Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of ((C22⋊C27)⋊C2)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| derived | C22⋊C27 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ((C22⋊C27)⋊C2)⋊C3 |
| commutator, cosocle, hypocenter | C22⋊C27 | C22×C3 |
| Frattini | C9 | C2×C3×S4 |
| Fitting | C23×C3×C9 | D3 |
| socle | C23×C3 | C3×D9 |
| LCS | G ▷ C22⋊C27 | |
| UCS | C1 ◁ C2 | |
174 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1817);