Finite Group DatabasebetaFinite Group Databasebeta| order | 29988 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#16809726⋊C3) |
| unambiguous semidirect product | (C3⋊#16809726)⋊C3 |
| Fitting, socle, derived | C3×C72×C17 ⋊ C3×C4 |
| Hall | C3×C17×#3861 ⋊ C4 |
| Hall | C72×C17 ⋊ C3×Dic3 |
| Hall | C3×#3861 ⋊ Dic17 |
| Sylow | C72 ⋊ C3×Dic51 |
| Sylow | C17 ⋊ C3⋊(#5997⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C72×C17 | C3×C4 |
| prime radical | C3×C17×#64424 | C2 |
| cosocle | C3×C17×C7⋊D7 | C2×C3 |
| LCS | G ▷ C3×C72×C17 | |
| UCS | C1 | |
152 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 714 |
Polycyclic presentation:
2,3,2,3,7^2,17|a^2=c,d^a=d^2,e^a=e^4f^3,f^a=e^6f^3,g^a=g^16,e^b=e^4,f^b=f^4,e^c=e^6,f^c=f^6366 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(29988, 418);