Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C16 · Schur multiplier (H2) = C23 |
| direct product | C22×C16×3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C16 | C32 |
| commutator | C3 | C22×C32×C16 |
| Frattini, cosocle | C3×C8 | C23×C32 |
| socle | C23×C3 | C32×C8 |
| prime radical | C23×C32 | C3×C8 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C16 ◁ G | |
392 conjugacy classes of subgroups; only the 343 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 343 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) = 576 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2)(21,22,24,23,26,28,27,25,29);(3,4)(22,25,26)(24,28,29);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)704 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11329);