Finite Group DatabasebetaFinite Group Databasebeta| order | 21756 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×ΓL(1,𝔽8)×#3866 |
| Fitting, socle, Hall | C72×C37 ⋊ C3×C4 |
| derived | C7×C37 ⋊ C3×C4×C7 |
| Hall | C72×#3866 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8)×C37 ⋊ C4 |
| Sylow | C72 ⋊ C3×#3866 |
| Sylow | C37 ⋊ C4×C7×ΓL(1,𝔽8) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | ΓL(1,𝔽8)×#3866 |
| commutator, hypocenter | C7×C37 | C3×C4×C7 |
| Frattini | C1 | G |
| Fitting, socle | C72×C37 | C3×C4 |
| prime radical | C7×ΓL(1,𝔽8)×D37 | C2 |
| cosocle | C7×D37 | C2×C3×C7 |
| LCS | G ▷ C7×C37 | |
| UCS | C1 ◁ C7 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 777 |
455 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21756, 177);