Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4×C8 · Schur multiplier (H2) = C23×C32 |
| direct product | H(3,𝔽3)×#336 |
| unambiguous semidirect product | (C32×#336)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C3×C4 | C22×C32 |
| commutator | C2×C3 | C32×C4×C8 |
| socle | C23×C3 | C2×C32×C4 |
| prime radical | C23×H(3,𝔽3) | C2×C4 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
649 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,4)(3,6,8)(5,9,7)(10,11)(12,13)(18,19,20,22,21,23,24,25);(2,5,6)(4,8,9)(11,13)(14,15,16,17)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2073);