Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊(C3⋊#1020) |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C24×C33 |
| direct product | C2×C3⋊(C3⋊#1020) |
| unambiguous semidirect product | (C2×C3)⋊(C3⋊#1020) |
| Fitting, socle | C25×C32 ⋊ D3 |
| derived | C32×C22⋊A4 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊(C3⋊#1020) |
| commutator, cosocle, hypocenter | C32×C22⋊A4 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C25×C32 | D3 |
| LCS | G ▷ C32×C22⋊A4 | |
| UCS | C1 ◁ C2 | |
1860 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 1860 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C3⋊(C3⋊#1020), C2×C3⋊D3×#1020, C2×#13166068 |
| maximal normal subgroup of | C2×C3⋊D3×#1020, C2×#13166068 |
| maximal quotient of | D3×C3⋊(C3⋊#1020), C2×C3⋊(C3⋊#5935) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47921);