Finite Group DatabasebetaFinite Group Databasebeta| order | 6780 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#14555933 |
| Fitting, socle, derived, Hall | C3×C5×C113 ⋊ C4 |
| Hall | C5×C113 ⋊ Dic3 |
| Hall | C3×C113 ⋊ AGL(1,𝔽5) |
| Sylow | C113 ⋊ ΓL(1,𝔽16) |
| Hall | C3×C5 ⋊ #90673 |
| Sylow | C5 ⋊ C3⋊#90673 |
| Sylow | C3 ⋊ #14555933 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C5×C113 | C4 |
| LCS | G ▷ C3×C5×C113 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,3,5,113|a^2=b,c^a=c^2,d^a=d^2,e^a=e^15,d^b=d^4,e^b=e^112429 conjugacy classes — view the full character table.
| maximal subgroup of | C9⋊#14555933, C3×C3⋊#14555933, C3⋊(C3⋊#14555933), … |
| maximal normal subgroup of | C3×C3⋊#14555933, C5×C3⋊#14555933 |
| maximal quotient of | C9⋊#14555933, C3×C3⋊#14555933, C3⋊(C3⋊#14555933), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6780, 20);