Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| isoclinism | isoclinic class of D5×(C3⋊D3)⋊D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | D5×(C3⋊D3)⋊D5 |
| unambiguous semidirect product | (D5×C3⋊D3)⋊D5 |
| unambiguous semidirect product | (D5×D15)⋊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 54 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 60 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | #72222549, D32×D52, AGL(1,𝔽5)×(C3⋊D3)⋊D5, … |
| maximal normal subgroup of | D32×D52, AGL(1,𝔽5)×(C3⋊D3)⋊D5, #72231129, … |
| maximal quotient of | #72222549, D5×D15⋊(C3⋊D3), C2×D5×(C3⋊D3)⋊D5 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 680);