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G = A4⋊D4order 96 = 25·3

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

non-abelian, soluble, monomial

Aliases: A42D4, C222S4, C241S3, 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

C1C22C2×A4 — A4⋊D4
C1C22A4C2×A4C2×S4 — A4⋊D4
A4C2×A4 — A4⋊D4
C1C2C22

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 [×5], C3, C4 [×3], C22 [×2], C22 [×11], S3, C6 [×2], C2×C4 [×3], D4 [×6], C23, C23 [×5], Dic3, A4, D6, C2×C6, C22⋊C4 [×3], C2×D4 [×3], C24, C3⋊D4, S4, C2×A4, C2×A4, C22≀C2, A4⋊C4, C2×S4, C22×A4, A4⋊D4
Quotients: C1, C2 [×3], C22, S3, D4, D6, C3⋊D4, S4, C2×S4, A4⋊D4

Character table of A4⋊D4

 class 12A2B2C2D2E2F34A4B4C6A6B6C
 size 112336128121212888
ρ111111111111111    trivial
ρ211-111-111-11-1-11-1    linear of order 2
ρ3111111-11-1-1-1111    linear of order 2
ρ411-111-1-111-11-11-1    linear of order 2
ρ52222220-1000-1-1-1    orthogonal lifted from S3
ρ62-20-220020000-20    orthogonal lifted from D4
ρ722-222-20-10001-11    orthogonal lifted from D6
ρ82-20-2200-1000-31--3    complex lifted from C3⋊D4
ρ92-20-2200-1000--31-3    complex lifted from C3⋊D4
ρ10333-1-1-1-1011-1000    orthogonal lifted from S4
ρ1133-3-1-11-10-111000    orthogonal lifted from C2×S4
ρ12333-1-1-110-1-11000    orthogonal lifted from S4
ρ1333-3-1-11101-1-1000    orthogonal lifted from C2×S4
ρ146-602-2000000000    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 9)(2 10 8)(3 5 11)(4 12 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), (1,7,9)(2,10,8)(3,5,11)(4,12,6), (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), (1,7,9)(2,10,8)(3,5,11)(4,12,6), (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)], [(1,7,9),(2,10,8),(3,5,11),(4,12,6)], [(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 12)(2 7)(3 10)(4 5)(6 16)(8 14)(9 13)(11 15)
(1 16)(2 13)(3 14)(4 15)(5 11)(6 12)(7 9)(8 10)
(5 15 11)(6 12 16)(7 13 9)(8 10 14)
(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)| (1,12)(2,7)(3,10)(4,5)(6,16)(8,14)(9,13)(11,15), (1,16)(2,13)(3,14)(4,15)(5,11)(6,12)(7,9)(8,10), (5,15,11)(6,12,16)(7,13,9)(8,10,14), (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( (1,12)(2,7)(3,10)(4,5)(6,16)(8,14)(9,13)(11,15), (1,16)(2,13)(3,14)(4,15)(5,11)(6,12)(7,9)(8,10), (5,15,11)(6,12,16)(7,13,9)(8,10,14), (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([(1,12),(2,7),(3,10),(4,5),(6,16),(8,14),(9,13),(11,15)], [(1,16),(2,13),(3,14),(4,15),(5,11),(6,12),(7,9),(8,10)], [(5,15,11),(6,12,16),(7,13,9),(8,10,14)], [(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 17)(2 20)(3 19)(4 18)(5 13)(6 8)(7 15)(9 11)(10 23)(12 21)(14 16)(22 24)
(1 3)(2 4)(5 15)(6 16)(7 13)(8 14)(9 24)(10 21)(11 22)(12 23)(17 19)(18 20)
(1 21 14)(2 15 22)(3 23 16)(4 13 24)(5 9 20)(6 17 10)(7 11 18)(8 19 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 9)(6 12)(7 11)(8 10)(13 22)(14 21)(15 24)(16 23)(17 19)

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

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

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

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

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

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

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

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  D42S4  D6⋊S4  A4⋊D12  (C2×C6)⋊4S4  PSO4+ (𝔽3)  C22⋊S5  D10⋊S4  A4⋊D20  C242D15
A4⋊D4 is a maximal quotient of
C24.3D6  C24.5D6  A4⋊SD16  D4⋊S4  A42Q16  Q83S4  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  C242D15

Polynomial with Galois group A4⋊D4 over ℚ
actionf(x)Disc(f)
12T49x12-4x11+4x10+6x9-14x8-4x7+12x6+4x5-12x3+4x2+4x-2-220·117·59392
12T50x12-24x10+219x8-954x6+1986x4-1584x2+24247·315·58·194
12T52x12-2x11-5x8+12x7+12x6-12x5-23x4-34x3-32x2-16x-7-220·117·4212·293832

Matrix representation of A4⋊D4 in GL5(𝔽13)

10000
01000
000112
001012
000012
,
10000
01000
001200
001201
001210
,
610000
106000
001210
001200
001201
,
01000
120000
00100
001120
001012
,
120000
01000
00100
001120
001012

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

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