metabelian, soluble, monomial, A-group
Aliases: C42⋊C3, C22.A4, SmallGroup(48,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C42⋊C3 |
Generators and relations for C42⋊C3
G = < a,b,c | a4=b4=c3=1, ab=ba, cac-1=ab-1, cbc-1=a-1b2 >
Character table of C42⋊C3
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 4D | |
| size | 1 | 3 | 16 | 16 | 3 | 3 | 3 | 3 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 3 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ5 | 3 | -1 | 0 | 0 | 1 | -1+2i | -1-2i | 1 | complex faithful |
| ρ6 | 3 | -1 | 0 | 0 | -1-2i | 1 | 1 | -1+2i | complex faithful |
| ρ7 | 3 | -1 | 0 | 0 | 1 | -1-2i | -1+2i | 1 | complex faithful |
| ρ8 | 3 | -1 | 0 | 0 | -1+2i | 1 | 1 | -1-2i | complex faithful |
►On 12 points - transitive group 12T31
Generators in S12
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 4 2 3)(5 8 7 6)
(1 11 5)(2 9 7)(3 10 8)(4 12 6)
G:=sub<Sym(12)| (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,4,2,3)(5,8,7,6), (1,11,5)(2,9,7)(3,10,8)(4,12,6)>;
G:=Group( (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,4,2,3)(5,8,7,6), (1,11,5)(2,9,7)(3,10,8)(4,12,6) );
G=PermutationGroup([[(1,2),(3,4),(5,6,7,8),(9,10,11,12)], [(1,4,2,3),(5,8,7,6)], [(1,11,5),(2,9,7),(3,10,8),(4,12,6)]])
G:=TransitiveGroup(12,31);
►On 16 points - transitive group 16T63
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 12 13 5)(2 9 14 6)(3 10 15 7)(4 11 16 8)
(2 10 6)(3 13 15)(4 7 11)(5 8 14)(9 16 12)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,12,13,5)(2,9,14,6)(3,10,15,7)(4,11,16,8), (2,10,6)(3,13,15)(4,7,11)(5,8,14)(9,16,12)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,12,13,5)(2,9,14,6)(3,10,15,7)(4,11,16,8), (2,10,6)(3,13,15)(4,7,11)(5,8,14)(9,16,12) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,12,13,5),(2,9,14,6),(3,10,15,7),(4,11,16,8)], [(2,10,6),(3,13,15),(4,7,11),(5,8,14),(9,16,12)]])
G:=TransitiveGroup(16,63);
►On 24 points - transitive group 24T58
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4 6 7)(2 3 5 8)(9 21)(10 22)(11 23)(12 24)(13 20 15 18)(14 17 16 19)
(1 22 13)(2 12 19)(3 21 20)(4 11 14)(5 10 17)(6 24 15)(7 9 16)(8 23 18)
G:=sub<Sym(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), (1,4,6,7)(2,3,5,8)(9,21)(10,22)(11,23)(12,24)(13,20,15,18)(14,17,16,19), (1,22,13)(2,12,19)(3,21,20)(4,11,14)(5,10,17)(6,24,15)(7,9,16)(8,23,18)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,6,7)(2,3,5,8)(9,21)(10,22)(11,23)(12,24)(13,20,15,18)(14,17,16,19), (1,22,13)(2,12,19)(3,21,20)(4,11,14)(5,10,17)(6,24,15)(7,9,16)(8,23,18) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4,6,7),(2,3,5,8),(9,21),(10,22),(11,23),(12,24),(13,20,15,18),(14,17,16,19)], [(1,22,13),(2,12,19),(3,21,20),(4,11,14),(5,10,17),(6,24,15),(7,9,16),(8,23,18)]])
G:=TransitiveGroup(24,58);
C42⋊C3 is a maximal subgroup of
C42⋊S3 C42⋊C6 C23.A4 C82⋊C3 C42⋊2A4 C42⋊A4 C42.A4 C42⋊(C7⋊C3)
C42⋊C3 is a maximal quotient of
C23.3A4 C42⋊C9 C82⋊C3 C42⋊2A4 C42⋊(C7⋊C3)
| action | f(x) | Disc(f) |
|---|---|---|
| 12T31 | x12-38x10+450x8-2106x6+3679x4-2028x2+169 | 224·720·1314·7974 |
Matrix representation of C42⋊C3 ►in GL3(𝔽5) generated by
| 2 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 2 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(5))| [2,0,0,0,2,0,0,0,4],[1,0,0,0,3,0,0,0,2],[0,1,0,0,0,1,1,0,0] >;
C42⋊C3 in GAP, Magma, Sage, TeX
C_4^2\rtimes C_3
% in TeX
G:=Group("C4^2:C3"); // GroupNames label
G:=SmallGroup(48,3);
// by ID
G=gap.SmallGroup(48,3);
# by ID
G:=PCGroup([5,-3,-2,2,-2,2,61,126,497,42,483,904]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^3=1,a*b=b*a,c*a*c^-1=a*b^-1,c*b*c^-1=a^-1*b^2>;
// generators/relations
Export
Subgroup lattice of C42⋊C3 in TeX
Character table of C42⋊C3 in TeX