Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊AGL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C4 |
| direct product | C4×C3⋊#3771 |
| derived, Sylow | C33 ⋊ C4×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C3⋊AGL(1,𝔽9) |
| commutator, hypocenter | C33 | C4×C16 |
| Frattini | C22 | C2×C3⋊AGL(1,𝔽9) |
| Fitting | C2×C33×C4 | C8 |
| socle, prime radical | C22×C33 | C2×C8 |
| cosocle | C2×C3×#622 | C22 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2×C4 | |
162 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32278);