Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊((C22⋊C9)⋊C2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C3 |
| direct product | C23×C3⋊((C22⋊C9)⋊C2) |
| derived | C3×C22⋊C9 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | C3⋊((C22⋊C9)⋊C2) |
| commutator, cosocle, hypocenter | C3×C22⋊C9 | C24 |
| Frattini | C3 | C23×C3⋊S4 |
| Fitting, socle | C25×C32 | D3 |
| LCS | G ▷ C3×C22⋊C9 | |
| UCS | C1 ◁ C23 | |
3938 conjugacy classes of subgroups; only the 307 normal subgroups are shown.
This group has 3938 conjugacy classes of subgroups — too many to draw. Only its 307 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
168 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#11722396, C23×C3⋊(((C22⋊C9)⋊C2)⋊C3) |
| maximal normal subgroup of | C22×#11722396, C23×C3⋊(((C22⋊C9)⋊C2)⋊C3) |
| maximal quotient of | C22×#11722396 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46146);