Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊D3×#120⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | C3⋊D3×#120⋊C3 |
| Fitting | C32×#120 ⋊ C2×C3 |
| derived | C24×C32 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C24×C32 | C22×C3 |
| Frattini | C22 | C2×A4×C3⋊D3 |
| Fitting | C32×#120 | C2×C3 |
| socle | C22×C32 | C22×A4 |
| LCS | G ▷ C24×C32 | |
| UCS | C1 | |
916 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 916 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | #72220666, D32×#120⋊C3, C3⋊D3×#1020⋊C2, … |
| maximal normal subgroup of | D32×#120⋊C3, C3⋊D3×#1020⋊C2, #72220928 |
| maximal quotient of | #72220666 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 39214);