non-abelian, soluble, monomial
Aliases: A4⋊C4, C2.1S4, C23.S3, C22⋊Dic3, (C2×A4).C2, SL2(ℤ/4ℤ), SmallGroup(48,30)
Series: Derived ►Chief ►Lower central ►Upper central
| A4 — A4⋊C4 |
Generators and relations for A4⋊C4
G = < a,b,c,d | a2=b2=c3=d4=1, cac-1=dad-1=ab=ba, cbc-1=a, bd=db, dcd-1=c-1 >
Character table of A4⋊C4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | |
| size | 1 | 1 | 3 | 3 | 8 | 6 | 6 | 6 | 6 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | -1 | 1 | -i | i | -i | i | -1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | -1 | 1 | i | -i | i | -i | -1 | linear of order 4 |
| ρ5 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ7 | 3 | 3 | -1 | -1 | 0 | 1 | 1 | -1 | -1 | 0 | orthogonal lifted from S4 |
| ρ8 | 3 | 3 | -1 | -1 | 0 | -1 | -1 | 1 | 1 | 0 | orthogonal lifted from S4 |
| ρ9 | 3 | -3 | -1 | 1 | 0 | i | -i | -i | i | 0 | complex faithful |
| ρ10 | 3 | -3 | -1 | 1 | 0 | -i | i | i | -i | 0 | complex faithful |
►On 12 points - transitive group 12T27
Generators in S12
(1 2)(3 4)(5 7)(6 12)(8 10)(9 11)
(1 3)(2 4)(5 9)(6 10)(7 11)(8 12)
(1 10 5)(2 6 11)(3 12 7)(4 8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
G:=sub<Sym(12)| (1,2)(3,4)(5,7)(6,12)(8,10)(9,11), (1,3)(2,4)(5,9)(6,10)(7,11)(8,12), (1,10,5)(2,6,11)(3,12,7)(4,8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)>;
G:=Group( (1,2)(3,4)(5,7)(6,12)(8,10)(9,11), (1,3)(2,4)(5,9)(6,10)(7,11)(8,12), (1,10,5)(2,6,11)(3,12,7)(4,8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12) );
G=PermutationGroup([[(1,2),(3,4),(5,7),(6,12),(8,10),(9,11)], [(1,3),(2,4),(5,9),(6,10),(7,11),(8,12)], [(1,10,5),(2,6,11),(3,12,7),(4,8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)]])
G:=TransitiveGroup(12,27);
►On 12 points - transitive group 12T30
Generators in S12
(1 3)(6 8)(9 11)(10 12)
(1 3)(2 4)(5 7)(6 8)
(1 7 9)(2 10 8)(3 5 11)(4 12 6)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
G:=sub<Sym(12)| (1,3)(6,8)(9,11)(10,12), (1,3)(2,4)(5,7)(6,8), (1,7,9)(2,10,8)(3,5,11)(4,12,6), (1,2,3,4)(5,6,7,8)(9,10,11,12)>;
G:=Group( (1,3)(6,8)(9,11)(10,12), (1,3)(2,4)(5,7)(6,8), (1,7,9)(2,10,8)(3,5,11)(4,12,6), (1,2,3,4)(5,6,7,8)(9,10,11,12) );
G=PermutationGroup([[(1,3),(6,8),(9,11),(10,12)], [(1,3),(2,4),(5,7),(6,8)], [(1,7,9),(2,10,8),(3,5,11),(4,12,6)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)]])
G:=TransitiveGroup(12,30);
►On 16 points - transitive group 16T62
Generators in S16
(1 12)(2 7)(3 10)(4 5)(6 16)(8 14)(9 13)(11 15)
(1 16)(2 13)(3 14)(4 15)(5 11)(6 12)(7 9)(8 10)
(5 15 11)(6 12 16)(7 13 9)(8 10 14)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
G:=sub<Sym(16)| (1,12)(2,7)(3,10)(4,5)(6,16)(8,14)(9,13)(11,15), (1,16)(2,13)(3,14)(4,15)(5,11)(6,12)(7,9)(8,10), (5,15,11)(6,12,16)(7,13,9)(8,10,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;
G:=Group( (1,12)(2,7)(3,10)(4,5)(6,16)(8,14)(9,13)(11,15), (1,16)(2,13)(3,14)(4,15)(5,11)(6,12)(7,9)(8,10), (5,15,11)(6,12,16)(7,13,9)(8,10,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );
G=PermutationGroup([[(1,12),(2,7),(3,10),(4,5),(6,16),(8,14),(9,13),(11,15)], [(1,16),(2,13),(3,14),(4,15),(5,11),(6,12),(7,9),(8,10)], [(5,15,11),(6,12,16),(7,13,9),(8,10,14)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)]])
G:=TransitiveGroup(16,62);
►On 24 points - transitive group 24T51
Generators in S24
(1 11)(2 4)(3 9)(5 7)(6 15)(8 13)(10 12)(14 16)(17 23)(18 22)(19 21)(20 24)
