Finite Group DatabasebetaFinite Group Databasebeta| order | 30420 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | #23778457⋊C3 |
| unambiguous semidirect product | #23778458⋊C3 |
| unambiguous semidirect product | #23778465⋊C3 |
| unambiguous semidirect product | #23778466⋊C3 |
| Fitting, socle, derived, Hall | C5×C132 ⋊ C32×C4 |
| Hall | C5×C13⋊C32 ⋊ C4 |
| Hall | #16810004 ⋊ C32 |
| Hall | C13⋊C32 ⋊ AGL(1,𝔽5) |
| Sylow | C132 ⋊ C32×AGL(1,𝔽5) |
| Sylow | C5 ⋊ #14523146⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C5×C132 | C32×C4 |
| prime radical | C5⋊((C13⋊C3)⋊(D13⋊C3)) | C2 |
| cosocle | C5⋊(C13⋊D13) | C2×C32 |
| LCS | G ▷ C5×C132 | |
| UCS | C1 | |
160 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 34 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:
2,3^2,2,5,13^2|a^2=d,e^a=e^2,f^a=f^5,g^a=g^8,g^b=g^3,f^c=f^3,e^d=e^4,f^d=f^12,g^d=g^1295 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30420, 475);