Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #63747⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×#63747⋊C3 |
| unambiguous semidirect product | (C2×#63747)⋊C3 |
| derived | #63747 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #63747⋊C3 |
| commutator, cosocle, hypocenter | #63747 | C2×C3 |
| Frattini | C22 | C2×PSU(3,𝔽2)⋊C3 |
| Fitting, socle | C23×C32 | SL(2,𝔽3) |
| LCS | G ▷ #63747 | |
| UCS | C1 ◁ C2 | |
197 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 197 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31098);