Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #1088 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×#88626 |
| derived, Sylow | H(3,𝔽3) ⋊ C2×C4×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C42 | #157 |
| commutator, hypocenter | H(3,𝔽3) | C2×C4×C8 |
| Frattini, socle | C22×C3 | C22×#157 |
| Fitting | C42×H(3,𝔽3) | C4 |
| prime radical | C22×#272 | C2×C4 |
| cosocle | C2×#1084 | C23 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3×C42 | |
459 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 459 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13371);