Q8xA4 - GroupNames

G = Q₈xA₄ order 96 = 2⁵·3

Direct product of Q₈ and A₄

direct product, metabelian, soluble, monomial

Aliases: Q₈xA₄, C₂²:(C₃xQ₈), C₄.₁(C₂xA₄), (C₂²xC₄).C₆, (C₄xA₄).₃C₂, (C₂²xQ₈):₂C₃, C₂³.₇(C₂xC₆), C₂.₄(C₂²xA₄), (C₂xA₄).₈C₂², SmallGroup(96,199)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — Q₈xA₄

Chief series

C₁ — C₂² — C₂³ — C₂xA₄ — C₄xA₄ — Q₈xA₄

Lower central

C₂² — C₂³ — Q₈xA₄

Upper central

C₁ — C₂ — Q₈

Generators and relations for Q₈xA₄
G = < a,b,c,d,e | a⁴=c²=d²=e³=1, b²=a², bab^-1=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece^-1=cd=dc, ede^-1=c >

Subgroups: 108 in 46 conjugacy classes, 18 normal (9 characteristic)
Quotients: C₁, C₂, C₃, C₂², C₆, Q₈, A₄, C₂xC₆, C₃xQ₈, C₂xA₄, C₂²xA₄, Q₈xA₄

C₁

C₂

₃C₂

₄C₃

C₄

C₂²

C₄

₃C₂²

₃C₄

₃C₂²

₃C₄

₄C₆

Q₈

C₂³

₃C₂xC₄

₃Q₈

₃C₂xC₄

₃Q₈

A₄

₄C₁₂

C₂²xC₄

₃C₂xQ₈

C₂xA₄

₄C₃xQ₈

C₂²xQ₈

C₄xA₄

Q₈xA₄

Character table of Q₈xA₄

class	1	2A	2B	2C	3A	3B	4A	4B	4C	4D	4E	4F	6A	6B	12A	12B	12C	12D	12E	12F
size	1	1	3	3	4	4	2	2	2	6	6	6	4	4	8	8	8	8	8	8
ρ₁	1	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	-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	-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	1	linear of order 2
ρ₅	1	1	1	1	ζ₃	ζ₃²	-1	1	-1	-1	-1	1	ζ₃	ζ₃²	ζ₆	ζ₃²	ζ₆⁵	ζ₆	ζ₆⁵	ζ₃	linear of order 6
ρ₆	1	1	1	1	ζ₃	ζ₃²	1	1	1	1	1	1	ζ₃	ζ₃²	ζ₃²	ζ₃²	ζ₃	ζ₃²	ζ₃	ζ₃	linear of order 3
ρ₇	1	1	1	1	ζ₃²	ζ₃	1	-1	-1	-1	1	-1	ζ₃²	ζ₃	ζ₆⁵	ζ₆⁵	ζ₃²	ζ₃	ζ₆	ζ₆	linear of order 6
ρ₈	1	1	1	1	ζ₃²	ζ₃	1	1	1	1	1	1	ζ₃²	ζ₃	ζ₃	ζ₃	ζ₃²	ζ₃	ζ₃²	ζ₃²	linear of order 3
ρ₉	1	1	1	1	ζ₃	ζ₃²	1	-1	-1	-1	1	-1	ζ₃	ζ₃²	ζ₆	ζ₆	ζ₃	ζ₃²	ζ₆⁵	ζ₆⁵	linear of order 6
ρ₁₀	1	1	1	1	ζ₃²	ζ₃	-1	-1	1	1	-1	-1	ζ₃²	ζ₃	ζ₃	ζ₆⁵	ζ₆	ζ₆⁵	ζ₃²	ζ₆	linear of order 6
ρ₁₁	1	1	1	1	ζ₃	ζ₃²	-1	-1	1	1	-1	-1	ζ₃	ζ₃²	ζ₃²	ζ₆	ζ₆⁵	ζ₆	ζ₃	ζ₆⁵	linear of order 6
ρ₁₂	1	1	1	1	ζ₃²	ζ₃	-1	1	-1	-1	-1	1	ζ₃²	ζ₃	ζ₆⁵	ζ₃	ζ₆	ζ₆⁵	ζ₆	ζ₃²	linear of order 6
ρ₁₃	2	-2	2	-2	2	2	0	0	0	0	0	0	-2	-2	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₄	2	-2	2	-2	-1+√-3	-1-√-3	0	0	0	0	0	0	1-√-3	1+√-3	0	0	0	0	0	0	complex lifted from C₃xQ₈
ρ₁₅	2	-2	2	-2	-1-√-3	-1+√-3	0	0	0	0	0	0	1+√-3	1-√-3	0	0	0	0	0	0	complex lifted from C₃xQ₈
ρ₁₆	3	3	-1	-1	0	0	-3	3	-3	1	1	-1	0	0	0	0	0	0	0	0	orthogonal lifted from C₂xA₄
ρ₁₇	3	3	-1	-1	0	0	3	3	3	-1	-1	-1	0	0	0	0	0	0	0	0	orthogonal lifted from A₄
ρ₁₈	3	3	-1	-1	0	0	-3	-3	3	-1	1	1	0	0	0	0	0	0	0	0	orthogonal lifted from C₂xA₄
ρ₁₉	3	3	-1	-1	0	0	3	-3	-3	1	-1	1	0	0	0	0	0	0	0	0	orthogonal lifted from C₂xA₄
ρ₂₀	6	-6	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic faithful, Schur index 2

Permutation representations of Q₈xA₄
►On 24 points - transitive group 24T86

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)

(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)

(1 3)(2 4)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)

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

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

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

G:=TransitiveGroup(24,86);

Q₈xA₄ is a maximal subgroup of A₄:₂Q₁₆ Q₈:₃S₄ Q₈:₄S₄
Q₈xA₄ is a maximal quotient of SL₂(F₃):₃Q₈

Matrix representation of Q₈xA₄ ►in GL₅(F₁₃)

12	12	0	0	0
2	1	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	0	0	12

8	0	0	0	0
10	5	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	0	0	12

1	0	0	0	0
0	1	0	0	0
0	0	12	0	0
0	0	12	0	1
0	0	12	1	0

1	0	0	0	0
0	1	0	0	0
0	0	0	12	1
0	0	0	12	0
0	0	1	12	0

9	0	0	0	0
0	9	0	0	0
0	0	0	0	9
0	0	9	0	0
0	0	0	9	0

G:=sub<GL(5,GF(13))| [12,2,0,0,0,12,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[8,10,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[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],[9,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,9,0,0,9,0,0] >;

Q₈xA₄ in GAP, Magma, Sage, TeX

Q_8\times A_4

% in TeX

G:=Group("Q8xA4");

// GroupNames label

G:=SmallGroup(96,199);

// by ID

G=gap.SmallGroup(96,199);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,2,72,169,79,376,665]);

// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^2=d^2=e^3=1,b^2=a^2,b*a*b^-1=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,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;

// generators/relations

Export

Subgroup lattice of Q₈xA₄ in TeX
Character table of Q₈xA₄ in TeX

G = Q8xA4 order 96 = 25·3

Direct product of Q8 and A4

G = Q₈xA₄ order 96 = 2⁵·3

Direct product of Q₈ and A₄