Finite Group DatabasebetaFinite Group Databasebeta| order | 1080 |
| isoclinism | isoclinic class of C27⋊AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×C27⋊AGL(1,𝔽5) |
| unambiguous semidirect product | D5⋊Dic27 |
| unambiguous semidirect product | (C2×C27)⋊AGL(1,𝔽5) |
| Fitting | C2×C5×C27 ⋊ C4 |
| derived, Hall | C5×C27 ⋊ C2×C4 |
| Sylow | C27 ⋊ C2×AGL(1,𝔽5) |
| Sylow | C5 ⋊ C2×Dic27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C27⋊AGL(1,𝔽5) |
| commutator, hypocenter | C5×C27 | C2×C4 |
| Frattini | C9 | C2×ΓL(1,𝔽16) |
| Fitting | C2×C5×C27 | C4 |
| socle | C2×C3×C5 | Dic9 |
| prime radical | C2×C3×D5 | D9 |
| cosocle | D5×C27 | C22 |
| LCS | G ▷ C5×C27 | |
| UCS | C1 ◁ C2 | |
64 conjugacy classes of subgroups.
| transitive degree | μt(G) = 270 |
114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1080, 56);