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G = Q8⋊S4order 192 = 26·3

1st semidirect product of Q8 and S4 acting via S4/C22=S3

non-abelian, soluble

Aliases: Q81S4, C23.12S4, C22⋊GL2(𝔽3), Q8⋊A42C2, (C22×Q8)⋊7S3, C2.3(C22⋊S4), SmallGroup(192,1490)

Series: Derived Chief Lower central Upper central

C1C2C22×Q8Q8⋊A4 — Q8⋊S4
C1C2Q8C22×Q8Q8⋊A4 — Q8⋊S4
Q8⋊A4 — Q8⋊S4
C1C2

Generators and relations for Q8⋊S4
 G = < a,b,c,d,e,f | a4=c2=d2=e3=f2=1, b2=a2, bab-1=fbf=a-1, ac=ca, ad=da, eae-1=ab, faf=a2b, bc=cb, bd=db, ebe-1=a, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 405 in 73 conjugacy classes, 9 normal (7 characteristic)
C1, C2, C2 [×3], C3, C4 [×4], C22, C22 [×5], S3 [×2], C6, C8 [×2], C2×C4 [×5], D4 [×4], Q8 [×2], Q8 [×4], C23, C23, A4, D6, C22⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×3], SL2(𝔽3) [×3], S4 [×2], C2×A4, C22⋊C8, Q8⋊C4 [×2], C4⋊D4, C2×SD16 [×2], C22×Q8, GL2(𝔽3) [×2], C2×S4, Q8⋊D4, Q8⋊A4, Q8⋊S4
Quotients: C1, C2, S3, S4 [×3], GL2(𝔽3), C22⋊S4, Q8⋊S4

Character table of Q8⋊S4

 class 12A2B2C2D34A4B4C4D68A8B8C8D
 size 113324326612243212121212
ρ1111111111111111    trivial
ρ21111-11111-11-1-1-1-1    linear of order 2
ρ322220-12220-10000    orthogonal lifted from S3
ρ42-2-220-100001--2-2-2--2    complex lifted from GL2(𝔽3)
ρ52-2-220-100001-2--2--2-2    complex lifted from GL2(𝔽3)
ρ6333310-1-1-110-1-1-1-1    orthogonal lifted from S4
ρ733-1-110-13-1-10-11-11    orthogonal lifted from S4
ρ833-1-1103-1-1-101-11-1    orthogonal lifted from S4
ρ933-1-1-10-13-1101-11-1    orthogonal lifted from S4
ρ1033-1-1-103-1-110-11-11    orthogonal lifted from S4
ρ113333-10-1-1-1-101111    orthogonal lifted from S4
ρ124-4-44010000-10000    orthogonal lifted from GL2(𝔽3)
ρ1366-2-200-2-22000000    orthogonal lifted from C22⋊S4
ρ146-62-20000000-2-2--2--2    complex faithful
ρ156-62-20000000--2--2-2-2    complex faithful

Permutation representations of Q8⋊S4
On 24 points - transitive group 24T314
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 12 3 10)(2 11 4 9)(5 22 7 24)(6 21 8 23)(13 17 15 19)(14 20 16 18)
(1 3)(2 4)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 17 21)(2 15 7)(3 19 23)(4 13 5)(6 9 20)(8 11 18)(10 16 24)(12 14 22)
(2 10)(4 12)(5 14)(6 18)(7 16)(8 20)(9 11)(13 22)(15 24)(17 21)(19 23)

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

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

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

G:=TransitiveGroup(24,314);

Matrix representation of Q8⋊S4 in GL5(𝔽73)

01000
720000
00001
00727272
00100
,
112000
1272000
00010
00100
00727272
,
10000
01000
00010
00100
00727272
,
10000
01000
00727272
00001
00010
,
677000
65000
00100
00001
00727272
,
677000
686000
00100
00001
00010

G:=sub<GL(5,GF(73))| [0,72,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,72,0,0,0,1,72,0],[1,12,0,0,0,12,72,0,0,0,0,0,0,1,72,0,0,1,0,72,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,1,0,72,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,72,0,1,0,0,72,1,0],[67,6,0,0,0,7,5,0,0,0,0,0,1,0,72,0,0,0,0,72,0,0,0,1,72],[67,68,0,0,0,7,6,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

Q8⋊S4 in GAP, Magma, Sage, TeX

Q_8\rtimes S_4
% in TeX

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

G:=SmallGroup(192,1490);
// by ID

G=gap.SmallGroup(192,1490);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,57,254,135,171,262,1684,1271,172,1013,2532,285,124]);
// 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*b*f=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a*b,f*a*f=a^2*b,b*c=c*b,b*d=d*b,e*b*e^-1=a,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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