Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊#3688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
| direct product | C22×C3⋊#3689 |
| derived | C3×Q8⋊C9 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | C3⋊((C22⋊C9)⋊C2) |
| commutator, cosocle, hypocenter | C3×Q8⋊C9 | C23 |
| Frattini | C2×C3 | C22×C3⋊S4 |
| Fitting | C22×C32×Q8 | D3 |
| socle | C23×C32 | S4 |
| LCS | G ▷ C3×Q8⋊C9 | |
| UCS | C1 ◁ C23 | |
1112 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 864 |
144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30546);