A4:D4 - GroupNames

G = A₄⋊D₄ order 96 = 2⁵·3

The semidirect product of A₄ and D₄ acting via D₄/C₂²=C₂

Aliases: A₄⋊₂D₄, C₂²⋊₂S₄, C₂⁴⋊₁S₃, C₂³.₅D₆, A₄⋊C₄⋊C₂, (C₂×S₄)⋊₂C₂, C₂.₁₁(C₂×S₄), C₂²⋊(C₃⋊D₄), (C₂²×A₄)⋊₂C₂, (C₂×A₄).₅C₂², Aut(C₄²), GL₂(ℤ/4ℤ), SmallGroup(96,195)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₂×A₄ — A₄⋊D₄

Chief series

C₁ — C₂² — A₄ — C₂×A₄ — C₂×S₄ — A₄⋊D₄

Lower central

A₄ — C₂×A₄ — A₄⋊D₄

Upper central

C₁ — C₂ — C₂²

Generators and relations for A₄⋊D₄
G = < a,b,c,d,e | a²=b²=c³=d⁴=e²=1, cac^-1=dad^-1=eae=ab=ba, cbc^-1=a, bd=db, be=eb, dcd^-1=ece=c^-1, ede=d^-1 >

Subgroups: 234 in 62 conjugacy classes, 12 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₂×C₄, D₄, C₂³, C₂³, Dic₃, A₄, D₆, C₂×C₆, C₂²⋊C₄, C₂×D₄, C₂⁴, C₃⋊D₄, S₄, C₂×A₄, C₂×A₄, C₂²≀C₂, A₄⋊C₄, C₂×S₄, C₂²×A₄, A₄⋊D₄
Quotients: C₁, C₂, C₂², S₃, D₄, D₆, C₃⋊D₄, S₄, C₂×S₄, A₄⋊D₄

Character table of A₄⋊D₄

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

Permutation representations of A₄⋊D₄
►On 12 points - transitive group 12T49

Generators in S₁₂

(1 4)(2 3)(5 10)(6 8)(7 12)(9 11)

(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)

(1 10 8)(2 5 11)(3 12 6)(4 7 9)

(1 2 3 4)(5 6 7 8)(9 10 11 12)

(2 4)(5 9)(6 12)(7 11)(8 10)

G:=sub<Sym(12)| (1,4)(2,3)(5,10)(6,8)(7,12)(9,11), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11), (1,10,8)(2,5,11)(3,12,6)(4,7,9), (1,2,3,4)(5,6,7,8)(9,10,11,12), (2,4)(5,9)(6,12)(7,11)(8,10)>;

G:=Group( (1,4)(2,3)(5,10)(6,8)(7,12)(9,11), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11), (1,10,8)(2,5,11)(3,12,6)(4,7,9), (1,2,3,4)(5,6,7,8)(9,10,11,12), (2,4)(5,9)(6,12)(7,11)(8,10) );

G=PermutationGroup([[(1,4),(2,3),(5,10),(6,8),(7,12),(9,11)], [(1,3),(2,4),(5,12),(6,9),(7,10),(8,11)], [(1,10,8),(2,5,11),(3,12,6),(4,7,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(2,4),(5,9),(6,12),(7,11),(8,10)]])

G:=TransitiveGroup(12,49);

►On 12 points - transitive group 12T50

Generators in S₁₂

(1 3)(6 8)(9 11)(10 12)

(1 3)(2 4)(5 7)(6 8)

(1 7 11)(2 12 8)(3 5 9)(4 10 6)

(1 2 3 4)(5 6 7 8)(9 10 11 12)

(1 4)(2 3)(5 8)(6 7)(9 12)(10 11)

G:=sub<Sym(12)| (1,3)(6,8)(9,11)(10,12), (1,3)(2,4)(5,7)(6,8), (1,7,11)(2,12,8)(3,5,9)(4,10,6), (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)>;

G:=Group( (1,3)(6,8)(9,11)(10,12), (1,3)(2,4)(5,7)(6,8), (1,7,11)(2,12,8)(3,5,9)(4,10,6), (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11) );

G=PermutationGroup([[(1,3),(6,8),(9,11),(10,12)], [(1,3),(2,4),(5,7),(6,8)], [(1,7,11),(2,12,8),(3,5,9),(4,10,6)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]])

G:=TransitiveGroup(12,50);

►On 12 points - transitive group 12T52

Generators in S₁₂

(1 3)(2 4)(5 7)(9 11)

(5 7)(6 8)(9 11)(10 12)

(1 11 8)(2 5 12)(3 9 6)(4 7 10)

