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G = C2≀A4order 192 = 26·3

Wreath product of C2 by A4

non-abelian, soluble, monomial

Aliases: C2A4, C241A4, 2+ 1+41C6, C2≀C22⋊C3, C23⋊A41C2, C23.1(C2×A4), C2.4(C24⋊C6), SmallGroup(192,201)

Series: Derived Chief Lower central Upper central

C1C22+ 1+4 — C2≀A4
C1C2C232+ 1+4C23⋊A4 — C2≀A4
2+ 1+4 — C2≀A4
C1C2

Generators and relations for C2≀A4
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g3=1, ab=ba, faf=ac=ca, ad=da, eae=abd, ag=ga, bc=cb, fbf=gcg-1=bd=db, be=eb, gbg-1=bcd, ece=cd=dc, cf=fc, de=ed, df=fd, dg=gd, geg-1=ef=fe, gfg-1=e >

Subgroups: 351 in 58 conjugacy classes, 8 normal (all characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C22 [×9], C6 [×3], C2×C4 [×3], D4 [×5], Q8, C23, C23 [×5], A4 [×2], C2×C6, C22⋊C4 [×2], C2×D4 [×3], C4○D4, C24, SL2(𝔽3), C2×A4 [×4], C23⋊C4, C22≀C2, 2+ 1+4, C22×A4, C2≀C22, C23⋊A4, C2≀A4
Quotients: C1, C2, C3, C6, A4, C2×A4, C24⋊C6, C2≀A4

Character table of C2≀A4

 class 12A2B2C2D2E3A3B4A4B6A6B6C6D6E6F
 size 114461216161224161616161616
ρ11111111111111111    trivial
ρ211-1-111111-1-11-1-1-11    linear of order 2
ρ311-1-111ζ32ζ31-1ζ65ζ32ζ6ζ6ζ65ζ3    linear of order 6
ρ411-1-111ζ3ζ321-1ζ6ζ3ζ65ζ65ζ6ζ32    linear of order 6
ρ5111111ζ3ζ3211ζ32ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ6111111ζ32ζ311ζ3ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ733-3-33-100-11000000    orthogonal lifted from C2×A4
ρ833333-100-1-1000000    orthogonal lifted from A4
ρ94-42-2001100-1-11-11-1    orthogonal faithful
ρ104-4-220011001-1-11-1-1    orthogonal faithful
ρ114-42-200ζ32ζ300ζ65ζ6ζ32ζ6ζ3ζ65    complex faithful
ρ124-4-2200ζ32ζ300ζ3ζ6ζ6ζ32ζ65ζ65    complex faithful
ρ134-42-200ζ3ζ3200ζ6ζ65ζ3ζ65ζ32ζ6    complex faithful
ρ144-4-2200ζ3ζ3200ζ32ζ65ζ65ζ3ζ6ζ6    complex faithful
ρ156600-2200-20000000    orthogonal lifted from C24⋊C6
ρ166600-2-20020000000    orthogonal lifted from C24⋊C6

Permutation representations of C2≀A4
On 8 points - transitive group 8T38
Generators in S8
(1 2)
(3 6)(4 7)
(1 2)(3 6)
(1 2)(3 6)(4 7)(5 8)
(1 5)(2 8)(3 4)(6 7)
(1 3)(2 6)(4 5)(7 8)
(3 4 5)(6 7 8)

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

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

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

G:=TransitiveGroup(8,38);

On 16 points - transitive group 16T425
Generators in S16
(1 4)(2 3)(5 13)(6 11)(7 12)(8 15)(9 16)(10 14)
(1 10)(2 5)(3 13)(4 14)(6 7)(8 9)(11 12)(15 16)
(1 6)(2 8)(3 15)(4 11)(5 9)(7 10)(12 14)(13 16)
(1 2)(3 4)(5 10)(6 8)(7 9)(11 15)(12 16)(13 14)
(3 14)(4 13)(6 8)(7 9)(11 12)(15 16)
(3 15)(4 11)(5 10)(7 9)(12 13)(14 16)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)

