Finite Group DatabasebetaFinite Group Databasebeta| order | 1080 |
| isoclinism | isoclinic class of AGL(1,𝔽5)×(C3⋊D3)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | AGL(1,𝔽5)×(C3⋊D3)⋊C3 |
| unambiguous semidirect product | (C3⋊D3×AGL(1,𝔽5))⋊C3 |
| Fitting, Hall | C5×H(3,𝔽3) ⋊ C2×C4 |
| derived | C32×C5 ⋊ C2×C3×C4 |
| Sylow | H(3,𝔽3) ⋊ C2×AGL(1,𝔽5) |
| Sylow | C5 ⋊ C4×(C3⋊D3)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C32×C5 | C2×C3×C4 |
| Frattini | C3 | C3×D3×AGL(1,𝔽5) |
| Fitting | C5×H(3,𝔽3) | C2×C4 |
| socle | C3×C5 | C3×C4×D3 |
| prime radical | D5×(C3⋊D3)⋊C3 | C2 |
| cosocle | C32×D5 | C22×C3 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 | |
124 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 45 |
50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1080, 270);