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## G = A4⋊C4order 48 = 24·3

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

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

 Derived series C1 — C22 — A4 — A4⋊C4
 Chief series C1 — C22 — A4 — C2×A4 — A4⋊C4
 Lower central A4 — A4⋊C4
 Upper central C1 — C2

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 >

Character table of A4⋊C4

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6 size 1 1 3 3 8 6 6 6 6 8 ρ1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ3 1 -1 1 -1 1 -i i -i i -1 linear of order 4 ρ4 1 -1 1 -1 1 i -i i -i -1 linear of order 4 ρ5 2 2 2 2 -1 0 0 0 0 -1 orthogonal lifted from S3 ρ6 2 -2 2 -2 -1 0 0 0 0 1 symplectic lifted from Dic3, Schur index 2 ρ7 3 3 -1 -1 0 1 1 -1 -1 0 orthogonal lifted from S4 ρ8 3 3 -1 -1 0 -1 -1 1 1 0 orthogonal lifted from S4 ρ9 3 -3 -1 1 0 i -i -i i 0 complex faithful ρ10 3 -3 -1 1 0 -i i i -i 0 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

 2 1 0 2 3 0 1 2 4
,
 4 4 2 0 2 4 0 3 3
,
 1 0 3 3 0 3 2 1 4
,
 3 3 4 3 1 2 4 2 3
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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