Finite Group DatabasebetaFinite Group Databasebeta| order | 780 |
| isoclinism | isoclinic class of #63121 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C3×#63121 |
| Fitting, socle, Hall | C3×C5×C13 ⋊ C4 |
| derived, Hall | C5×C13 ⋊ C3×C4 |
| Hall | C3×C13 ⋊ AGL(1,𝔽5) |
| Hall | C3×C5 ⋊ #261 |
| Sylow | C13 ⋊ C3×AGL(1,𝔽5) |
| Sylow | C5 ⋊ C3×#261 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #63121 |
| commutator, hypocenter | C5×C13 | C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C3×C5×C13 | C4 |
| prime radical | C3×D65 | C2 |
| cosocle | D65 | C2×C3 |
| LCS | G ▷ C5×C13 | |
| UCS | C1 ◁ C3 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 195 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3)(5,6,7,9)(8,11,12,16)(10,14,15,13)(18,19,20,21);(4,5,8,11,15,9,13,14,6,10,16,12,7)(17,18,19,21,20)60 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D5⋊#261, D3×#63121, #63121⋊D3, … |
| maximal normal subgroup of | C3×D5⋊#261, D3×#63121, #63121⋊D3, … |
| maximal quotient of | C3×#10587065, C2×C3×#63121, C9×#63121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(780, 27);