-11-11-111-1-10000    orthogonal lifted from C2×S4
ρ1933-1-1-1-1-10333-1-1-1-11-111110000    orthogonal lifted from S4
ρ2033-1-11-1-10-33-31-111-11-1-1110000    orthogonal lifted from C2×S4
ρ2133-1-11-110-3-33-111-11-11-1-110000    orthogonal lifted from C2×S4
ρ2233-1-1-1-1103-3-311-11-11-11-110000    orthogonal lifted from C2×S4
ρ234-44-4000-200000000000002000    orthogonal lifted from Q8⋊3S3, Schur index 2
ρ246-6-2200000000002i-2i-2i2i0000000    complex faithful
ρ256-6-220000000000-2i2i2i-2i0000000    complex faithful

Permutation representations of Q8⋊4S4
►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 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 4)(2 3)(5 20)(6 19)(7 18)(8 17)(9 13)(10 16)(11 15)(12 14)(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)(9,19,11,17)(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), (1,4)(2,3)(5,20)(6,19)(7,18)(8,17)(9,13)(10,16)(11,15)(12,14)(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)(9,19,11,17)(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), (1,4)(2,3)(5,20)(6,19)(7,18)(8,17)(9,13)(10,16)(11,15)(12,14)(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),(9,19,11,17),(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)], [(1,4),(2,3),(5,20),(6,19),(7,18),(8,17),(9,13),(10,16),(11,15),(12,14),(21,22),(23,24)]])
 
G:=TransitiveGroup(24,320);
 

Matrix representation of Q8⋊4S4 ►in GL5(𝔽13)

87000
05000
001200
000120
000012
,
80000
45000
00100
00010
00001
,
10000
01000
000121
000120
001120
,
10000
01000
000112
001012
000012
,
10000
01000
001210
001200
001201
,
120000
61000
00010
00100
00001

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] >;
 

Q8⋊4S4 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
 

Export

Character table of Q8⋊4S4 in TeX

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