Finite Group DatabasebetaFinite Group Databasebeta| order | 30420 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3×C13 |
| unambiguous semidirect product | C3⋊#23778486 |
| Fitting, socle | C3×C5×C132 ⋊ Dic3 |
| derived, 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⋊#14523161 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C3×C5×#92789 | C4 |
| Fitting, socle | C3×C5×C132 | Dic3 |
| prime radical, cosocle | C3×C5⋊#11759816 | C2 |
| LCS | G ▷ C3×C5×#92789 | |
| UCS | C1 | |
128 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 390 |
Polycyclic presentation:
2^2,3^2,5,13^2|a^2=b,c^a=c^2,d^a=d^2,e^a=e^2,f^a=g^12,g^a=f,e^b=e^4,f^b=f^12,g^b=g^12,f^c=f^3,g^c=g^9231 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30420, 617);