Finite Group DatabasebetaFinite Group Databasebeta| order | 21756 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Hall | C72×C37 ⋊ C3×C4 |
| Hall | C7×C7⋊#3866 ⋊ C3 |
| Hall | #19129589 ⋊ C4 |
| Sylow | C72 ⋊ #3866⋊C3 |
| Sylow | C37 ⋊ #10596873 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C72×C37 | C3×C4 |
| prime radical | #24925975 | C2 |
| cosocle | C72×D37 | C2×C3 |
| LCS | G ▷ C72×C37 | |
| UCS | C1 | |
72 conjugacy classes of subgroups.
| transitive degree | μt(G) = 518 |
Polycyclic presentation:
2,3,2,7^2,37|a^2=c,d^a=d^6,f^a=f^31,d^b=d^4,e^b=e^4,f^b=f^26,f^c=f^36181 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21756, 220);