Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #6258 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C23×#6258 |
| Fitting | C25×C32 ⋊ D3 |
| derived | C22⋊H(3,𝔽3) ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | C3⋊S4 |
| commutator, cosocle, hypocenter | C22⋊H(3,𝔽3) | C24 |
| Frattini | C3 | C23×C3⋊S4 |
| Fitting | C25×C32 | D3 |
| socle | C25×C3 | C3⋊D3 |
| LCS | G ▷ C22⋊H(3,𝔽3) | |
| UCS | C1 ◁ C23×C3 | |
3320 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 3320 conjugacy classes of subgroups — too many to draw. Only its 211 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
152 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#11722398 |
| maximal normal subgroup of | C22×#11722398 |
| maximal quotient of | C22×#11722398 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46147);