Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #11720106 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | C2×#11720106 |
| derived | C42⋊C27 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11720106 |
| commutator, cosocle, hypocenter | C42⋊C27 | C22 |
| Frattini | C22×C9 | C2×S4 |
| Fitting | C2×C42×C9 | D3 |
| socle | C23×C3 | (C22⋊C9)⋊C2 |
| LCS | G ▷ C42⋊C27 | |
| UCS | C1 ◁ C2 | |
222 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 23 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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2186);