Finite Group DatabasebetaFinite Group Databasebeta| order | 6708 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C43⋊C3)⋊Dic13 |
| derived, Hall | C13×C43 ⋊ C3×C4 |
| Hall | Dic559 ⋊ C3 |
| Hall | C13×C43⋊C3 ⋊ C4 |
| Hall | C43⋊C3 ⋊ Dic13 |
| Sylow | C43 ⋊ C3×Dic13 |
| Sylow | C13 ⋊ (C43⋊C3)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | (C43⋊C3)⋊D13 |
| commutator, hypocenter | C13×C43 | C3×C4 |
| Fitting, socle | C2×C13×C43 | C2×C3 |
| LCS | G ▷ C13×C43 | |
| UCS | C1 ◁ C2 | |
24 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,13,43,2|a^2=e,c^a=c^12,d^a=d^42,d^b=d^36230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6708, 21);