Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D5×C3⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
| direct product | D5×C3⋊S4 |
| unambiguous semidirect product | (C3×D5)⋊S4 |
| unambiguous semidirect product | (D5×A4)⋊D3 |
| Fitting, socle | C22×C3×C5 ⋊ C2×D3 |
| derived | C3×C5×A4 ⋊ C22 |
| Sylow | C5 ⋊ C2×C3⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C5×A4 | C22 |
| Fitting, socle | C22×C3×C5 | C2×D3 |
| LCS | G ▷ C3×C5×A4 | |
| UCS | C1 | |
172 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 60 |
36 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×C3⋊S4, ΓL(1,𝔽16)⋊S4, D3×D5×S4, … |
| maximal normal subgroup of | AGL(1,𝔽5)×C3⋊S4, ΓL(1,𝔽16)⋊S4, D3×D5×S4, … |
| maximal quotient of | #13088576, #13088577, #13088578, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 796);