Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #6258 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×C3×#6258 |
| Fitting | C23×C33 ⋊ D3 |
| derived | C22⋊H(3,𝔽3) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | C3⋊S4 |
| commutator, cosocle, hypocenter | C22⋊H(3,𝔽3) | C22×C3 |
| Frattini | C3 | C2×C3×C3⋊S4 |
| Fitting | C23×C33 | D3 |
| socle | C23×C32 | C3⋊D3 |
| LCS | G ▷ C22⋊H(3,𝔽3) | |
| UCS | C1 ◁ C2×C32 | |
556 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
114 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#11722398, C2×#13049222, #72220621, … |
| maximal normal subgroup of | C3×#11722398, C2×#13049222, #72220621, … |
| maximal quotient of | C3×#11722398 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3002);