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G = A4⋊C4order 48 = 24·3

The semidirect product of A4 and C4 acting via C4/C2=C2

non-abelian, soluble, monomial

Aliases: A4⋊C4, C2.1S4, C23.S3, C22⋊Dic3, (C2×A4).C2, SL2(ℤ/4ℤ), SmallGroup(48,30)

Series: Derived Chief Lower central Upper central

C1C22A4 — A4⋊C4
C1C22A4C2×A4 — A4⋊C4
A4 — A4⋊C4
C1C2

Generators and relations for A4⋊C4
 G = < a,b,c,d | a2=b2=c3=d4=1, cac-1=dad-1=ab=ba, cbc-1=a, bd=db, dcd-1=c-1 >

3C2
3C2
4C3
3C22
3C22
6C4
6C4
4C6
3C2×C4
3C2×C4
4Dic3
3C22⋊C4

Character table of A4⋊C4

 class 12A2B2C34A4B4C4D6
 size 1133866668
ρ11111111111    trivial
ρ211111-1-1-1-11    linear of order 2
ρ31-11-11-ii-ii-1    linear of order 4
ρ41-11-11i-ii-i-1    linear of order 4
ρ52222-10000-1    orthogonal lifted from S3
ρ62-22-2-100001    symplectic lifted from Dic3, Schur index 2
ρ733-1-1011-1-10    orthogonal lifted from S4
ρ833-1-10-1-1110    orthogonal lifted from S4
ρ93-3-110i-i-ii0    complex faithful
ρ103-3-110-iii-i0    complex faithful

Permutation representations of A4⋊C4
On 12 points - transitive group 12T27
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)

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)>;

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) );

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)])

G:=TransitiveGroup(12,27);

On 12 points - transitive group 12T30
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)

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)>;

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) );

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)])

G:=TransitiveGroup(12,30);

On 16 points - transitive group 16T62
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)

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)>;

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) );

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)])

G:=TransitiveGroup(16,62);

On 24 points - transitive group 24T51
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)

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)>;

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) );

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)])

G:=TransitiveGroup(24,51);

On 24 points - transitive group 24T57
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)

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)>;

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) );

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)])

G:=TransitiveGroup(24,57);

Polynomial with Galois group A4⋊C4 over ℚ
actionf(x)Disc(f)
12T27x12-4x11-20x10+68x9+154x8-384x7-496x6+880x5+524x4-816x3-16x2+144x-8276·36·52·377
12T30x12-20x10+131x8-384x6+527x4-296x2+37236·377·674

Matrix representation of A4⋊C4 in GL3(𝔽5) generated by

210
230
124
,
442
024
033
,
103
303
214
,
334
312
423
G:=sub<GL(3,GF(5))| [2,2,1,1,3,2,0,0,4],[4,0,0,4,2,3,2,4,3],[1,3,2,0,0,1,3,3,4],[3,3,4,3,1,2,4,2,3] >;

A4⋊C4 in GAP, Magma, Sage, TeX

A_4\rtimes C_4
% in TeX

G:=Group("A4:C4");
// GroupNames label

G:=SmallGroup(48,30);
// by ID

G=gap.SmallGroup(48,30);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,2,10,122,483,133,304,239]);
// Polycyclic

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

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