Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C32 |
| direct product | H(3,𝔽3)×#547 |
| unambiguous semidirect product | (C32×#547)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22×C3 | C24×C32 |
| LCS | G ▷ C22×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
1606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4)(3,6,8)(5,9,7)(11,13)(14,15,18,19)(16,17,20,21);(2,5,6)(4,8,9)(15,19)(17,21);(10,11)(12,13)(14,16,18,20)(15,17,19,21);(14,17)(15,20)(16,19)(18,21)275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 20687);