Copied to
clipboard

G = C4×A4order 48 = 24·3

Direct product of C4 and A4

direct product, metabelian, soluble, monomial, A-group

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

C1C22 — C4×A4
C1C22C23C2×A4 — C4×A4
C22 — C4×A4
C1C4

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 >

3C2
3C2
4C3
3C22
3C4
3C22
4C6
3C2×C4
3C2×C4
4C12

Character table of C4×A4

 class 12A2B2C3A3B4A4B4C4D6A6B12A12B12C12D
 size 1133441133444444
ρ11111111111111111    trivial
ρ2111111-1-1-1-111-1-1-1-1    linear of order 2
ρ31111ζ3ζ32-1-1-1-1ζ32ζ3ζ6ζ65ζ6ζ65    linear of order 6
ρ41111ζ32ζ31111ζ3ζ32ζ3ζ32ζ3ζ32    linear of order 3
ρ51111ζ32ζ3-1-1-1-1ζ3ζ32ζ65ζ6ζ65ζ6    linear of order 6
ρ61111ζ3ζ321111ζ32ζ3ζ32ζ3ζ32ζ3    linear of order 3
ρ71-1-1111i-ii-i-1-1-i-iii    linear of order 4
ρ81-1-1111-ii-ii-1-1ii-i-i    linear of order 4
ρ91-1-11ζ32ζ62ζ2ζ2ζ2ζ2ζ65ζ6ζ4ζ3ζ4ζ32ζ43ζ3ζ43ζ32    linear of order 12
ρ101-1-11ζ62ζ32ζ2ζ2ζ2ζ2ζ6ζ65ζ43ζ32ζ43ζ3ζ4ζ32ζ4ζ3    linear of order 12
ρ111-1-11ζ32ζ62ζ2ζ2ζ2ζ2ζ65ζ6ζ43ζ3ζ43ζ32ζ4ζ3ζ4ζ32    linear of order 12
ρ121-1-11ζ62ζ32ζ2ζ2ζ2ζ2ζ6ζ65ζ4ζ32ζ4ζ3ζ43ζ32ζ43ζ3    linear of order 12
ρ1333-1-10033-1-1000000    orthogonal lifted from A4
ρ1433-1-100-3-311000000    orthogonal lifted from C2×A4
ρ153-31-100-3i3ii-i000000    complex faithful
ρ163-31-1003i-3i-ii000000    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 9)(2 8 10)(3 5 11)(4 6 12)

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

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

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

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

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

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

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

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);

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

300
030
003
,
400
010
004
,
100
040
004
,
030
004
300
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

׿
×
𝔽