Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C33 |
| direct product | #67×C3⋊(C3⋊D3) |
| Fitting | C22×C34 ⋊ C22 |
| Sylow | C34 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C2×C34 | C23 |
| Fitting | C22×C34 | C22 |
| hypercenter | C22 | D3×C3⋊(C3⋊D3) |
| hypocenter | C34 | C2×D4 |
| LCS | G ▷ C2×C34 ▷ C34 | |
| UCS | C1 ◁ C2 ◁ C22 | |
2136 conjugacy classes of subgroups; only the 270 normal subgroups are shown.
This group has 2136 conjugacy classes of subgroups — too many to draw. Only its 270 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
135 conjugacy classes — view the full character table.
| maximal subgroup of | #72259876, #67×C3⋊((C3⋊D3)⋊C3) |
| maximal normal subgroup of | #67×C3⋊((C3⋊D3)⋊C3) |
| maximal quotient of | #67×C3⋊#272, #72259972 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3401);