Finite Group DatabasebetaFinite Group Databasebeta| order | 5400 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C3×C5 |
| direct product | C3×C3⋊D3×#1044 |
| Fitting, socle, Hall | C33×C52 ⋊ C2×C4 |
| derived | C32×C52 ⋊ C2×C3×C4 |
| Sylow | C33 ⋊ C2×#1044 |
| Sylow | C52 ⋊ C3×C4×C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊D3×#1044 |
| commutator, hypocenter | C32×C52 | C2×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C33×C52 | C2×C4 |
| prime radical | C3×C3⋊D3×C5⋊D5 | C2 |
| cosocle | C32×C5⋊D5 | C22×C3 |
| LCS | G ▷ C32×C52 | |
| UCS | C1 ◁ C3 | |
632 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 632 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 180 |
180 conjugacy classes — view the full character table.
| maximal subgroup of | C3×C3⋊D3×D5⋊AGL(1,𝔽5), D3×C3⋊D3×#1044, C3⋊D3×#10597183 |
| maximal normal subgroup of | C3×C3⋊D3×D5⋊AGL(1,𝔽5), D3×C3⋊D3×#1044, C3⋊D3×#10597183 |
Past the SmallGroups library coverage — no SmallGroups id.