Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | SL(2,𝔽3)×#272 |
| Fitting, derived | Q8×H(3,𝔽3) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2×C3 | A4×C3⋊D3 |
| commutator, Fitting, cosocle, hypocenter | Q8×H(3,𝔽3) | C2×C3 |
| LCS | G ▷ Q8×H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
304 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
70 conjugacy classes — view the full character table.
| maximal subgroup of | SL(2,𝔽3)×((C3⋊D3)⋊C3)⋊C2, GL(2,𝔽3)×#272, #72244580 |
| maximal normal subgroup of | SL(2,𝔽3)×((C3⋊D3)⋊C3)⋊C2, GL(2,𝔽3)×#272, #72244580 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2183);