(1 2 3 4)(5 6 7 8)(9 10 11 12)

(2 4)(5 10)(6 9)(7 12)(8 11)

G:=sub<Sym(12)| (1,3)(2,4)(5,7)(9,11), (5,7)(6,8)(9,11)(10,12), (1,11,8)(2,5,12)(3,9,6)(4,7,10), (1,2,3,4)(5,6,7,8)(9,10,11,12), (2,4)(5,10)(6,9)(7,12)(8,11)>;

G:=Group( (1,3)(2,4)(5,7)(9,11), (5,7)(6,8)(9,11)(10,12), (1,11,8)(2,5,12)(3,9,6)(4,7,10), (1,2,3,4)(5,6,7,8)(9,10,11,12), (2,4)(5,10)(6,9)(7,12)(8,11) );

G=PermutationGroup([[(1,3),(2,4),(5,7),(9,11)], [(5,7),(6,8),(9,11),(10,12)], [(1,11,8),(2,5,12),(3,9,6),(4,7,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(2,4),(5,10),(6,9),(7,12),(8,11)]])

G:=TransitiveGroup(12,52);

►On 16 points - transitive group 16T186

Generators in S₁₆

(1 14)(2 5)(3 16)(4 7)(6 12)(8 10)(9 13)(11 15)

(1 10)(2 11)(3 12)(4 9)(5 15)(6 16)(7 13)(8 14)

(5 11 15)(6 16 12)(7 9 13)(8 14 10)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(2 4)(5 13)(6 16)(7 15)(8 14)(9 11)

G:=sub<Sym(16)| (1,14)(2,5)(3,16)(4,7)(6,12)(8,10)(9,13)(11,15), (1,10)(2,11)(3,12)(4,9)(5,15)(6,16)(7,13)(8,14), (5,11,15)(6,16,12)(7,9,13)(8,14,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (2,4)(5,13)(6,16)(7,15)(8,14)(9,11)>;

G:=Group( (1,14)(2,5)(3,16)(4,7)(6,12)(8,10)(9,13)(11,15), (1,10)(2,11)(3,12)(4,9)(5,15)(6,16)(7,13)(8,14), (5,11,15)(6,16,12)(7,9,13)(8,14,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (2,4)(5,13)(6,16)(7,15)(8,14)(9,11) );

G=PermutationGroup([[(1,14),(2,5),(3,16),(4,7),(6,12),(8,10),(9,13),(11,15)], [(1,10),(2,11),(3,12),(4,9),(5,15),(6,16),(7,13),(8,14)], [(5,11,15),(6,16,12),(7,9,13),(8,14,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(2,4),(5,13),(6,16),(7,15),(8,14),(9,11)]])

G:=TransitiveGroup(16,186);

►On 16 points - transitive group 16T193

Generators in S₁₆

(1 6)(2 11)(3 8)(4 9)(5 15)(7 13)(10 16)(12 14)

(1 16)(2 13)(3 14)(4 15)(5 9)(6 10)(7 11)(8 12)

(5 9 15)(6 16 10)(7 11 13)(8 14 12)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 4)(2 3)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)

G:=sub<Sym(16)| (1,6)(2,11)(3,8)(4,9)(5,15)(7,13)(10,16)(12,14), (1,16)(2,13)(3,14)(4,15)(5,9)(6,10)(7,11)(8,12), (5,9,15)(6,16,10)(7,11,13)(8,14,12), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)>;

G:=Group( (1,6)(2,11)(3,8)(4,9)(5,15)(7,13)(10,16)(12,14), (1,16)(2,13)(3,14)(4,15)(5,9)(6,10)(7,11)(8,12), (5,9,15)(6,16,10)(7,11,13)(8,14,12), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16) );

G=PermutationGroup([[(1,6),(2,11),(3,8),(4,9),(5,15),(7,13),(10,16),(12,14)], [(1,16),(2,13),(3,14),(4,15),(5,9),(6,10),(7,11),(8,12)], [(5,9,15),(6,16,10),(7,11,13),(8,14,12)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]])

G:=TransitiveGroup(16,193);

►On 24 points - transitive group 24T153

Generators in S₂₄

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

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

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

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

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

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

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

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

G:=TransitiveGroup(24,153);

►On 24 points - transitive group 24T154

Generators in S₂₄

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

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

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

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

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

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

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

G:=TransitiveGroup(24,154);

►On 24 points - transitive group 24T155

Generators in S₂₄

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

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

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

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

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

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

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

G:=TransitiveGroup(24,155);

►On 24 points - transitive group 24T156

Generators in S₂₄

(1 3)(5 7)(6 8)(9 11)(13 15)(14 16)(18 20)(22 24)

