Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C34 |
| direct product | H(3,𝔽3)×#41⋊C3 |
| Fitting | #41×H(3,𝔽3) ⋊ C3 |
| Sylow | #41 ⋊ C3×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C32×A4 |
| commutator, cosocle | C3×Q8 | C2×C33 |
| Frattini, socle | C2×C3 | C2×C32×A4 |
| Fitting | #41×H(3,𝔽3) | C3 |
| hypercenter | C4×H(3,𝔽3) | A4 |
| hypocenter | Q8 | C2×C3×H(3,𝔽3) |
| LCS | G ▷ C3×Q8 ▷ Q8 | |
| UCS | C1 ◁ C3×C4 ◁ C4×H(3,𝔽3) | |
335 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
154 conjugacy classes — view the full character table.
| maximal subgroup of | #72242713, H(3,𝔽3)×2−1+4⋊C3 |
| maximal normal subgroup of | H(3,𝔽3)×2−1+4⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2527);