Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C9⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C27×C32 |
| direct product | C24×C22⋊H(3,𝔽3) |
| Fitting | C26×C32 ⋊ C3 |
| Sylow | C26 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center | C24×C3 | C3×A4 |
| commutator, cosocle | C22×C3 | C24×C32 |
| Frattini | C3 | C24×C3×A4 |
| Fitting | C26×C32 | C3 |
| socle | C26×C3 | C32 |
| hypercenter | C24×C32 | A4 |
| hypocenter | C22 | C24×H(3,𝔽3) |
| LCS | G ▷ C22×C3 ▷ C22 | |
| UCS | C1 ◁ C24×C3 ◁ C24×C32 | |
6856 conjugacy classes of subgroups; only the 670 normal subgroups are shown.
This group has 6856 conjugacy classes of subgroups — too many to draw. Only its 670 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×(C22⋊A4)⋊H(3,𝔽3), C22×#64269⋊C3, C24×(C3⋊D3)⋊A4, … |
| maximal normal subgroup of | C22×(C22⋊A4)⋊H(3,𝔽3), C22×#64269⋊C3, C24×(C3⋊D3)⋊A4, … |
| maximal quotient of | #72252733 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46170);