Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of C3⋊#1020 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C23×C3 |
| direct product | C5×C3⋊#1020 |
| unambiguous semidirect product | C3⋊(C5×#1020) |
| Fitting, socle | C24×C3×C5 ⋊ D3 |
| derived | C3×C22⋊A4 ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | C3⋊#1020 |
| commutator, cosocle, hypocenter | C3×C22⋊A4 | C2×C5 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3×C5 | D3 |
| LCS | G ▷ C3×C22⋊A4 | |
| UCS | C1 ◁ C5 | |
268 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,4,5)(6,7,9,8)(10,11,13,12)(15,16);(14,15,16);(7,8,9)(11,13,12)90 conjugacy classes — view the full character table.
| maximal subgroup of | D5×C3⋊#1020, C5×D3×#1020, #72231058, … |
| maximal normal subgroup of | D5×C3⋊#1020, C5×D3×#1020, #72231058, … |
| maximal quotient of | C5×C3⋊#5935, C3×C5×C3⋊#1020, C5×C3⋊#5934 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5931);