Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#88955 |
| Sylow | C34 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D33 |
| commutator, cosocle | C2×C33 | C23×C3 |
| Frattini | C2 | C3×D33 |
| Fitting, socle | C2×C34 | C23 |
| hypocenter | C33 | C2×C3×D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2×C3 | |
753 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 753 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
126 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#11724440, #72220344, #72220348, … |
| maximal normal subgroup of | C3×#11724440, #72220344, #72220348, … |
| maximal quotient of | #72228864, #72259526 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3194);