Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊(#120⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C3×C3⋊(#120⋊C3) |
| unambiguous semidirect product | (C2×C3)⋊(C3×#120⋊C3) |
| unambiguous semidirect product | (C2×#953)⋊C32 |
| unambiguous semidirect product | (C2×C3×#953)⋊C3 |
| Fitting | C25×C32 ⋊ C2×C3 |
| derived | C24×C3 ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C3⋊(#120⋊C3) |
| commutator, cosocle, hypocenter | C24×C3 | C22×C32 |
| Frattini | C22 | C2×C3×D3×A4 |
| Fitting | C25×C32 | C2×C3 |
| socle | C23×C32 | C2×A4 |
| LCS | G ▷ C24×C3 | |
| UCS | C1 ◁ C2×C3 | |
876 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 876 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×C3⋊(#120⋊C3), C2×#10595527⋊C3, C2×C3×#10593244, … |
| maximal normal subgroup of | C2×#10595527⋊C3, C2×C3×#10593244, C2×#13166068 |
| maximal quotient of | C3×D3×C3⋊(#120⋊C3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 39200);