Finite Group DatabasebetaFinite Group Databasebeta| order | 21756 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×#24925918 |
| Fitting | C2×C37×C49 ⋊ C2×C3 |
| derived, Hall | C37×C49 ⋊ C22×C3 |
| Hall | C2×C37×D49 ⋊ C3 |
| Hall | #19129581 ⋊ C22 |
| Hall | C2×D49 ⋊ C37⋊C3 |
| Sylow | C49 ⋊ C22×C37⋊C3 |
| Sylow | C37 ⋊ C2×D49⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #24925918 |
| commutator, cosocle, hypocenter | C37×C49 | C22×C3 |
| Frattini | C7 | C2×#13095535 |
| Fitting | C2×C37×C49 | C2×C3 |
| socle | C2×C7×C37 | AGL(1,𝔽7) |
| LCS | G ▷ C37×C49 | |
| UCS | C1 ◁ C2 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 3626 |
652 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21756, 42);