Finite Group DatabasebetaFinite Group Databasebeta| order | 675 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C9⋊(C52⋊C3) |
| Fitting | C52×C9 ⋊ C3 |
| Sylow | C52 ⋊ 3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×C52⋊C3 |
| commutator, socle, cosocle | C3×C52 | C32 |
| Fitting | C52×C9 | C3 |
| prime radical | C3×C52⋊C3 | C3 |
| hypercenter | C9 | C52⋊C3 |
| hypocenter | C52 | 3−1+2 |
| LCS | G ▷ C3×C52 ▷ C52 | |
| UCS | C1 ◁ C3 ◁ C9 | |
22 conjugacy classes of subgroups.
| transitive degree | μt(G) = 135 |
Polycyclic presentation:
3^3,5^2|b^3=c,b^a=bc,d^a=d^3e^2,e^a=de83 conjugacy classes — view the full character table.
| maximal subgroup of | #13062489, D9⋊(C52⋊C3), #13062497, … |
| maximal normal subgroup of | #13062489, D9⋊(C52⋊C3), #13062497, … |
| maximal quotient of | C2×C9⋊(C52⋊C3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(675, 9);