Finite Group DatabasebetaFinite Group Databasebeta| order | 1620 |
| isoclinism | isoclinic class of #1113 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2 |
| direct product | C3×C5×#1113 |
| Fitting, socle, Hall | C34×C5 ⋊ C22 |
| derived | C33 ⋊ C22×C3×C5 |
| Sylow | C34 ⋊ C22×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5 | #1113 |
| commutator, cosocle, hypocenter | C33 | C22×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C34×C5 | C22 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C3×C5 | |
348 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 60 |
225 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D5×#1113, C5×(((C3⋊D3)⋊C9)⋊C2)⋊C3, C5×#11757730, … |
| maximal normal subgroup of | C3×D5×#1113, C5×#10620215, #72254848 |
| maximal quotient of | C5×(#4207⋊C2)⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1620, 454);