Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊(C3⋊D3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C33 |
| direct product | C32×C3⋊(C3⋊D3) |
| Fitting | C33×C32 ⋊ C2 |
| derived, Sylow | C33 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | C3⋊(C3⋊D3) |
| commutator, hypocenter | C33 | C2×C32 |
| Frattini | C16 | C2×C3⋊(C3⋊D3) |
| Fitting | C33×C32 | C2 |
| socle | C2×C33 | C2×C16 |
| prime radical | C2×C3⋊(C3⋊D3) | C16 |
| cosocle | C33×C16 | C22 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C32 | |
476 conjugacy classes of subgroups.
| transitive degree | μt(G) = 864 |
480 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12055);