Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C9⋊SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C32 |
| direct product | C23×Q8⋊H(3,𝔽3) |
| Fitting | C23×C32×Q8 ⋊ C3 |
| Sylow | C23×Q8 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C3×A4 |
| commutator, cosocle | C3×Q8 | C23×C32 |
| Frattini | C2×C3 | C23×C3×A4 |
| Fitting | C23×C32×Q8 | C3 |
| hypercenter | C24×C32 | A4 |
| hypocenter | Q8 | C23×H(3,𝔽3) |
| LCS | G ▷ C3×Q8 ▷ Q8 | |
| UCS | C1 ◁ C24×C3 ◁ C24×C32 | |
1597 conjugacy classes of subgroups; only the 313 normal subgroups are shown.
This group has 1597 conjugacy classes of subgroups — too many to draw. Only its 313 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
248 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30912);