Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of D7×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×D7×S4 |
| unambiguous semidirect product | (C7×S4)⋊(C2×C5) |
| unambiguous semidirect product | (C7⋊S4)⋊(C2×C5) |
| unambiguous semidirect product | (A4×D7)⋊(C2×C5) |
| Fitting, socle | C22×C5×C7 ⋊ C2×D3 |
| derived | C7×A4 ⋊ C22×C5 |
| Hall | C5×C7 ⋊ C2×S4 |
| Sylow | C7 ⋊ C2×C5×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | D7×S4 |
| commutator, cosocle, hypocenter | C7×A4 | C22×C5 |
| Frattini | C1 | G |
| Fitting, socle | C22×C5×C7 | C2×D3 |
| LCS | G ▷ C7×A4 | |
| UCS | C1 ◁ C5 | |
132 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 140 |
125 conjugacy classes — view the full character table.
| maximal subgroup of | C5×S4×AGL(1,𝔽7), D5×D7×S4 |
| maximal normal subgroup of | C5×S4×AGL(1,𝔽7), D5×D7×S4 |
| maximal quotient of | C5×D7×GL(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 951);