Q8:3S4 - GroupNames

G = Q₈⋊₃S₄ order 192 = 2⁶·3

The semidirect product of Q₈ and S₄ acting via S₄/A₄=C₂

Aliases: Q₈⋊₃S₄, A₄⋊₃SD₁₆, A₄⋊C₈⋊₃C₂, C₄⋊S₄.₂C₂, C₄.₄(C₂×S₄), (Q₈×A₄)⋊₁C₂, (C₂×A₄).₁₀D₄, (C₂²×Q₈)⋊₁S₃, (C₂²×C₄).₄D₆, (C₄×A₄).₄C₂², C₂²⋊(Q₈⋊₂S₃), C₂.₇(A₄⋊D₄), C₂³.₂₀(C₃⋊D₄), SmallGroup(192,976)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₄×A₄ — Q₈⋊₃S₄

Chief series

C₁ — C₂² — A₄ — C₂×A₄ — C₄×A₄ — C₄⋊S₄ — Q₈⋊₃S₄

Lower central

A₄ — C₂×A₄ — C₄×A₄ — Q₈⋊₃S₄

Upper central

C₁ — C₂ — C₄ — Q₈

Generators and relations for Q₈⋊₃S₄
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, fbf=a^-1b, ece^-1=fcf=cd=dc, ede^-1=c, df=fd, fef=e^-1 >

Subgroups: 354 in 77 conjugacy classes, 15 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₈, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₂³, C₁₂, A₄, D₆, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, SD₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₃⋊C₈, D₁₂, C₃×Q₈, S₄, C₂×A₄, C₂²⋊C₈, Q₈⋊C₄, C₄⋊D₄, C₂×SD₁₆, C₂²×Q₈, Q₈⋊₂S₃, C₄×A₄, C₄×A₄, C₂×S₄, Q₈⋊D₄, A₄⋊C₈, C₄⋊S₄, Q₈×A₄, Q₈⋊₃S₄
Quotients: C₁, C₂, C₂², S₃, D₄, D₆, SD₁₆, C₃⋊D₄, S₄, Q₈⋊₂S₃, C₂×S₄, A₄⋊D₄, Q₈⋊₃S₄

Character table of Q₈⋊₃S₄

class	1	2A	2B	2C	2D	3	4A	4B	4C	4D	4E	6	8A	8B	8C	8D	12A	12B	12C
size	1	1	3	3	24	8	2	4	6	12	24	8	12	12	12	12	16	16	16
ρ₁	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	linear of order 2
ρ₃	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	linear of order 2
ρ₅	2	2	2	2	0	2	-2	0	-2	0	0	2	0	0	0	0	0	0	-2	orthogonal lifted from D₄
ρ₆	2	2	2	2	0	-1	2	2	2	2	0	-1	0	0	0	0	-1	-1	-1	orthogonal lifted from S₃
ρ₇	2	2	2	2	0	-1	2	-2	2	-2	0	-1	0	0	0	0	1	1	-1	orthogonal lifted from D₆
ρ₈	2	2	2	2	0	-1	-2	0	-2	0	0	-1	0	0	0	0	-√-3	√-3	1	complex lifted from C₃⋊D₄
ρ₉	2	2	2	2	0	-1	-2	0	-2	0	0	-1	0	0	0	0	√-3	-√-3	1	complex lifted from C₃⋊D₄
ρ₁₀	2	-2	-2	2	0	2	0	0	0	0	0	-2	√-2	-√-2	-√-2	√-2	0	0	0	complex lifted from SD₁₆
ρ₁₁	2	-2	-2	2	0	2	0	0	0	0	0	-2	-√-2	√-2	√-2	-√-2	0	0	0	complex lifted from SD₁₆
ρ₁₂	3	3	-1	-1	-1	0	3	-3	-1	1	1	0	-1	1	-1	1	0	0	0	orthogonal lifted from C₂×S₄
ρ₁₃	3	3	-1	-1	-1	0	3	3	-1	-1	1	0	1	-1	1	-1	0	0	0	orthogonal lifted from S₄
ρ₁₄	3	3	-1	-1	1	0	3	3	-1	-1	-1	0	-1	1	-1	1	0	0	0	orthogonal lifted from S₄
ρ₁₅	3	3	-1	-1	1	0	3	-3	-1	1	-1	0	1	-1	1	-1	0	0	0	orthogonal lifted from C₂×S₄
ρ₁₆	4	-4	-4	4	0	-2	0	0	0	0	0	2	0	0	0	0	0	0	0	orthogonal lifted from Q₈⋊₂S₃
ρ₁₇	6	6	-2	-2	0	0	-6	0	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from A₄⋊D₄
ρ₁₈	6	-6	2	-2	0	0	0	0	0	0	0	0	-√-2	-√-2	√-2	√-2	0	0	0	complex faithful
ρ₁₉	6	-6	2	-2	0	0	0	0	0	0	0	0	√-2	√-2	-√-2	-√-2	0	0	0	complex faithful

Permutation representations of Q₈⋊₃S₄
►On 24 points - transitive group 24T326

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)

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

G:=TransitiveGroup(24,326);

Matrix representation of Q₈⋊₃S₄ ►in GL₅(𝔽₇₃)

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

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

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

1	0	0	0	0
61	72	0	0	0
0	0	0	1	0
0	0	1	0	0
0	0	0	0	1

G:=sub<GL(5,GF(73))| [72,12,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,72,72,72,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,72,72,72],[1,0,0,0,0,0,1,0,0,0,0,0,72,72,72,0,0,1,0,0,0,0,0,0,1],[1,61,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

Q₈⋊₃S₄ in GAP, Magma, Sage, TeX

Q_8\rtimes_3S_4

% in TeX

G:=Group("Q8:3S4");

// GroupNames label

G:=SmallGroup(192,976);

// by ID

G=gap.SmallGroup(192,976);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,85,64,254,135,58,1124,4037,285,2358,475]);

// 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,f*b*f=a^-1*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⋊₃S₄ in TeX

G = Q8⋊3S4 order 192 = 26·3

The semidirect product of Q8 and S4 acting via S4/A4=C2

G = Q₈⋊₃S₄ order 192 = 2⁶·3

The semidirect product of Q₈ and S₄ acting via S₄/A₄=C₂