Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of C3⋊#226 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C3 |
| direct product | C5×C3⋊GL(2,𝔽3) |
| unambiguous semidirect product | C3⋊(C5×GL(2,𝔽3)) |
| Fitting | C3×C5×Q8 ⋊ D3 |
| derived | C3×SL(2,𝔽3) ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | C3⋊S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C2×C5 |
| Frattini | C2 | C5×C3⋊S4 |
| Fitting | C3×C5×Q8 | D3 |
| socle | C2×C3×C5 | S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C5 | |
92 conjugacy classes of subgroups.
| transitive degree | μt(G) = 120 |
75 conjugacy classes — view the full character table.
| maximal subgroup of | C5×#64198, C5×#64199, C5×#64202, … |
| maximal normal subgroup of | C5×#64198, C5×#64199, C5×#64202, … |
| maximal quotient of | C5×C3⋊#859, C2×C5×C3⋊GL(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 533);