Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of H(3,𝔽3)×#327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C1 |
| direct product | 3−1+2×#330 |
| unambiguous semidirect product | (C9×#330)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C32×C22⋊C4 |
| commutator | C2×C3×C4 | C2×C32×C4 |
| Frattini, cosocle | C3×C42 | C22×C32 |
| prime radical | C22×C32 | C3×C22⋊C4 |
| LCS | G ▷ C2×C3×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C42×3−1+2 ◁ G | |
240 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,4,3,6,8,7,5,9)(11,14,15,16)(18,19,22,24,23,20,21,25);(2,5,6)(4,8,9)(10,11)(12,16)(13,15)(14,17)(18,20,23,19)(21,24,22,25)209 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2056);