Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊#157 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C3⋊#63540 |
| unambiguous semidirect product | (C2×C3)⋊#63540 |
| derived, Sylow | C33 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | C3⋊#157 |
| commutator, hypocenter | C33 | C2×C32 |
| Frattini | C8 | C2×C3⋊#157 |
| Fitting | C2×C33×C8 | C4 |
| socle, prime radical | C22×C33 | C16 |
| cosocle | C3×C3⋊(C3⋊C16) | C22 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2×C8 | |
168 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 23245);