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

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

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

G:=TransitiveGroup(16,425);

On 16 points - transitive group 16T427
Generators in S16
(1 3)(2 4)(5 13)(6 11)(7 12)(8 14)(9 15)(10 16)
(6 8)(7 9)(11 14)(12 15)
(1 2)(3 4)(6 8)(11 14)
(1 2)(3 4)(5 10)(6 8)(7 9)(11 14)(12 15)(13 16)
(1 5)(2 10)(3 16)(4 13)(6 7)(8 9)(11 12)(14 15)
(1 6)(2 8)(3 14)(4 11)(5 7)(9 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)(5,13)(6,11)(7,12)(8,14)(9,15)(10,16), (6,8)(7,9)(11,14)(12,15), (1,2)(3,4)(6,8)(11,14), (1,2)(3,4)(5,10)(6,8)(7,9)(11,14)(12,15)(13,16), (1,5)(2,10)(3,16)(4,13)(6,7)(8,9)(11,12)(14,15), (1,6)(2,8)(3,14)(4,11)(5,7)(9,10)(12,13)(15,16), (5,6,7)(8,9,10)(11,12,13)(14,15,16)>;

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

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

G:=TransitiveGroup(16,427);

On 24 points - transitive group 24T288
Generators in S24
(1 23)(2 24)(3 22)(10 15)(11 13)(12 14)(16 19)(17 20)(18 21)
(2 24)(3 22)(11 13)(12 14)(16 19)(17 20)
(2 24)(4 8)(5 9)(6 7)(11 13)(16 19)
(1 23)(2 24)(3 22)(4 8)(5 9)(6 7)(10 15)(11 13)(12 14)(16 19)(17 20)(18 21)
(1 8)(2 20)(3 11)(4 23)(5 15)(6 18)(7 21)(9 10)(12 19)(13 22)(14 16)(17 24)
(1 12)(2 9)(3 21)(4 16)(5 24)(6 13)(7 11)(8 19)(10 20)(14 23)(15 17)(18 22)
(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)| (1,23)(2,24)(3,22)(10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (2,24)(3,22)(11,13)(12,14)(16,19)(17,20), (2,24)(4,8)(5,9)(6,7)(11,13)(16,19), (1,23)(2,24)(3,22)(4,8)(5,9)(6,7)(10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (1,8)(2,20)(3,11)(4,23)(5,15)(6,18)(7,21)(9,10)(12,19)(13,22)(14,16)(17,24), (1,12)(2,9)(3,21)(4,16)(5,24)(6,13)(7,11)(8,19)(10,20)(14,23)(15,17)(18,22), (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( (1,23)(2,24)(3,22)(10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (2,24)(3,22)(11,13)(12,14)(16,19)(17,20), (2,24)(4,8)(5,9)(6,7)(11,13)(16,19), (1,23)(2,24)(3,22)(4,8)(5,9)(6,7)(10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (1,8)(2,20)(3,11)(4,23)(5,15)(6,18)(7,21)(9,10)(12,19)(13,22)(14,16)(17,24), (1,12)(2,9)(3,21)(4,16)(5,24)(6,13)(7,11)(8,19)(10,20)(14,23)(15,17)(18,22), (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([(1,23),(2,24),(3,22),(10,15),(11,13),(12,14),(16,19),(17,20),(18,21)], [(2,24),(3,22),(11,13),(12,14),(16,19),(17,20)], [(2,24),(4,8),(5,9),(6,7),(11,13),(16,19)], [(1,23),(2,24),(3,22),(4,8),(5,9),(6,7),(10,15),(11,13),(12,14),(16,19),(17,20),(18,21)], [(1,8),(2,20),(3,11),(4,23),(5,15),(6,18),(7,21),(9,10),(12,19),(13,22),(14,16),(17,24)], [(1,12),(2,9),(3,21),(4,16),(5,24),(6,13),(7,11),(8,19),(10,20),(14,23),(15,17),(18,22)], [(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(24,288);

