Finite Group DatabasebetaFinite Group Databasebeta| order | 21756 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #13095537⋊D7 |
| Fitting, socle, derived, Hall | C72×C37 ⋊ C22×C3 |
| Hall | (C7⋊D7)⋊D37 ⋊ C3 |
| Hall | C7×#11715913 ⋊ C22 |
| Sylow | C72 ⋊ C2×D37⋊C3 |
| Sylow | C37 ⋊ D7×AGL(1,𝔽7) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C37 | C22×C3 |
| LCS | G ▷ C72×C37 | |
| UCS | C1 | |
84 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1554 |
Polycyclic presentation:
2^2,3,7^2,37|d^a=d^6,f^a=f^36,d^b=d^6,e^b=e^6,e^c=e^2,f^c=f^26191 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21756, 195);