Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊((C22⋊C9)⋊C2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C3 |
| direct product | C22×C3⋊((C22⋊C9)⋊C4) |
| derived | C3×C22⋊C9 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | C3⋊((C22⋊C9)⋊C2) |
| commutator, hypocenter | C3×C22⋊C9 | C22×C4 |
| Frattini | C2×C3 | C22×C3⋊S4 |
| Fitting, socle, prime radical | C25×C32 | D3 |
| cosocle | C2×C3×C22⋊C9 | C23 |
| LCS | G ▷ C3×C22⋊C9 | |
| UCS | C1 ◁ C23 | |
1554 conjugacy classes of subgroups; only the 267 normal subgroups are shown.
This group has 1554 conjugacy classes of subgroups — too many to draw. Only its 267 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
168 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#11722396, C22×C3⋊(#1094⋊C4) |
| maximal normal subgroup of | C22×#11722396, C22×C3⋊(#1094⋊C4) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30571);