Aliases: C4.A4, Q8.C6, C4○SL2(𝔽3), SL2(𝔽3)⋊2C2, C4○D4⋊C3, C2.3(C2×A4), SmallGroup(48,33)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C4.A4 |
Generators and relations for C4.A4
G = < a,b,c,d | a4=d3=1, b2=c2=a2, ab=ba, ac=ca, ad=da, cbc-1=a2b, dbd-1=a2bc, dcd-1=b >
Character table of C4.A4
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 6 | 4 | 4 | 1 | 1 | 6 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | -1 | ζ32 | ζ3 | -1 | -1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | -1 | ζ3 | ζ32 | -1 | -1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ7 | 2 | -2 | 0 | -1 | -1 | 2i | -2i | 0 | 1 | 1 | i | i | -i | -i | complex faithful |
| ρ8 | 2 | -2 | 0 | -1 | -1 | -2i | 2i | 0 | 1 | 1 | -i | -i | i | i | complex faithful |
| ρ9 | 2 | -2 | 0 | ζ65 | ζ6 | -2i | 2i | 0 | ζ32 | ζ3 | ζ43ζ3 | ζ43ζ32 | ζ4ζ3 | ζ4ζ32 | complex faithful |
| ρ10 | 2 | -2 | 0 | ζ65 | ζ6 | 2i | -2i | 0 | ζ32 | ζ3 | ζ4ζ3 | ζ4ζ32 | ζ43ζ3 | ζ43ζ32 | complex faithful |
| ρ11 | 2 | -2 | 0 | ζ6 | ζ65 | 2i | -2i | 0 | ζ3 | ζ32 | ζ4ζ32 | ζ4ζ3 | ζ43ζ32 | ζ43ζ3 | complex faithful |
| ρ12 | 2 | -2 | 0 | ζ6 | ζ65 | -2i | 2i | 0 | ζ3 | ζ32 | ζ43ζ32 | ζ43ζ3 | ζ4ζ32 | ζ4ζ3 | complex faithful |
| ρ13 | 3 | 3 | 1 | 0 | 0 | -3 | -3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ14 | 3 | 3 | -1 | 0 | 0 | 3 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
►On 16 points - transitive group 16T60
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 8 3 6)(2 5 4 7)(9 13 11 15)(10 14 12 16)
(1 15 3 13)(2 16 4 14)(5 12 7 10)(6 9 8 11)
(5 16 10)(6 13 11)(7 14 12)(8 15 9)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8,3,6)(2,5,4,7)(9,13,11,15)(10,14,12,16), (1,15,3,13)(2,16,4,14)(5,12,7,10)(6,9,8,11), (5,16,10)(6,13,11)(7,14,12)(8,15,9)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8,3,6)(2,5,4,7)(9,13,11,15)(10,14,12,16), (1,15,3,13)(2,16,4,14)(5,12,7,10)(6,9,8,11), (5,16,10)(6,13,11)(7,14,12)(8,15,9) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,8,3,6),(2,5,4,7),(9,13,11,15),(10,14,12,16)], [(1,15,3,13),(2,16,4,14),(5,12,7,10),(6,9,8,11)], [(5,16,10),(6,13,11),(7,14,12),(8,15,9)]])
G:=TransitiveGroup(16,60);
►On 24 points - transitive group 24T21
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 3 2)(5 23 7 21)(6 24 8 22)(9 10 11 12)(13 17 15 19)(14 18 16 20)
(1 12 3 10)(2 9 4 11)(5 6 7 8)(13 18 15 20)(14 19 16 17)(21 24 23 22)
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 18 10)(6 19 11)(7 20 12)(8 17 9)
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), (1,4,3,2)(5,23,7,21)(6,24,8,22)(9,10,11,12)(13,17,15,19)(14,18,16,20), (1,12,3,10)(2,9,4,11)(5,6,7,8)(13,18,15,20)(14,19,16,17)(21,24,23,22), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,18,10)(6,19,11)(7,20,12)(8,17,9)>;
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), (1,4,3,2)(5,23,7,21)(6,24,8,22)(9,10,11,12)(13,17,15,19)(14,18,16,20), (1,12,3,10)(2,9,4,11)(5,6,7,8)(13,18,15,20)(14,19,16,17)(21,24,23,22), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,18,10)(6,19,11)(7,20,12)(8,17,9) );
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)], [(1,4,3,2),(5,23,7,21),(6,24,8,22),(9,10,11,12),(13,17,15,19),(14,18,16,20)], [(1,12,3,10),(2,9,4,11),(5,6,7,8),(13,18,15,20),(14,19,16,17),(21,24,23,22)], [(1,24,16),(2,21,13),(3,22,14),(4,23,15),(5,18,10),(6,19,11),(7,20,12),(8,17,9)]])
G:=TransitiveGroup(24,21);
C4.A4 is a maximal subgroup of
U2(𝔽3) C8.A4 C4.S4 C4.6S4 C4.3S4 Q8.A4 D4.A4 Dic3.A4 C4○D4⋊A4 2+ 1+4.3C6 C4.A5 Dic5.A4 Dic7.2A4 Q8.F7 C28.A4
C4.A4 is a maximal quotient of
C4×SL2(𝔽3) Q8.C18 Dic3.A4 C42⋊4C4⋊C3 C23⋊2D4⋊C3 (C22×C4).A4 C23.19(C2×A4) C4○D4⋊A4 Dic5.A4 Dic7.2A4 Q8.F7 C28.A4
Matrix representation of C4.A4 ►in GL2(𝔽5) generated by
| 2 | 0 |
| 0 | 2 |
| 3 | 3 |
| 0 | 2 |
| 2 | 0 |
| 1 | 3 |
| 0 | 4 |
| 1 | 4 |
G:=sub<GL(2,GF(5))| [2,0,0,2],[3,0,3,2],[2,1,0,3],[0,1,4,4] >;
C4.A4 in GAP, Magma, Sage, TeX
C_4.A_4
% in TeX
G:=Group("C4.A4"); // GroupNames label
G:=SmallGroup(48,33);
// by ID
G=gap.SmallGroup(48,33);
# by ID
G:=PCGroup([5,-2,-3,-2,2,-2,120,97,72,188,133,58]);
// Polycyclic
G:=Group<a,b,c,d|a^4=d^3=1,b^2=c^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b*c,d*c*d^-1=b>;
// generators/relations
Export
Subgroup lattice of C4.A4 in TeX
Character table of C4.A4 in TeX