Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #88614 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C4×D9⋊SL(2,𝔽3) |
| unambiguous semidirect product | D9⋊(C4×SL(2,𝔽3)) |
| unambiguous semidirect product | (C4×D9)⋊SL(2,𝔽3) |
| Fitting | C4×Q8×C9 ⋊ C2×C3 |
| derived | Q8×C9 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D9⋊A4 |
| commutator, hypocenter | Q8×C9 | C2×C3×C4 |
| Frattini, socle | C22×C3 | C2×D3×A4 |
| Fitting | C4×Q8×C9 | C2×C3 |
| prime radical | C2×D9⋊SL(2,𝔽3) | C2 |
| cosocle | C2×Q8×C9 | C22×C3 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C2×C4 | |
335 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 335 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13168);