direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×M4(2), C8⋊3C6, C4.C12, C24⋊7C2, C12.4C4, C22.C12, C12.22C22, (C2×C6).1C4, (C2×C4).2C6, C4.6(C2×C6), C6.12(C2×C4), (C2×C12).8C2, C2.3(C2×C12), SmallGroup(48,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×M4(2)
G = < a,b,c | a3=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Character table of C3×M4(2)
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | -1 | 1 | 1 | -1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ6 | ζ65 | ζ3 | ζ3 | ζ32 | ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | -1 | -1 | -1 | -1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | -1 | -1 | -1 | -1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ10 | 1 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | 1 | -1 | -1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 6 |
| ρ11 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | 1 | -1 | -1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 6 |
| ρ12 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | -1 | 1 | 1 | -1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ65 | ζ6 | ζ32 | ζ32 | ζ3 | ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ13 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | i | -i | i | -i | -1 | -1 | -1 | -1 | -1 | -1 | -i | i | -i | i | -i | i | -i | i | linear of order 4 |
| ρ14 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | i | i | -i | -i | -1 | -1 | -1 | -1 | 1 | 1 | i | -i | i | -i | -i | i | -i | i | linear of order 4 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | i | -i | i | -1 | -1 | -1 | -1 | -1 | -1 | i | -i | i | -i | i | -i | i | -i | linear of order 4 |
| ρ16 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | -i | i | i | -1 | -1 | -1 | -1 | 1 | 1 | -i | i | -i | i | i | -i | i | -i | linear of order 4 |
| ρ17 | 1 | 1 | -1 | ζ32 | ζ3 | -1 | -1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | -i | -i | i | i | ζ65 | ζ65 | ζ6 | ζ6 | ζ3 | ζ32 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | linear of order 12 |
| ρ18 | 1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | -i | i | -i | i | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | ζ6 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | linear of order 12 |
| ρ19 | 1 | 1 | -1 | ζ3 | ζ32 | -1 | -1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | -i | -i | i | i | ζ6 | ζ6 | ζ65 | ζ65 | ζ32 | ζ3 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | linear of order 12 |
| ρ20 | 1 | 1 | -1 | ζ32 | ζ3 | -1 | -1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | i | i | -i | -i | ζ65 | ζ65 | ζ6 | ζ6 | ζ3 | ζ32 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | linear of order 12 |
| ρ21 | 1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | i | -i | i | -i | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | ζ6 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | linear of order 12 |
| ρ22 | 1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | i | -i | i | -i | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | ζ65 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | linear of order 12 |
| ρ23 | 1 | 1 | -1 | ζ3 | ζ32 | -1 | -1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | i | i | -i | -i | ζ6 | ζ6 | ζ65 | ζ65 | ζ32 | ζ3 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | linear of order 12 |
| ρ24 | 1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | -i | i | -i | i | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | ζ65 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | linear of order 12 |
| ρ25 | 2 | -2 | 0 | 2 | 2 | -2i | 2i | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ26 | 2 | -2 | 0 | 2 | 2 | 2i | -2i | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ27 | 2 | -2 | 0 | -1-√-3 | -1+√-3 | 2i | -2i | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ4ζ3 | 2ζ43ζ3 | 2ζ4ζ32 | 2ζ43ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ28 | 2 | -2 | 0 | -1+√-3 | -1-√-3 | -2i | 2i | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ43ζ32 | 2ζ4ζ32 | 2ζ43ζ3 | 2ζ4ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ29 | 2 | -2 | 0 | -1+√-3 | -1-√-3 | 2i | -2i | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ4ζ32 | 2ζ43ζ32 | 2ζ4ζ3 | 2ζ43ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ30 | 2 | -2 | 0 | -1-√-3 | -1+√-3 | -2i | 2i | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ43ζ3 | 2ζ4ζ3 | 2ζ43ζ32 | 2ζ4ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T16
Generators in S24
(1 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 17)(8 9 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)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)
G:=sub<Sym(24)| (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,17)(8,9,18), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)>;
G:=Group( (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,17)(8,9,18), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24) );
G=PermutationGroup([[(1,10,19),(2,11,20),(3,12,21),(4,13,22),(5,14,23),(6,15,24),(7,16,17),(8,9,18)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24)]])
G:=TransitiveGroup(24,16);
C3×M4(2) is a maximal subgroup of
C12.53D4 C12.46D4 C12.47D4 D12⋊C4 D12.C4 C8⋊D6 C8.D6 M4(2).A4 C8⋊F7 C28.C12 He3⋊1M4(2) He3⋊4M4(2)
C3×M4(2) is a maximal quotient of
C8⋊F7 C28.C12
Matrix representation of C3×M4(2) ►in GL2(𝔽13) generated by
| 3 | 0 |
| 0 | 3 |
| 0 | 6 |
| 3 | 0 |
| 12 | 0 |
| 0 | 1 |
G:=sub<GL(2,GF(13))| [3,0,0,3],[0,3,6,0],[12,0,0,1] >;
C3×M4(2) in GAP, Magma, Sage, TeX
C_3\times M_4(2)
% in TeX
G:=Group("C3xM4(2)"); // GroupNames label
G:=SmallGroup(48,24);
// by ID
G=gap.SmallGroup(48,24);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-2,60,261,58]);
// Polycyclic
G:=Group<a,b,c|a^3=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations
Export
Subgroup lattice of C3×M4(2) in TeX
Character table of C3×M4(2) in TeX