Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C22×C32×#226 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23×C32 | S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C23×C32 |
| Frattini | C2 | C22×C32×S4 |
| Fitting | C22×C32×Q8 | D3 |
| prime radical | C22×C32×SL(2,𝔽3) | C2 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C23×C32 | |
1086 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 222 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, 47420);