Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3×C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C3×C4 |
| direct product | C2×C3⋊C8×C3⋊Dic3 |
| derived, Sylow | C33 ⋊ C2×C4×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | D3×C3⋊D3 |
| commutator, hypocenter | C33 | C2×C4×C8 |
| Frattini | C2×C4 | C2×D3×C3⋊D3 |
| Fitting | C22×C33×C4 | C22 |
| socle, prime radical | C23×C33 | C2×C4 |
| cosocle | C2×C33×C4 | C23 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C22×C4 | |
1276 conjugacy classes of subgroups.
| transitive degree | μt(G) = 576 |
288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 26333);