Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| isoclinism | isoclinic class of D3×D5×D15 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | D3×D5×D15 |
| unambiguous semidirect product | (C3×D3×D5)⋊D5 |
| unambiguous semidirect product | (C5×D3×D5)⋊D3 |
| Fitting, socle, derived, Hall | C32×C52 ⋊ C23 |
| Sylow | C52 ⋊ C2×D32 |
| Sylow | C32 ⋊ C2×D52 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C32×C52 | C23 |
| LCS | G ▷ C32×C52 | |
| UCS | C1 | |
320 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 60 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D152, #72222546, D32×D52, … |
| maximal normal subgroup of | D32×D52, D3×AGL(1,𝔽5)×D15, D3×ΓL(1,𝔽16)⋊D5, … |
| maximal quotient of | D3×D152, #72222546, D3×D5×C3⋊D15, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 678);