Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C8 · Schur multiplier (H2) = C2×C32 |
| direct product | H(3,𝔽3)×#334 |
| unambiguous semidirect product | (C32×#334)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C32×D4 |
| commutator | C3×C4 | C2×C32×C8 |
| Frattini, cosocle | C3×C42 | C22×C32 |
| socle | C22×C3 | C32×C4⋊C4 |
| prime radical | C22×H(3,𝔽3) | C4⋊C4 |
| LCS | G ▷ C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C42×H(3,𝔽3) ◁ G | |
297 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 297 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) = 25 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,4)(3,6,8)(5,9,7)(11,14)(12,16)(15,17)(18,19,20,22,21,23,24,25);(2,5,6)(4,8,9)(10,11,12,14,13,15,16,17)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2067);