Finite Group DatabasebetaFinite Group Databasebeta| order | 21756 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C49⋊C3)⋊#3866 |
| Fitting, derived, Hall | C37×C49 ⋊ C3×C4 |
| Hall | C49⋊#3866 ⋊ C3 |
| Hall | C37×C49⋊C3 ⋊ C4 |
| Hall | C49⋊C3 ⋊ #3866 |
| Sylow | C49 ⋊ C3×#3866 |
| Sylow | C37 ⋊ (C49⋊C3)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C37×C49 | C3×C4 |
| Frattini | C7 | ΓL(1,𝔽8)⋊#3866 |
| socle | C7×C37 | ΓL(1,𝔽8)⋊C4 |
| prime radical | D37×C49⋊C3 | C2 |
| cosocle | C49×D37 | C2×C3 |
| LCS | G ▷ C37×C49 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1813 |
Polycyclic presentation:
2,3,2,7,37,7|a^2=c,d^7=f^6,d^a=d^6f,e^a=e^31,f^a=f^6,d^b=d^2f^3,f^b=f^2,e^c=e^36199 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21756, 31);