Q8:S4 - GroupNames

G = Q₈⋊S₄ order 192 = 2⁶·3

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

non-abelian, soluble

Aliases: Q₈⋊₁S₄, C₂³.₁₂S₄, C₂²⋊GL₂(𝔽₃), Q₈⋊A₄⋊₂C₂, (C₂²×Q₈)⋊₇S₃, C₂.₃(C₂²⋊S₄), SmallGroup(192,1490)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₂²×Q₈ — Q₈⋊A₄ — Q₈⋊S₄

Chief series

C₁ — C₂ — Q₈ — C₂²×Q₈ — Q₈⋊A₄ — Q₈⋊S₄

Lower central

Q₈⋊A₄ — Q₈⋊S₄

Upper central

C₁ — C₂

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, C₃, C₄, C₂², C₂², S₃, C₆, C₈, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₂³, A₄, D₆, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, SD₁₆, C₂²×C₄, C₂×D₄, C₂×Q₈, SL₂(𝔽₃), S₄, C₂×A₄, C₂²⋊C₈, Q₈⋊C₄, C₄⋊D₄, C₂×SD₁₆, C₂²×Q₈, GL₂(𝔽₃), C₂×S₄, Q₈⋊D₄, Q₈⋊A₄, Q₈⋊S₄
Quotients: C₁, C₂, S₃, S₄, GL₂(𝔽₃), C₂²⋊S₄, Q₈⋊S₄

Character table of Q₈⋊S₄

class	1	2A	2B	2C	2D	3	4A	4B	4C	4D	6	8A	8B	8C	8D
size	1	1	3	3	24	32	6	6	12	24	32	12	12	12	12
ρ₁	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	linear of order 2
ρ₃	2	2	2	2	0	-1	2	2	2	0	-1	0	0	0	0	orthogonal lifted from S₃
ρ₄	2	-2	-2	2	0	-1	0	0	0	0	1	-√-2	√-2	√-2	-√-2	complex lifted from GL₂(𝔽₃)
ρ₅	2	-2	-2	2	0	-1	0	0	0	0	1	√-2	-√-2	-√-2	√-2	complex lifted from GL₂(𝔽₃)
ρ₆	3	3	3	3	1	0	-1	-1	-1	1	0	-1	-1	-1	-1	orthogonal lifted from S₄
ρ₇	3	3	-1	-1	1	0	-1	3	-1	-1	0	-1	1	-1	1	orthogonal lifted from S₄
ρ₈	3	3	-1	-1	1	0	3	-1	-1	-1	0	1	-1	1	-1	orthogonal lifted from S₄
ρ₉	3	3	-1	-1	-1	0	-1	3	-1	1	0	1	-1	1	-1	orthogonal lifted from S₄
ρ₁₀	3	3	-1	-1	-1	0	3	-1	-1	1	0	-1	1	-1	1	orthogonal lifted from S₄
ρ₁₁	3	3	3	3	-1	0	-1	-1	-1	-1	0	1	1	1	1	orthogonal lifted from S₄
ρ₁₂	4	-4	-4	4	0	1	0	0	0	0	-1	0	0	0	0	orthogonal lifted from GL₂(𝔽₃)
ρ₁₃	6	6	-2	-2	0	0	-2	-2	2	0	0	0	0	0	0	orthogonal lifted from C₂²⋊S₄
ρ₁₄	6	-6	2	-2	0	0	0	0	0	0	0	√-2	√-2	-√-2	-√-2	complex faithful
ρ₁₅	6	-6	2	-2	0	0	0	0	0	0	0	-√-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)

0	1	0	0	0
72	0	0	0	0
0	0	0	0	1
0	0	72	72	72
0	0	1	0	0

1	12	0	0	0
12	72	0	0	0
0	0	0	1	0
0	0	1	0	0
0	0	72	72	72

1	0	0	0	0
0	1	0	0	0
0	0	0	1	0
0	0	1	0	0
0	0	72	72	72

1	0	0	0	0
0	1	0	0	0
0	0	72	72	72
0	0	0	0	1
0	0	0	1	0

67	7	0	0	0
6	5	0	0	0
0	0	1	0	0
0	0	0	0	1
0	0	72	72	72

67	7	0	0	0
68	6	0	0	0
0	0	1	0	0
0	0	0	0	1
0	0	0	1	0

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

G = Q8⋊S4 order 192 = 26·3

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

G = Q₈⋊S₄ order 192 = 2⁶·3

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