Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C5 · Schur multiplier (H2) = C3 |
| Fitting | C2×C3×C5×Q8 ⋊ C3 |
| derived | Q8 ⋊ C2×C32×C5 |
| Hall | C2×C5×Q8 ⋊ C32 |
| Sylow | C2×Q8 ⋊ C32×C5 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C3×C5 | A4 |
| commutator, cosocle, hypocenter | Q8 | C2×C32×C5 |
| Frattini | C2 | C2×C3×C5×A4 |
| Fitting | C2×C3×C5×Q8 | C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C3×C5 | |
100 conjugacy classes of subgroups.
| transitive degree | μt(G) = 240 |
210 conjugacy classes — view the full character table.
| maximal subgroup of | C3×C5×#859, C3×#91184, C5×C3⋊#859, … |
| maximal normal subgroup of | C3×C5×#859, C3×#91184, C5×C3⋊#859, … |
| maximal quotient of | C3×C4×C5×SL(2,𝔽3), C22×C3×C5×SL(2,𝔽3), C5×SL(2,𝔽3)×AGL(1,𝔽7), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 707);