Finite Group DatabasebetaFinite Group Databasebeta| order | 780 |
| isoclinism | isoclinic class of C3⋊#63122 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#63122 |
| Fitting, socle, derived, Hall | C3×C5×C13 ⋊ C4 |
| Hall | C5×C13 ⋊ Dic3 |
| Hall | C3×C13 ⋊ AGL(1,𝔽5) |
| Hall | C3×C5 ⋊ #261 |
| Sylow | C13 ⋊ ΓL(1,𝔽16) |
| Sylow | C5 ⋊ C3⋊#261 |
| Sylow | C3 ⋊ #63122 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C5×C13 | C4 |
| prime radical, cosocle | C3×D65 | C2 |
| LCS | G ▷ C3×C5×C13 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 195 |
Polycyclic presentation:
2^2,3,5,13|a^2=b,c^a=c^2,d^a=d^2,e^a=e^5,d^b=d^4,e^b=e^1254 conjugacy classes — view the full character table.
| maximal subgroup of | D5⋊(C3⋊#261), D3×#63122, #63121⋊D3, … |
| maximal normal subgroup of | D5⋊(C3⋊#261), D3×#63122, #63121⋊D3, … |
| maximal quotient of | C3⋊#10587066, C2×C3⋊#63122, C9⋊#63122, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(780, 33);