Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C64 · Schur multiplier (H2) = C32 |
| direct product | H(3,𝔽3)×C64 |
| unambiguous semidirect product | (C32×C64)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C64 | C32 |
| commutator | C3 | C32×C64 |
| Frattini, cosocle | C3×C32 | C2×C32 |
| socle | C2×C3 | C32×C32 |
| prime radical | C2×H(3,𝔽3) | C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C64 ◁ G | |
77 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 73 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(2,5,6)(4,8,9)704 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 187);