On 24 points - transitive group 24T425
Generators in S24
(4 16)(5 17)(6 18)(7 21)(8 19)(9 20)
(1 23)(2 10)(3 13)(5 20)(6 21)(7 18)(9 17)(11 22)(12 14)(15 24)
(1 12)(3 13)(4 16)(5 9)(6 18)(7 21)(8 19)(11 22)(14 23)(17 20)
(1 23)(2 24)(3 22)(4 8)(5 9)(6 7)(10 15)(11 13)(12 14)(16 19)(17 20)(18 21)
(2 9)(3 7)(5 24)(6 22)(10 17)(11 21)(12 14)(13 18)(15 20)(16 19)
(1 8)(3 7)(4 23)(6 22)(10 15)(11 18)(12 19)(13 21)(14 16)(17 20)
(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)| (4,16)(5,17)(6,18)(7,21)(8,19)(9,20), (1,23)(2,10)(3,13)(5,20)(6,21)(7,18)(9,17)(11,22)(12,14)(15,24), (1,12)(3,13)(4,16)(5,9)(6,18)(7,21)(8,19)(11,22)(14,23)(17,20), (1,23)(2,24)(3,22)(4,8)(5,9)(6,7)(10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (2,9)(3,7)(5,24)(6,22)(10,17)(11,21)(12,14)(13,18)(15,20)(16,19), (1,8)(3,7)(4,23)(6,22)(10,15)(11,18)(12,19)(13,21)(14,16)(17,20), (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( (4,16)(5,17)(6,18)(7,21)(8,19)(9,20), (1,23)(2,10)(3,13)(5,20)(6,21)(7,18)(9,17)(11,22)(12,14)(15,24), (1,12)(3,13)(4,16)(5,9)(6,18)(7,21)(8,19)(11,22)(14,23)(17,20), (1,23)(2,24)(3,22)(4,8)(5,9)(6,7)(10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (2,9)(3,7)(5,24)(6,22)(10,17)(11,21)(12,14)(13,18)(15,20)(16,19), (1,8)(3,7)(4,23)(6,22)(10,15)(11,18)(12,19)(13,21)(14,16)(17,20), (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([(4,16),(5,17),(6,18),(7,21),(8,19),(9,20)], [(1,23),(2,10),(3,13),(5,20),(6,21),(7,18),(9,17),(11,22),(12,14),(15,24)], [(1,12),(3,13),(4,16),(5,9),(6,18),(7,21),(8,19),(11,22),(14,23),(17,20)], [(1,23),(2,24),(3,22),(4,8),(5,9),(6,7),(10,15),(11,13),(12,14),(16,19),(17,20),(18,21)], [(2,9),(3,7),(5,24),(6,22),(10,17),(11,21),(12,14),(13,18),(15,20),(16,19)], [(1,8),(3,7),(4,23),(6,22),(10,15),(11,18),(12,19),(13,21),(14,16),(17,20)], [(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(24,425);

Polynomial with Galois group C2≀A4 over ℚ
actionf(x)Disc(f)
8T38x8-3x7-13x6+18x5+42x4-17x3-31x2+2x+422·89·1634·6532

Matrix representation of C2≀A4 in GL4(ℤ) generated by

1000
0-100
00-10
000-1
,
1000
0-100
0010
000-1
,
-1000
0-100
0010
0001
,
-1000
0-100
00-10
000-1
,
0010
0001
1000
0100
,
0100
1000
0001
0010
,
1000
0010
0001
0100
G:=sub<GL(4,Integers())| [1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1],[-1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,1],[-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;

C2≀A4 in GAP, Magma, Sage, TeX

C_2\wr A_4
% in TeX

G:=Group("C2wrA4");
// GroupNames label

G:=SmallGroup(192,201);
// by ID

G=gap.SmallGroup(192,201);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,632,135,1683,262,851,375,3540,1027]);
// Polycyclic

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

Export

Character table of C2≀A4 in TeX

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