Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#89049 |
| Fitting | C22×C34 ⋊ C22 |
| Sylow | C34 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×#1113 |
| commutator, cosocle | C2×C33 | C23×C3 |
| Frattini | C2 | C2×C3×#1113 |
| Fitting | C22×C34 | C22 |
| socle | C2×C34 | C23 |
| hypercenter | C22×C3 | #1113 |
| hypocenter | C33 | C2×C3×D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 | |
922 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 922 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | #72219714, #72219727, #72219728, … |
| maximal normal subgroup of | #72219714, C2×C3×#89049, C3×#11724354, … |
| maximal quotient of | C2×C3×#89049, C3×#11723729, C3×#11723728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3376);