Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8×C9 · Schur multiplier (H2) = C3 |
| direct product | C3×C8×Q8⋊C9 |
| unambiguous semidirect product | (C3×C8×Q8)⋊C9 |
| derived | Q8 ⋊ C3×C8×C9 |
| Sylow | C8×Q8 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32×C8 | A4 |
| commutator, hypocenter | Q8 | C3×C8×C9 |
| Frattini | C2×C3×C4 | C2×C3×A4 |
| Fitting | C32×C8×Q8 | C3 |
| socle, prime radical | C22×C32 | C4×A4 |
| cosocle | C3×C4×Q8 | C2×C32 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C32×C8 | |
222 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 1728 |
504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7316);