Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of H(3,𝔽3)×#465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C4 |
| direct product | 3−1+2×#498 |
| unambiguous semidirect product | (C9×#498)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×C32×D4 |
| commutator, Frattini, cosocle | C2×C3×C4 | C23×C32 |
| prime radical | C22×C32 | C2×C3×D4 |
| LCS | G ▷ C2×C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C42×3−1+2 ◁ G | |
392 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 231 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 — 3 generators on 25 points:
(1,2,4,3,6,8,7,5,9)(10,11,13,15)(12,16,17,14)(19,22)(20,24)(23,25);(2,5,6)(4,8,9)(10,12,13,17)(11,14,15,16);(18,19,20,22,21,23,24,25)242 conjugacy classes — view the full character table.
| Fitting subgroup of | AGL(1,ℤ/9)×#498, #72245390, #72252816, … |
| maximal subgroup of | AGL(1,ℤ/9)×#498, 3−1+2×#1715, #72245390, … |
| maximal normal subgroup of | AGL(1,ℤ/9)×#498, 3−1+2×#1715, #72245390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11317);