Finite Group DatabasebetaFinite Group Databasebeta| order | 30420 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#16809979⋊C3) |
| unambiguous semidirect product | (C3⋊#16809979)⋊C3 |
| Fitting, derived | C3×C5×C169 ⋊ C3×C4 |
| Hall | C3×C5×C169⋊C3 ⋊ C4 |
| Hall | C3×C169⋊C3 ⋊ AGL(1,𝔽5) |
| Hall | C5×C169 ⋊ C3×Dic3 |
| Sylow | C169 ⋊ C3×ΓL(1,𝔽16) |
| Sylow | C5 ⋊ C3⋊(Dic169⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C3×C5×C169 | C3×C4 |
| Frattini | C13 | C3⋊(#63120⋊C3) |
| socle | C3×C5×C13 | Dic13⋊C3 |
| prime radical | C3×D5×C169⋊C3 | C2 |
| cosocle | C3×D5×C169 | C2×C3 |
| LCS | G ▷ C3×C5×C169 | |
| UCS | C1 | |
84 conjugacy classes of subgroups.
| transitive degree | μt(G) = 2535 |
Polycyclic presentation:
2,3,2,3,5,13^2|a^2=c,f^13=g^12,d^a=d^2,e^a=e^2,f^a=f^12g,g^a=g^12,f^b=f^3g^2,g^b=g^3,e^c=e^4363 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30420, 85);