Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C16 · Schur multiplier (H2) = C22 |
| direct product | 3−1+2×C22⋊C16 |
| unambiguous semidirect product | (C9×C22⋊C16)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C3×C8 | C22×C32 |
| commutator | C2×C3 | C2×C32×C16 |
| socle | C22×C3 | C2×C32×C8 |
| prime radical | C23×C32 | C3×C8 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ G | |
264 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,4,3,6,8,7,5,9)(10,11)(12,13)(14,15,16,19,17,20,22,25,18,21,23,26,24,27,28,29);(2,5,6)(4,8,9)(11,13)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2110);