Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊(C3⋊S4) |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C33×C4 |
| direct product | C4×C3⋊(C3⋊SL(2,ℤ/4)) |
| derived | C32×A4 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C3⋊(C3⋊S4) |
| commutator, hypocenter | C32×A4 | C42 |
| Frattini | C22 | C2×C3⋊(C3⋊S4) |
| Fitting | C23×C32×C4 | D3 |
| socle | C24×C32 | C2×D3 |
| prime radical, cosocle | C22×C32×A4 | C22 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 ◁ C2×C4 | |
1230 conjugacy classes of subgroups; only the 279 normal subgroups are shown.
This group has 1230 conjugacy classes of subgroups — too many to draw. Only its 279 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 39123);