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

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

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

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

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

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

G:=TransitiveGroup(24,156);

►On 24 points - transitive group 24T157

Generators in S₂₄

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

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

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

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

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

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

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

G:=TransitiveGroup(24,157);

►On 24 points - transitive group 24T158

Generators in S₂₄

(1 9)(3 11)(5 22)(6 23)(7 24)(8 21)(14 20)(16 18)

(1 9)(2 10)(3 11)(4 12)(13 19)(14 20)(15 17)(16 18)

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

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

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

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

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

G:=TransitiveGroup(24,158);

►On 24 points - transitive group 24T159

Generators in S₂₄

(1 16)(3 14)(5 10)(6 11)(7 12)(8 9)(17 22)(19 24)

(1 16)(2 13)(3 14)(4 15)(17 22)(18 23)(19 24)(20 21)

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

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

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

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

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

G:=TransitiveGroup(24,159);

►On 24 points - transitive group 24T165

Generators in S₂₄

(1 8)(2 5)(3 6)(4 7)(10 13)(12 15)(18 22)(20 24)

(9 16)(10 13)(11 14)(12 15)(17 21)(18 22)(19 23)(20 24)

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

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

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

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

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

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

G:=TransitiveGroup(24,165);

►On 24 points - transitive group 24T166

Generators in S₂₄

(1 23)(2 24)(3 21)(4 22)(6 17)(8 19)(10 14)(12 16)

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

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

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

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

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

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

G:=TransitiveGroup(24,166);

A₄⋊D₄ is a maximal subgroup of
C₂⁴.₁₀D₆ D₄×S₄ D₄⋊₂S₄ D₆⋊S₄ A₄⋊D₁₂ (C₂×C₆)⋊₄S₄ PSO₄⁺ (𝔽₃) C₂²⋊S₅ D₁₀⋊S₄ A₄⋊D₂₀ C₂⁴⋊₂D₁₅
A₄⋊D₄ is a maximal quotient of
C₂⁴.₃D₆ C₂⁴.₅D₆ A₄⋊SD₁₆ D₄⋊S₄ A₄⋊₂Q₁₆ Q₈⋊₃S₄ C₂³.₁₄S₄ C₂³.₁₆S₄ SL₂(𝔽₃).D₄ SL₂(𝔽₃)⋊D₄ Q₈.₄S₄ Q₈.₅S₄ D₄.S₄ D₄.₃S₄ C₂⁵.S₃ C₂³.D₁₈ D₆⋊S₄ A₄⋊D₁₂ (C₂×C₆)⋊₄S₄ D₁₀⋊S₄ A₄⋊D₂₀ C₂⁴⋊₂D₁₅

Polynomial with Galois group A₄⋊D₄ over ℚ

action	f(x)	Disc(f)
12T49	x¹²-4x¹¹+4x¹⁰+6x⁹-14x⁸-4x⁷+12x⁶+4x⁵-12x³+4x²+4x-2	-2²⁰·11⁷·5939²
12T50	x¹²-24x¹⁰+219x⁸-954x⁶+1986x⁴-1584x²+24	2⁴⁷·3¹⁵·5⁸·19⁴
12T52	x¹²-2x¹¹-5x⁸+12x⁷+12x⁶-12x⁵-23x⁴-34x³-32x²-16x-7	-2²⁰·11⁷·421²·29383²

Matrix representation of A₄⋊D₄ ►in GL₅(𝔽₁₃)

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

6	10	0	0	0
10	6	0	0	0
0	0	12	1	0
0	0	12	0	0
0	0	12	0	1

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

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

G:=sub<GL(5,GF(13))| [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],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[6,10,0,0,0,10,6,0,0,0,0,0,12,12,12,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,1,0,0,0,0,0,0,1,1,1,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,1,0,0,0,0,0,1,1,1,0,0,0,12,0,0,0,0,0,12] >;

A₄⋊D₄ in GAP, Magma, Sage, TeX

A_4\rtimes D_4

% in TeX

G:=Group("A4:D4");

// GroupNames label

G:=SmallGroup(96,195);

// by ID

G=gap.SmallGroup(96,195);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,2,73,387,1444,202,869,347]);

// Polycyclic

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

// generators/relations

Export

Character table of A₄⋊D₄ in TeX

G = A4⋊D4 order 96 = 25·3

The semidirect product of A4 and D4 acting via D4/C22=C2

G = A₄⋊D₄ order 96 = 2⁵·3

The semidirect product of A₄ and D₄ acting via D₄/C₂²=C₂