(1 9)(2 10)(3 11)(4 12)(5 16)(6 13)(7 14)(8 15)(17 19)(18 20)(21 23)(22 24)
(1 7 19)(2 20 8)(3 5 17)(4 18 6)(9 16 21)(10 22 13)(11 14 23)(12 24 15)
(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)| (1,11)(2,4)(3,9)(5,7)(6,15)(8,13)(10,12)(14,16)(17,23)(18,22)(19,21)(20,24), (1,9)(2,10)(3,11)(4,12)(5,16)(6,13)(7,14)(8,15)(17,19)(18,20)(21,23)(22,24), (1,7,19)(2,20,8)(3,5,17)(4,18,6)(9,16,21)(10,22,13)(11,14,23)(12,24,15), (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( (1,11)(2,4)(3,9)(5,7)(6,15)(8,13)(10,12)(14,16)(17,23)(18,22)(19,21)(20,24), (1,9)(2,10)(3,11)(4,12)(5,16)(6,13)(7,14)(8,15)(17,19)(18,20)(21,23)(22,24), (1,7,19)(2,20,8)(3,5,17)(4,18,6)(9,16,21)(10,22,13)(11,14,23)(12,24,15), (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([[(1,11),(2,4),(3,9),(5,7),(6,15),(8,13),(10,12),(14,16),(17,23),(18,22),(19,21),(20,24)], [(1,9),(2,10),(3,11),(4,12),(5,16),(6,13),(7,14),(8,15),(17,19),(18,20),(21,23),(22,24)], [(1,7,19),(2,20,8),(3,5,17),(4,18,6),(9,16,21),(10,22,13),(11,14,23),(12,24,15)], [(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(24,51);
►On 24 points - transitive group 24T57
Generators in S24
(1 16)(3 14)(5 10)(6 11)(7 12)(8 9)(17 22)(19 24)
(1 16)(2 13)(3 14)(4 15)(17 22)(18 23)(19 24)(20 21)
(1 23 9)(2 10 24)(3 21 11)(4 12 22)(5 19 13)(6 14 20)(7 17 15)(8 16 18)
(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)| (1,16)(3,14)(5,10)(6,11)(7,12)(8,9)(17,22)(19,24), (1,16)(2,13)(3,14)(4,15)(17,22)(18,23)(19,24)(20,21), (1,23,9)(2,10,24)(3,21,11)(4,12,22)(5,19,13)(6,14,20)(7,17,15)(8,16,18), (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( (1,16)(3,14)(5,10)(6,11)(7,12)(8,9)(17,22)(19,24), (1,16)(2,13)(3,14)(4,15)(17,22)(18,23)(19,24)(20,21), (1,23,9)(2,10,24)(3,21,11)(4,12,22)(5,19,13)(6,14,20)(7,17,15)(8,16,18), (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([[(1,16),(3,14),(5,10),(6,11),(7,12),(8,9),(17,22),(19,24)], [(1,16),(2,13),(3,14),(4,15),(17,22),(18,23),(19,24),(20,21)], [(1,23,9),(2,10,24),(3,21,11),(4,12,22),(5,19,13),(6,14,20),(7,17,15),(8,16,18)], [(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(24,57);
A4⋊C4 is a maximal subgroup of
A4⋊Q8 C4×S4 A4⋊D4 C6.7S4 C23.7S4 C23.9S4 C24⋊Dic3 C42⋊Dic3 Q8.1S4 C23.S4 Q8.S4 C24⋊4Dic3 A5⋊C4 A4⋊Dic5 A4⋊F5 A4⋊Dic7 C62⋊Dic3
A4⋊C4 is a maximal quotient of
A4⋊C8 Q8⋊Dic3 U2(𝔽3) C6.S4 C6.7S4 C23.9S4 C24⋊Dic3 C42⋊Dic3 C24⋊4Dic3 A4⋊Dic5 A4⋊F5 A4⋊Dic7 C62⋊Dic3
| action | f(x) | Disc(f) |
|---|---|---|
| 12T27 | x12-4x11-20x10+68x9+154x8-384x7-496x6+880x5+524x4-816x3-16x2+144x-8 | 276·36·52·377 |
| 12T30 | x12-20x10+131x8-384x6+527x4-296x2+37 | 236·377·674 |
Matrix representation of A4⋊C4 ►in GL3(𝔽5) generated by
| 2 | 1 | 0 |
| 2 | 3 | 0 |
| 1 | 2 | 4 |
| 4 | 4 | 2 |
| 0 | 2 | 4 |
| 0 | 3 | 3 |
| 1 | 0 | 3 |
| 3 | 0 | 3 |
| 2 | 1 | 4 |
| 3 | 3 | 4 |
| 3 | 1 | 2 |
| 4 | 2 | 3 |
G:=sub<GL(3,GF(5))| [2,2,1,1,3,2,0,0,4],[4,0,0,4,2,3,2,4,3],[1,3,2,0,0,1,3,3,4],[3,3,4,3,1,2,4,2,3] >;
A4⋊C4 in GAP, Magma, Sage, TeX
A_4\rtimes C_4
% in TeX
G:=Group("A4:C4"); // GroupNames label
G:=SmallGroup(48,30);
// by ID
G=gap.SmallGroup(48,30);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,2,10,122,483,133,304,239]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^3=d^4=1,c*a*c^-1=d*a*d^-1=a*b=b*a,c*b*c^-1=a,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of A4⋊C4 in TeX
Character table of A4⋊C4 in TeX