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## G = C4×A4order 48 = 24·3

### Direct product of C4 and A4

Aliases: C4×A4, C22⋊C12, C23.C6, (C22×C4)⋊C3, C2.1(C2×A4), (C2×A4).2C2, SmallGroup(48,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C4×A4
 Chief series C1 — C22 — C23 — C2×A4 — C4×A4
 Lower central C22 — C4×A4
 Upper central C1 — C4

Generators and relations for C4×A4
G = < a,b,c,d | a4=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Character table of C4×A4

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 12A 12B 12C 12D size 1 1 3 3 4 4 1 1 3 3 4 4 4 4 4 4 ρ1 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 linear of order 2 ρ3 1 1 1 1 ζ3 ζ32 -1 -1 -1 -1 ζ32 ζ3 ζ6 ζ65 ζ6 ζ65 linear of order 6 ρ4 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ5 1 1 1 1 ζ32 ζ3 -1 -1 -1 -1 ζ3 ζ32 ζ65 ζ6 ζ65 ζ6 linear of order 6 ρ6 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ7 1 -1 -1 1 1 1 i -i i -i -1 -1 -i -i i i linear of order 4 ρ8 1 -1 -1 1 1 1 -i i -i i -1 -1 i i -i -i linear of order 4 ρ9 1 -1 -1 1 ζ32 ζ3 -i i -i i ζ65 ζ6 ζ4ζ3 ζ4ζ32 ζ43ζ3 ζ43ζ32 linear of order 12 ρ10 1 -1 -1 1 ζ3 ζ32 i -i i -i ζ6 ζ65 ζ43ζ32 ζ43ζ3 ζ4ζ32 ζ4ζ3 linear of order 12 ρ11 1 -1 -1 1 ζ32 ζ3 i -i i -i ζ65 ζ6 ζ43ζ3 ζ43ζ32 ζ4ζ3 ζ4ζ32 linear of order 12 ρ12 1 -1 -1 1 ζ3 ζ32 -i i -i i ζ6 ζ65 ζ4ζ32 ζ4ζ3 ζ43ζ32 ζ43ζ3 linear of order 12 ρ13 3 3 -1 -1 0 0 3 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ14 3 3 -1 -1 0 0 -3 -3 1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ15 3 -3 1 -1 0 0 -3i 3i i -i 0 0 0 0 0 0 complex faithful ρ16 3 -3 1 -1 0 0 3i -3i -i i 0 0 0 0 0 0 complex faithful

Permutation representations of C4×A4
On 12 points - transitive group 12T29
Generators in S12
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(5 7)(6 8)(9 11)(10 12)
(1 3)(2 4)(9 11)(10 12)
(1 7 11)(2 8 12)(3 5 9)(4 6 10)

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

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

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

G:=TransitiveGroup(12,29);

On 16 points - transitive group 16T57
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 10)(2 11)(3 12)(4 9)(5 15)(6 16)(7 13)(8 14)
(1 6)(2 7)(3 8)(4 5)(9 15)(10 16)(11 13)(12 14)
(5 15 9)(6 16 10)(7 13 11)(8 14 12)

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

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

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

G:=TransitiveGroup(16,57);

On 24 points - transitive group 24T55
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 3)(2 4)(5 14)(6 15)(7 16)(8 13)(9 11)(10 12)(17 21)(18 22)(19 23)(20 24)
(1 11)(2 12)(3 9)(4 10)(5 7)(6 8)(13 15)(14 16)(17 23)(18 24)(19 21)(20 22)
(1 7 19)(2 8 20)(3 5 17)(4 6 18)(9 16 21)(10 13 22)(11 14 23)(12 15 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), (1,3)(2,4)(5,14)(6,15)(7,16)(8,13)(9,11)(10,12)(17,21)(18,22)(19,23)(20,24), (1,11)(2,12)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16)(17,23)(18,24)(19,21)(20,22), (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,16,21)(10,13,22)(11,14,23)(12,15,24)>;

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

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

G:=TransitiveGroup(24,55);

On 24 points - transitive group 24T56
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)
(5 16)(6 13)(7 14)(8 15)(17 21)(18 22)(19 23)(20 24)
(1 11)(2 12)(3 9)(4 10)(17 21)(18 22)(19 23)(20 24)
(1 7 19)(2 8 20)(3 5 17)(4 6 18)(9 16 21)(10 13 22)(11 14 23)(12 15 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), (5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (1,11)(2,12)(3,9)(4,10)(17,21)(18,22)(19,23)(20,24), (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,16,21)(10,13,22)(11,14,23)(12,15,24)>;

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

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

G:=TransitiveGroup(24,56);

C4×A4 is a maximal subgroup of
A4⋊C8  A4⋊Q8  C4⋊S4  C23.SL2(𝔽3)  C424C4⋊C3  C24⋊C12  C42⋊C12  C422C12  C232D4⋊C3  (C22×C4).A4  C23.19(C2×A4)  2+ 1+4.C6  C4○D4⋊A4  2+ 1+4.3C6  Dic7⋊A4
C4×A4 is a maximal quotient of
C8.A4  C24⋊C12  C42⋊C12  C422C12  Dic7⋊A4

Polynomial with Galois group C4×A4 over ℚ
actionf(x)Disc(f)
12T29x12-26x10+195x8-663x6+1144x4-975x2+325212·514·1311

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

 3 0 0 0 3 0 0 0 3
,
 4 0 0 0 1 0 0 0 4
,
 1 0 0 0 4 0 0 0 4
,
 0 3 0 0 0 4 3 0 0
G:=sub<GL(3,GF(5))| [3,0,0,0,3,0,0,0,3],[4,0,0,0,1,0,0,0,4],[1,0,0,0,4,0,0,0,4],[0,0,3,3,0,0,0,4,0] >;

C4×A4 in GAP, Magma, Sage, TeX

C_4\times A_4
% in TeX

G:=Group("C4xA4");
// GroupNames label

G:=SmallGroup(48,31);
// by ID

G=gap.SmallGroup(48,31);
# by ID

G:=PCGroup([5,-2,-3,-2,-2,2,30,248,459]);
// Polycyclic

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

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