Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊#64188 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C24×C3 |
| direct product | C2×C3⋊#64188 |
| unambiguous semidirect product | (C2×C3)⋊#64188 |
| derived | C3×C22⋊(C22⋊C9) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊#64188 |
| commutator, cosocle, hypocenter | C3×C22⋊(C22⋊C9) | C22 |
| Frattini | C3 | C2×C3⋊#1020 |
| Fitting, socle | C25×C32 | D3 |
| LCS | G ▷ C3×C22⋊(C22⋊C9) | |
| UCS | C1 ◁ C2 | |
1374 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 1374 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72220146, C2×#72220148, C22×C3⋊#64188 |
| maximal normal subgroup of | C2×#72220148, C22×C3⋊#64188 |
| maximal quotient of | C22×C3⋊#64188, C2×C3⋊#10593110 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46153);