Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C27 · Schur multiplier (H2) = C2 |
| direct product | C2×C4×Q8⋊C27 |
| unambiguous semidirect product | (C2×C4×Q8)⋊C27 |
| derived | Q8 ⋊ C2×C4×C27 |
| Sylow | C2×C4×Q8 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4×C9 | A4 |
| commutator, hypocenter | Q8 | C2×C4×C27 |
| Frattini | C22×C9 | C22×A4 |
| Fitting | C2×C4×Q8×C9 | C3 |
| socle, prime radical | C23×C3 | C2×C22⋊C9 |
| cosocle | C2×Q8×C9 | C22×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C4×C9 | |
254 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 113 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, 2238);