Finite Group DatabasebetaFinite Group Databasebeta| order | 675 |
| isoclinism | isoclinic class of C52⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C52⋊C27 |
| derived, Sylow | C52 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C9 | C52⋊C3 |
| commutator, hypocenter | C52 | C27 |
| Fitting, cosocle | C52×C9 | C3 |
| socle, prime radical | C3×C52 | C9 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C9 | |
14 conjugacy classes of subgroups.
| transitive degree | μt(G) = 135 |
Polycyclic presentation:
3^3,5^2|a^3=b,b^3=c,d^a=d^3e^2,e^a=de99 conjugacy classes — view the full character table.
| derived subgroup of | #13062464 |
| maximal subgroup of | #13062464, (C5⋊D5)⋊C27, C2×C52⋊C27 |
| maximal normal subgroup of | #13062464, (C5⋊D5)⋊C27, C2×C52⋊C27 |
| maximal quotient of | C2×C52⋊C27 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(675, 5);