Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of C5⋊#226 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×C5⋊GL(2,𝔽3) |
| unambiguous semidirect product | (C2×C5)⋊(C3×GL(2,𝔽3)) |
| Fitting | C2×C3×C5×Q8 ⋊ D3 |
| derived | C5×SL(2,𝔽3) ⋊ C22×C3 |
| Sylow | C5 ⋊ C2×C3×GL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | C5⋊S4 |
| commutator, cosocle, hypocenter | C5×SL(2,𝔽3) | C22×C3 |
| Frattini | C2 | C2×C3×C5⋊S4 |
| Fitting | C2×C3×C5×Q8 | D3 |
| socle | C22×C3×C5 | S4 |
| LCS | G ▷ C5×SL(2,𝔽3) | |
| UCS | C1 ◁ C22×C3 | |
218 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 240 |
132 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×#72213709, C2×C3×D5×GL(2,𝔽3), AGL(1,𝔽7)×C5⋊GL(2,𝔽3) |
| maximal normal subgroup of | C2×C3×D5×GL(2,𝔽3) |
| maximal quotient of | C2×C3×#72213709, AGL(1,𝔽7)×C5⋊GL(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5377);