Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of D3×D5×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| Fitting, socle | C22×C3×C5 ⋊ C22×D3 |
| derived | C3×C5×A4 ⋊ C23 |
| Sylow | C5 ⋊ C2×D3×S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C5×A4 | C23 |
| Fitting, socle | C22×C3×C5 | C22×D3 |
| LCS | G ▷ C3×C5×A4 | |
| UCS | C1 | |
544 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 60 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | D3×AGL(1,𝔽5)×S4, C2×D3×D5×S4, D3×D5×#1020, … |
| maximal normal subgroup of | D3×AGL(1,𝔽5)×S4, C2×D3×D5×S4, C3×D3×D5×S4 |
| maximal quotient of | C2×D3×D5×S4, D3×D5×GL(2,𝔽3), D3×D5×#1020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5871);