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G = C22×A4order 48 = 24·3

Direct product of C22 and A4

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

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

Series: Derived Chief Lower central Upper central

C1C22 — C22×A4
C1C22A4C2×A4 — C22×A4
C22 — C22×A4
C1C22

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 >

3C2
3C2
3C2
3C2
4C3
3C22
3C22
3C22
3C22
3C22
3C22
3C22
3C22
3C22
3C22
3C22
4C6
4C6
4C6
3C23
3C23
3C23
3C23
4C2×C6

Character table of C22×A4

 class 12A2B2C2D2E2F2G3A3B6A6B6C6D6E6F
 size 1111333344444444
ρ11111111111111111    trivial
ρ21-11-111-1-111-1-111-1-1    linear of order 2
ρ31-1-111-1-11111-1-1-11-1    linear of order 2
ρ411-1-11-11-111-11-1-1-11    linear of order 2
ρ51-1-111-1-11ζ32ζ3ζ32ζ6ζ65ζ6ζ3ζ65    linear of order 6
ρ61-11-111-1-1ζ32ζ3ζ6ζ6ζ3ζ32ζ65ζ65    linear of order 6
ρ711-1-11-11-1ζ3ζ32ζ65ζ3ζ6ζ65ζ6ζ32    linear of order 6
ρ811111111ζ3ζ32ζ3ζ3ζ32ζ3ζ32ζ32    linear of order 3
ρ911-1-11-11-1ζ32ζ3ζ6ζ32ζ65ζ6ζ65ζ3    linear of order 6
ρ101-11-111-1-1ζ3ζ32ζ65ζ65ζ32ζ3ζ6ζ6    linear of order 6
ρ1111111111ζ32ζ3ζ32ζ32ζ3ζ32ζ3ζ3    linear of order 3
ρ121-1-111-1-11ζ3ζ32ζ3ζ65ζ6ζ65ζ32ζ6    linear of order 6
ρ133-3-33-111-100000000    orthogonal lifted from C2×A4
ρ143-33-3-1-11100000000    orthogonal lifted from C2×A4
ρ153333-1-1-1-100000000    orthogonal lifted from A4
ρ1633-3-3-11-1100000000    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 3)(2 4)(5 8)(6 9)(7 10)(11 14)(12 15)(13 16)
(1 4)(2 3)(5 16)(6 14)(7 15)(8 13)(9 11)(10 12)
(1 16)(2 8)(3 13)(4 5)(6 7)(9 10)(11 12)(14 15)
(1 14)(2 9)(3 11)(4 6)(5 7)(8 10)(12 13)(15 16)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)

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

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

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

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

6000
0100
0010
0001
,
1000
0600
0060
0006
,
1000
0600
0060
0001
,
1000
0600
0010
0006
,
4000
0010
0001
0100
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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