Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of H(3,𝔽3)×#99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C32 |
| direct product | C2×H(3,𝔽3)×#100 |
| unambiguous semidirect product | (C2×C32×#100)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C32×C22⋊C4 |
| commutator | C22×C3 | C22×C32×C4 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| prime radical | C22×D4×H(3,𝔽3) | C2 |
| LCS | G ▷ C22×C3 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C4×H(3,𝔽3) ◁ G | |
1023 conjugacy classes of subgroups; only the 287 normal subgroups are shown.
This group has 1023 conjugacy classes of subgroups — too many to draw. Only its 287 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 — 3 generators on 19 points:
(1,2)(11,12,14)(13,16,18)(15,19,17);(3,4,5,8,6,7,9,10)(12,15,16)(14,18,19);(4,7)(5,9)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11055);