Finite Group DatabasebetaFinite Group Databasebeta| order | 30420 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#23778456 |
| Fitting, socle, derived | C3×C5×C132 ⋊ C3×C4 |
| Hall | C3×C5×#92789 ⋊ C4 |
| Hall | C3×#92789 ⋊ AGL(1,𝔽5) |
| Hall | C5×C132 ⋊ C3×Dic3 |
| Sylow | C132 ⋊ C3×ΓL(1,𝔽16) |
| Sylow | C5 ⋊ C3⋊#14523145 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C5×C132 | C3×C4 |
| prime radical | C3×C5⋊#11759816 | C2 |
| cosocle | C3×C5⋊(C13⋊D13) | C2×C3 |
| LCS | G ▷ C3×C5×C132 | |
| UCS | C1 | |
160 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 585 |
Polycyclic presentation:
2,3,2,3,5,13^2|a^2=c,d^a=d^2,e^a=e^2,f^a=f^8,g^a=g^8,f^b=f^3,g^b=g^9,e^c=e^4,f^c=f^12,g^c=g^12237 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30420, 652);