Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #272 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C23×C3×#272 |
| Fitting | C23×C3×H(3,𝔽3) ⋊ C2 |
| derived | H(3,𝔽3) ⋊ C24×C3 |
| Sylow | C3×H(3,𝔽3) ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23×C32 | C3⋊D3 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C24×C3 |
| Frattini | C3 | C23×C3×C3⋊D3 |
| Fitting | C23×C3×H(3,𝔽3) | C2 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C23×C32 | |
2336 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
240 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×D4×#272 |
| maximal normal subgroup of | C2×C3×D4×#272 |
| maximal quotient of | C23×C3×#4183, C2×C3×D4×#272 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3071);