Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C32 |
| direct product | C2×#11722284 |
| Sylow | C2×H(5,𝔽2) ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C3 | A42 |
| commutator, cosocle | C3×H(5,𝔽2) | C2×C32 |
| Frattini | C2×C3 | C2×A42 |
| Fitting | C2×C3×H(5,𝔽2) | C32 |
| hypocenter | H(5,𝔽2) | C2×H(3,𝔽3) |
| LCS | G ▷ C3×H(5,𝔽2) ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C22×C3 | |
357 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
86 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72220368, C2×#13164078, #72252740 |
| maximal normal subgroup of | C2×#72220368, C2×#13164078, #72252740 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30348);