Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | SL(2,𝔽3)×#624 |
| unambiguous semidirect product | (Q8×#624)⋊C3 |
| derived | C32×Q8 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D32×A4 |
| commutator, hypocenter | C32×Q8 | C2×C3×C4 |
| Fitting, cosocle | C2×C32×Q8 | C22×C3 |
| socle | C22×C32 | C22×A4 |
| prime radical | C2×C3⋊D3×SL(2,𝔽3) | C2 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C22 | |
551 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 551 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
126 conjugacy classes — view the full character table.
| maximal subgroup of | #624×PSU(3,𝔽2)⋊C3 |
| maximal quotient of | #624×PSU(3,𝔽2)⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32212);