Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of C3⋊((C22⋊C9)⋊C2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×C3×C3⋊((C22⋊C9)⋊C2) |
| unambiguous semidirect product | (C2×C3)⋊(C3×(C22⋊C9)⋊C2) |
| derived | C3×C22⋊C9 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C3⋊((C22⋊C9)⋊C2) |
| commutator, cosocle, hypocenter | C3×C22⋊C9 | C22×C3 |
| Frattini | C3 | C2×C3×C3⋊S4 |
| Fitting, socle | C23×C33 | D3 |
| LCS | G ▷ C3×C22⋊C9 | |
| UCS | C1 ◁ C2×C3 | |
448 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 448 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
126 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#11722396 |
| maximal normal subgroup of | C3×#11722396 |
| maximal quotient of | C3×#11722396 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3000);