Finite Group DatabasebetaFinite Group Databasebeta| order | 7533 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C27 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | (C31⋊C9)⋊C27 |
| Sylow | C31 ⋊ C9⋊C27 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3×C9 | C3×C31⋊C3 |
| commutator | C3×C31 | C3×C27 |
| Fitting | C3×C27×C31 | C3 |
| socle | C32×C31 | C3×C9 |
| hypercenter | C3×C27 | C31⋊C3 |
| hypocenter | C31 | C9⋊C27 |
| LCS | G ▷ C3×C31 ▷ C31 | |
| UCS | C1 ◁ C3×C9 ◁ C3×C27 | |
52 conjugacy classes of subgroups.
Polycyclic presentation:
3^5,31|a^3=d,b^3=c,d^3=e,b^a=bc,f^b=f^25909 conjugacy classes — view the full character table.
| maximal subgroup of | C31⋊C3×C9⋊C27, C3×(C31⋊C9)⋊C27, #36381066, … |
| maximal normal subgroup of | C31⋊C3×C9⋊C27, C3×(C31⋊C9)⋊C27, #36381066, … |
| maximal quotient of | #36379536, #36379539, (C31⋊C9)⋊C81, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7533, 116);