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## G = C22×A4order 48 = 24·3

### Direct product of C22 and A4

Aliases: C22×A4, C23⋊C6, C241C3, C22⋊(C2×C6), SmallGroup(48,49)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22×A4
G = < a,b,c,d,e | a2=b2=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Character table of C22×A4

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 6A 6B 6C 6D 6E 6F size 1 1 1 1 3 3 3 3 4 4 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 1 -1 -1 1 1 1 1 -1 -1 -1 1 -1 linear of order 2 ρ4 1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 -1 1 linear of order 2 ρ5 1 -1 -1 1 1 -1 -1 1 ζ32 ζ3 ζ32 ζ6 ζ65 ζ6 ζ3 ζ65 linear of order 6 ρ6 1 -1 1 -1 1 1 -1 -1 ζ32 ζ3 ζ6 ζ6 ζ3 ζ32 ζ65 ζ65 linear of order 6 ρ7 1 1 -1 -1 1 -1 1 -1 ζ3 ζ32 ζ65 ζ3 ζ6 ζ65 ζ6 ζ32 linear of order 6 ρ8 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 linear of order 3 ρ9 1 1 -1 -1 1 -1 1 -1 ζ32 ζ3 ζ6 ζ32 ζ65 ζ6 ζ65 ζ3 linear of order 6 ρ10 1 -1 1 -1 1 1 -1 -1 ζ3 ζ32 ζ65 ζ65 ζ32 ζ3 ζ6 ζ6 linear of order 6 ρ11 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 linear of order 3 ρ12 1 -1 -1 1 1 -1 -1 1 ζ3 ζ32 ζ3 ζ65 ζ6 ζ65 ζ32 ζ6 linear of order 6 ρ13 3 -3 -3 3 -1 1 1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ14 3 -3 3 -3 -1 -1 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ15 3 3 3 3 -1 -1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from A4 ρ16 3 3 -3 -3 -1 1 -1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4

Permutation representations of C22×A4
On 12 points - transitive group 12T25
Generators in S12
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)
(2 8)(3 9)(5 11)(6 12)
(1 7)(3 9)(4 10)(6 12)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)

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

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

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

G:=TransitiveGroup(12,25);

On 12 points - transitive group 12T26
Generators in S12
(1 10)(2 11)(3 12)(4 8)(5 9)(6 7)
(1 4)(2 5)(3 6)(7 12)(8 10)(9 11)
(1 4)(2 9)(3 12)(5 11)(6 7)(8 10)
(1 10)(2 5)(3 7)(4 8)(6 12)(9 11)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)

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

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

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

G:=TransitiveGroup(12,26);

On 16 points - transitive group 16T58
Generators in S16
(1 4)(2 3)(5 8)(6 9)(7 10)(11 15)(12 16)(13 14)
(1 3)(2 4)(5 14)(6 15)(7 16)(8 13)(9 11)(10 12)
(1 8)(2 14)(3 13)(4 5)(6 7)(9 10)(11 12)(15 16)
(1 9)(2 15)(3 11)(4 6)(5 7)(8 10)(12 13)(14 16)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)

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

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

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

G:=TransitiveGroup(16,58);

On 24 points - transitive group 24T49
Generators in S24
(1 15)(2 13)(3 14)(4 20)(5 21)(6 19)(7 24)(8 22)(9 23)(10 18)(11 16)(12 17)
(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)
(1 15)(2 16)(3 12)(4 20)(5 9)(6 24)(7 19)(8 22)(10 18)(11 13)(14 17)(21 23)
(1 10)(2 13)(3 17)(4 22)(5 21)(6 7)(8 20)(9 23)(11 16)(12 14)(15 18)(19 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)

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

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

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

G:=TransitiveGroup(24,49);

On 24 points - transitive group 24T50
Generators in S24
(1 22)(2 23)(3 24)(4 7)(5 8)(6 9)(10 16)(11 17)(12 18)(13 19)(14 20)(15 21)
(1 10)(2 11)(3 12)(4 19)(5 20)(6 21)(7 13)(8 14)(9 15)(16 22)(17 23)(18 24)
(2 8)(3 9)(5 23)(6 24)(11 14)(12 15)(17 20)(18 21)
(1 7)(3 9)(4 22)(6 24)(10 13)(12 15)(16 19)(18 21)
(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)| (1,22)(2,23)(3,24)(4,7)(5,8)(6,9)(10,16)(11,17)(12,18)(13,19)(14,20)(15,21), (1,10)(2,11)(3,12)(4,19)(5,20)(6,21)(7,13)(8,14)(9,15)(16,22)(17,23)(18,24), (2,8)(3,9)(5,23)(6,24)(11,14)(12,15)(17,20)(18,21), (1,7)(3,9)(4,22)(6,24)(10,13)(12,15)(16,19)(18,21), (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( (1,22)(2,23)(3,24)(4,7)(5,8)(6,9)(10,16)(11,17)(12,18)(13,19)(14,20)(15,21), (1,10)(2,11)(3,12)(4,19)(5,20)(6,21)(7,13)(8,14)(9,15)(16,22)(17,23)(18,24), (2,8)(3,9)(5,23)(6,24)(11,14)(12,15)(17,20)(18,21), (1,7)(3,9)(4,22)(6,24)(10,13)(12,15)(16,19)(18,21), (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([(1,22),(2,23),(3,24),(4,7),(5,8),(6,9),(10,16),(11,17),(12,18),(13,19),(14,20),(15,21)], [(1,10),(2,11),(3,12),(4,19),(5,20),(6,21),(7,13),(8,14),(9,15),(16,22),(17,23),(18,24)], [(2,8),(3,9),(5,23),(6,24),(11,14),(12,15),(17,20),(18,21)], [(1,7),(3,9),(4,22),(6,24),(10,13),(12,15),(16,19),(18,21)], [(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(24,50);

C22×A4 is a maximal subgroup of
A4⋊D4  C24.A4  C24.2A4  C24.3A4  C2≀A4  C24.6A4  C24⋊A4  C24.7A4  C245A4
C22×A4 is a maximal quotient of
Q8.A4  D4.A4  C24.6A4  C24⋊A4

Polynomial with Galois group C22×A4 over ℚ
actionf(x)Disc(f)
12T25x12-12x10+48x8-77x6+48x4-12x2+1212·318·1992
12T26x12+2x10-5x8-4x6-5x4+2x2+1248·78

Matrix representation of C22×A4 in GL4(𝔽7) generated by

 6 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 6 0 0 0 0 6 0 0 0 0 6
,
 1 0 0 0 0 6 0 0 0 0 6 0 0 0 0 1
,
 1 0 0 0 0 6 0 0 0 0 1 0 0 0 0 6
,
 4 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
G:=sub<GL(4,GF(7))| [6,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[1,0,0,0,0,6,0,0,0,0,6,0,0,0,0,1],[1,0,0,0,0,6,0,0,0,0,1,0,0,0,0,6],[4,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;

C22×A4 in GAP, Magma, Sage, TeX

C_2^2\times A_4
% in TeX

G:=Group("C2^2xA4");
// GroupNames label

G:=SmallGroup(48,49);
// by ID

G=gap.SmallGroup(48,49);
# by ID

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

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

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