Finite Group DatabasebetaFinite Group Databasebeta| order | 30324 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊#21439055 |
| Fitting, derived, Hall | C22×C7×C361 ⋊ C3 |
| Hall | C7×C361 ⋊ A4 |
| Hall | C22×C361 ⋊ ΓL(1,𝔽8) |
| Sylow | C361 ⋊ C22⋊ΓL(1,𝔽8) |
| Hall | C22×C7 ⋊ C361⋊C3 |
| Sylow | C7 ⋊ C22⋊(C361⋊C3) |
| Sylow | C22 ⋊ #21439055 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C22×C7×C361 | C3 |
| Frattini | C19 | C22⋊#87704 |
| socle | C22×C7×C19 | C19⋊C3 |
| LCS | G ▷ C22×C7×C361 | |
| UCS | C1 | |
30 conjugacy classes of subgroups.
| transitive degree | μt(G) = 10108 |
Polycyclic presentation:
3,2^2,7,19^2|e^19=f^13,b^a=c,c^a=bc,d^a=d^4,e^a=e^11f,f^a=f^113372 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30324, 56);