Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C27 · Schur multiplier (H2) = C1 |
| direct product | C8×Q8⋊C27 |
| unambiguous semidirect product | (C8×Q8)⋊C27 |
| derived | Q8 ⋊ C8×C27 |
| Sylow | C8×Q8 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8×C9 | A4 |
| commutator, hypocenter | Q8 | C8×C27 |
| Frattini | C2×C4×C9 | C2×A4 |
| Fitting | C8×Q8×C9 | C3 |
| socle, prime radical | C22×C3 | C4×C22⋊C9 |
| cosocle | C4×Q8×C9 | C2×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C8×C9 | |
96 conjugacy classes of subgroups.
| 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, 209);