Finite Group DatabasebetaFinite Group Databasebeta| order | 1080 |
| isoclinism | isoclinic class of C3⋊(C5⋊S4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×C3⋊(C5⋊S4) |
| unambiguous semidirect product | C3⋊(C3×C5⋊S4) |
| unambiguous semidirect product | C5⋊(C3×C3⋊S4) |
| unambiguous semidirect product | A4⋊(C3×D15) |
| Fitting, socle | C22×C32×C5 ⋊ D3 |
| derived | C3×C5×A4 ⋊ C2×C3 |
| Sylow | C5 ⋊ C3×C3⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊(C5⋊S4) |
| commutator, cosocle, hypocenter | C3×C5×A4 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C22×C32×C5 | D3 |
| LCS | G ▷ C3×C5×A4 | |
| UCS | C1 ◁ C3 | |
154 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 120 |
99 conjugacy classes — view the full character table.
| maximal subgroup of | #72247618, C3×D5×C3⋊S4, C3×S4×D15, … |
| maximal normal subgroup of | C3×D5×C3⋊S4, C3×S4×D15, C3×D3×C5⋊S4 |
| maximal quotient of | #72247618, C3×C3⋊(C5⋊#1020), C3×C5⋊(A4⋊S4) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1080, 545);