Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C22⋊(C22⋊A4) |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C29×C33 |
| direct product | C32×C22⋊(C22⋊A4) |
| Fitting, socle | C26×C32 ⋊ C3 |
| derived, Sylow | C26 ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | C22⋊(C22⋊A4) |
| commutator, cosocle, hypocenter | C26 | C33 |
| Frattini | C1 | G |
| Fitting, socle | C26×C32 | C3 |
| LCS | G ▷ C26 | |
| UCS | C1 ◁ C32 | |
6794 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
216 conjugacy classes — view the full character table.
| maximal subgroup of | #72252810, C32×A4×C22⋊A4, #72252829 |
| maximal normal subgroup of | #72252810, C32×A4×C22⋊A4, #72252829 |
| maximal quotient of | C32×C22⋊#997 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47931);