Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊(C3⋊(#126⋊C3)) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×C3⋊(C3⋊(#126⋊C3)) |
| unambiguous semidirect product | (C2×C3)⋊(C3⋊(#126⋊C3)) |
| Fitting | C2×C32×C42 ⋊ C2×C3 |
| derived | C32×C42 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊(C3⋊(#126⋊C3)) |
| commutator, cosocle, hypocenter | C32×C42 | C22×C3 |
| Frattini | C22 | C2×A4×C3⋊D3 |
| Fitting | C2×C32×C42 | C2×C3 |
| socle | C23×C32 | C2×A4 |
| LCS | G ▷ C32×C42 | |
| UCS | C1 ◁ C2 | |
444 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 39203);