Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D3×A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C22 |
| Fitting, socle | C23×C3×C5 ⋊ C2×C3 |
| derived | C22×C3 ⋊ C22×C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D3×A4 |
| commutator, cosocle, hypocenter | C22×C3 | C22×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C23×C3×C5 | C2×C3 |
| LCS | G ▷ C22×C3 | |
| UCS | C1 ◁ C2×C5 | |
164 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 164 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) = 90 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D3×SL(2,ℤ/4), C5×#64209, C5×#64210, … |
| maximal normal subgroup of | C5×D3×SL(2,ℤ/4), C5×#64209, C5×#64210, … |
| maximal quotient of | C5×A4×Dic6, C4×C5×D3×A4, C5×A4×D12, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 819);