Finite Group DatabasebetaFinite Group Databasebeta| order | 4320 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×C5×A4⋊S4 |
| Fitting, socle | C24×C3×C5 ⋊ C3⋊D3 |
| derived | A42 ⋊ C2×C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5 | A4⋊S4 |
| commutator, cosocle, hypocenter | A42 | C2×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3×C5 | C3⋊D3 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C3×C5 | |
456 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 180 |
210 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D5×A4⋊S4 |
| maximal normal subgroup of | C3×D5×A4⋊S4 |
| maximal quotient of | C5×#72213207 |
Past the SmallGroups library coverage — no SmallGroups id.