Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×#10596399 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C3 | A4⋊S4 |
| commutator, cosocle, hypocenter | Ω+(4,𝔽3) | C2×C3 |
| Frattini | C2 | C3×A4⋊S4 |
| Fitting | C3×H(5,𝔽2) | C3⋊D3 |
| prime radical | C3×Ω+(4,𝔽3) | C2 |
| LCS | G ▷ Ω+(4,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
262 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | #72242719, C3×#11923520, C3×#11923518 |
| maximal normal subgroup of | C3×#11923520, C3×#11923518 |
| maximal quotient of | #72242714, #72245023 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46398);