Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊(#1020⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×C3⋊(#1020⋊C3) |
| unambiguous semidirect product | (C2×C3)⋊(#1020⋊C3) |
| Fitting, socle | C25×C3 ⋊ C3×D3 |
| derived | C3×C22⋊A4 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊(#1020⋊C3) |
| commutator, cosocle, hypocenter | C3×C22⋊A4 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C25×C3 | C3×D3 |
| LCS | G ▷ C3×C22⋊A4 | |
| UCS | C1 ◁ C2 | |
422 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13181792, C2×D3×#1020⋊C3, C2×(#1020⋊C3)⋊D3 |
| maximal normal subgroup of | C2×#13181792, C2×D3×#1020⋊C3, C2×(#1020⋊C3)⋊D3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47878);