Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C32 · Schur multiplier (H2) = C2×C32 |
| direct product | C2×H(3,𝔽3)×C32 |
| unambiguous semidirect product | (C2×C32×C32)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C32 | C32 |
| commutator | C3 | C2×C32×C32 |
| Frattini, cosocle | C3×C16 | C22×C32 |
| socle | C22×C3 | C32×C16 |
| prime radical | C22×H(3,𝔽3) | C16 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C32 ◁ G | |
187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2)(35,36,38)(37,40,42)(39,43,41);(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)(36,39,40)(38,42,43)704 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2172);