Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of (C3×A4)⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C23×C32 |
| direct product | C22×(C3×A4)⋊A4 |
| Sylow | C26 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | A42 |
| commutator, cosocle | C24×C3 | C22×C32 |
| Frattini | C3 | C22×A42 |
| Fitting, socle | C26×C3 | C32 |
| hypocenter | C24 | C22×H(3,𝔽3) |
| LCS | G ▷ C24×C3 ▷ C24 | |
| UCS | C1 ◁ C22×C3 | |
1302 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 1302 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
128 conjugacy classes — view the full character table.
| maximal subgroup of | C22×(C22⋊A4)⋊H(3,𝔽3), C22×#11724029 |
| maximal normal subgroup of | C22×(C22⋊A4)⋊H(3,𝔽3), C22×#11724029 |
| maximal quotient of | #72252740 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46123);