non-abelian, soluble, monomial
Aliases: Q8⋊2A4, C23⋊2A4, 2+ 1+4⋊2C3, C2.2(C22⋊A4), SmallGroup(96,204)
Series: Derived ►Chief ►Lower central ►Upper central
| 2+ 1+4 — C23⋊A4 |
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
3C23
3D4
3D4
3D4
3D4
4A4
4A4
4A4
Character table of C23⋊A4
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 6A | 6B | |
| size | 1 | 1 | 6 | 6 | 6 | 16 | 16 | 6 | 6 | 16 | 16 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 3 | 3 | -1 | -1 | 3 | 0 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ5 | 3 | 3 | -1 | -1 | -1 | 0 | 0 | -1 | 3 | 0 | 0 | orthogonal lifted from A4 |
| ρ6 | 3 | 3 | -1 | 3 | -1 | 0 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ7 | 3 | 3 | -1 | -1 | -1 | 0 | 0 | 3 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ8 | 3 | 3 | 3 | -1 | -1 | 0 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ9 | 4 | -4 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | -1 | -1 | orthogonal faithful |
| ρ10 | 4 | -4 | 0 | 0 | 0 | ζ32 | ζ3 | 0 | 0 | ζ65 | ζ6 | complex faithful |
| ρ11 | 4 | -4 | 0 | 0 | 0 | ζ3 | ζ32 | 0 | 0 | ζ6 | ζ65 | complex faithful |
►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 Q8⋊2S4 2+ 1+4.3C6 Ω4+ (𝔽3)
C23⋊A4 is a maximal quotient of
C24.7A4 Q8⋊SL2(𝔽3) C24⋊5A4 2+ 1+4⋊2C9
| action | f(x) | Disc(f) |
|---|---|---|
| 8T32 | x8+2x7-27x6-93x5-3x4+272x3+263x2+35x-2 | 212·34·532·614·3892 |
Matrix representation of C23⋊A4 ►in GL4(ℤ) generated by
| 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 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
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