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G = C23⋊A4order 96 = 25·3

2nd semidirect product of C23 and A4 acting faithfully

non-abelian, soluble, monomial

Aliases: Q82A4, C232A4, 2+ 1+42C3, C2.2(C22⋊A4), SmallGroup(96,204)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C23⋊A4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f3=1, faf-1=ab=ba, eae=ac=ca, ad=da, dbd=bc=cb, be=eb, fbf-1=a, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

6C2
6C2
6C2
16C3
3C22
3C4
3C22
3C4
3C22
4C22
4C22
4C22
12C22
16C6
3D4
3D4
3C2×C4
3C23
3D4
3D4
3D4
3C2×C4
3D4
3C2×C4
4A4
4A4
4A4
3C2×D4
3C2×D4
3C4○D4
3C2×D4
3C4○D4
4SL2(𝔽3)
4C2×A4
4C2×A4
4SL2(𝔽3)
4C2×A4

Character table of C23⋊A4

 class 12A2B2C2D3A3B4A4B6A6B
 size 116661616661616
ρ111111111111    trivial
ρ211111ζ32ζ311ζ3ζ32    linear of order 3
ρ311111ζ3ζ3211ζ32ζ3    linear of order 3
ρ433-1-1300-1-100    orthogonal lifted from A4
ρ533-1-1-100-1300    orthogonal lifted from A4
ρ633-13-100-1-100    orthogonal lifted from A4
ρ733-1-1-1003-100    orthogonal lifted from A4
ρ8333-1-100-1-100    orthogonal lifted from A4
ρ94-40001100-1-1    orthogonal faithful
ρ104-4000ζ32ζ300ζ65ζ6    complex faithful
ρ114-4000ζ3ζ3200ζ6ζ65    complex faithful

Permutation representations of C23⋊A4
On 8 points - transitive group 8T32
Generators in S8
(1 5)(2 8)(3 4)(6 7)
(1 3)(2 6)(4 5)(7 8)
(1 2)(3 6)(4 7)(5 8)
(3 6)(4 7)
(4 7)(5 8)
(3 4 5)(6 7 8)

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

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

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

G:=TransitiveGroup(8,32);

On 24 points - transitive group 24T97
Generators in S24
(5 9)(6 7)(10 15)(12 14)(16 19)(18 21)
(4 8)(6 7)(10 15)(11 13)(16 19)(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)
(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)| (5,9)(6,7)(10,15)(12,14)(16,19)(18,21), (4,8)(6,7)(10,15)(11,13)(16,19)(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), (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( (5,9)(6,7)(10,15)(12,14)(16,19)(18,21), (4,8)(6,7)(10,15)(11,13)(16,19)(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), (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([(5,9),(6,7),(10,15),(12,14),(16,19),(18,21)], [(4,8),(6,7),(10,15),(11,13),(16,19),(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)], [(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(24,97);

On 24 points - transitive group 24T149
Generators in S24
(2 10)(3 11)(4 8)(5 20)(6 18)(7 21)(9 17)(13 22)(15 24)(16 19)
(1 12)(3 11)(4 16)(5 9)(6 21)(7 18)(8 19)(13 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)| (2,10)(3,11)(4,8)(5,20)(6,18)(7,21)(9,17)(13,22)(15,24)(16,19), (1,12)(3,11)(4,16)(5,9)(6,21)(7,18)(8,19)(13,22)(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( (2,10)(3,11)(4,8)(5,20)(6,18)(7,21)(9,17)(13,22)(15,24)(16,19), (1,12)(3,11)(4,16)(5,9)(6,21)(7,18)(8,19)(13,22)(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([(2,10),(3,11),(4,8),(5,20),(6,18),(7,21),(9,17),(13,22),(15,24),(16,19)], [(1,12),(3,11),(4,16),(5,9),(6,21),(7,18),(8,19),(13,22),(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(24,149);

C23⋊A4 is a maximal subgroup of
C2≀A4  2+ 1+4.C6  C23.S4  Q8.S4  C23⋊S4  Q82S4  2+ 1+4.3C6  Ω4+ (𝔽3)
C23⋊A4 is a maximal quotient of
C24.7A4  Q8⋊SL2(𝔽3)  C245A4  2+ 1+42C9

Polynomial with Galois group C23⋊A4 over ℚ
actionf(x)Disc(f)
8T32x8+2x7-27x6-93x5-3x4+272x3+263x2+35x-2212·34·532·614·3892

Matrix representation of C23⋊A4 in GL4(ℤ) generated by

1000
0-100
0010
000-1
,
1000
0100
00-10
000-1
,
-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],[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] >;

C23⋊A4 in GAP, Magma, Sage, TeX

C_2^3\rtimes A_4
% in TeX

G:=Group("C2^3:A4");
// GroupNames label

G:=SmallGroup(96,204);
// by ID

G=gap.SmallGroup(96,204);
# by ID

G:=PCGroup([6,-3,-2,2,-2,2,-2,73,164,579,255,1084,730]);
// Polycyclic

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

Export

Subgroup lattice of C23⋊A4 in TeX
Character table of C23⋊A4 in TeX

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