Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C33 · Schur multiplier (H2) = C25×C33 |
| direct product | C22×C33×SD16 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C33 | D4 |
| commutator, Frattini, cosocle | C4 | C24×C33 |
| prime radical | C22×C33×D4 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C33 ◁ C22×C33×C4 ◁ G | |
4172 conjugacy classes of subgroups; only the 2492 normal subgroups are shown.
This group has 4172 conjugacy classes of subgroups — too many to draw. Only its 2492 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(13,14,15);(3,4)(16,17,18);(5,6,7,8,9,10,11,12)(19,20,21);(6,8)(7,11)(10,12)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46088);