Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3)×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C27×C32 |
| direct product | C2×H(3,𝔽3)×#120 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C23×C32 |
| commutator, Frattini, cosocle | C22×C3 | C24×C32 |
| LCS | G ▷ C22×C3 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
3641 conjugacy classes of subgroups; only the 735 normal subgroups are shown.
This group has 3641 conjugacy classes of subgroups — too many to draw. Only its 735 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(11,12,14)(13,16,18)(15,19,17);(3,4)(5,7)(6,8)(9,10)(12,15,16)(14,18,19);(4,7)(8,10);(4,8)(7,10)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 20609);