Finite Group DatabasebetaFinite Group Databasebeta| order | 33300 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#16814885⋊C3) |
| unambiguous semidirect product | (C3⋊#16814885)⋊C3 |
| Fitting, socle, derived | C3×C52×C37 ⋊ C3×C4 |
| Hall | C3×C52×C37⋊C3 ⋊ C4 |
| Hall | C52×C37 ⋊ C3×Dic3 |
| Hall | C3×C37⋊C3 ⋊ #1042 |
| Sylow | C37 ⋊ C3×C3⋊#1042 |
| Sylow | C52 ⋊ C3⋊(#3866⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C52×C37 | C3×C4 |
| prime radical | C3×C5×C5⋊(D37⋊C3) | C2 |
| cosocle | C3×C5×D185 | C2×C3 |
| LCS | G ▷ C3×C52×C37 | |
| UCS | C1 | |
124 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 124 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) = 2220 |
Polycyclic presentation:
2,3,2,3,5^2,37|a^2=c,d^a=d^2,e^a=e^2,f^a=f^4,g^a=g^6,g^b=g^26,e^c=e^4,g^c=g^36324 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33300, 569);