Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C4×C9×#226 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4×C9 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C4×C9 |
| Frattini, socle | C22×C3 | C2×C3×S4 |
| Fitting | C4×Q8×C9 | D3 |
| prime radical, cosocle | C2×C3×SL(2,𝔽3) | C22×C3 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C4×C9 | |
232 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12814);