Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #10596782 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C23×C33 |
| direct product | C3×#10596782 |
| Fitting, socle | C26×C3 ⋊ C32 |
| derived, Sylow | C26 ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #10596782 |
| commutator, cosocle, hypocenter | C26 | C33 |
| Frattini | C1 | G |
| Fitting, socle | C26×C3 | C32 |
| LCS | G ▷ C26 | |
| UCS | C1 ◁ C3 | |
1058 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | C3×A43, C3×#72215828, C2×C3×#10596782, … |
| maximal normal subgroup of | C3×A43, C2×C3×#10596782, C3×#1020⋊A4, … |
| maximal quotient of | #72252823, #72252829, C3×#72215828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47906);