Finite Group DatabasebetaFinite Group Databasebeta| order | 30420 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #15690146⋊D5 |
| derived, Hall | C5×#13095201 ⋊ C22 |
| Hall | #13095201 ⋊ C2×D5 |
| Hall | C5×C132 ⋊ C2×D9 |
| Sylow | C132 ⋊ D5×D9 |
| Sylow | C5 ⋊ #15690146⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C5×#13095201 | C22 |
| Frattini | C3 | #11759813⋊D5 |
| Fitting, socle | C3×C5×C132 | C2×D3 |
| LCS | G ▷ C5×#13095201 | |
| UCS | C1 | |
120 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 17 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:
2^2,3,5,13^2,3|c^3=g^2,c^a=c^2g,d^a=d^4,e^a=f,f^a=e,g^a=g^2,c^b=c^2g,e^b=f^12,f^b=e^12,g^b=g^2,e^c=e^3,f^c=f^9288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30420, 286);