Finite Group DatabasebetaFinite Group Databasebeta| order | 30420 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #23778486⋊C3 |
| Fitting, socle, Hall | C5×C132 ⋊ C3×Dic3 |
| derived | C5×#92789 ⋊ C3×C4 |
| Hall | C5×C13⋊C32 ⋊ C4 |
| Hall | C13⋊C32 ⋊ AGL(1,𝔽5) |
| Sylow | C132 ⋊ C3×ΓL(1,𝔽16) |
| Sylow | C5 ⋊ #14523161⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C5×#92789 | C3×C4 |
| Fitting, socle | C5×C132 | C3×Dic3 |
| prime radical | C5⋊((C13⋊C3)⋊(D13⋊C3)) | C2 |
| cosocle | C5⋊#11759816 | C2×C3 |
| LCS | G ▷ C5×#92789 | |
| UCS | C1 | |
112 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 130 |
Polycyclic presentation:
2,3,2,3,5,13^2|a^2=c,d^a=d^2,e^a=e^2,f^a=f^10g^3,g^a=fg^3,f^b=f^9,g^b=g^9,e^c=e^4,f^c=f^12,g^c=g^12,f^d=f^9g^5,g^d=g^377 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30420, 465);