Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of SL(2,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | C3×#6732 |
| derived | SL(2,𝔽5) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C3×C4 | A5 |
| commutator, hypocenter | SL(2,𝔽5) | C2×C3 |
| Frattini, cosocle | C2 | C2×C3×A5 |
| socle | C2×C3 | C2×A5 |
| LCS | G ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C3×C4 | |
67 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
54 conjugacy classes — view the full character table.
| maximal subgroup of | C3×GL(2,𝔽5), C3⋊GL(2,𝔽5), C3⋊#91145, … |
| maximal normal subgroup of | C3×GL(2,𝔽5), C3⋊GL(2,𝔽5), C3⋊#91145, … |
| maximal quotient of | C3×C4×SL(2,𝔽5), C2×C3×#6732, C3×C5×#6732, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 420);