Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#10587944 |
| Hall | C3×C5×C7 ⋊ #41 |
| Hall | C5×C7 ⋊ C3×#41 |
| Hall | C3×C7 ⋊ #709 |
| Hall | C3×C5 ⋊ #1157 |
| Sylow | C7 ⋊ C3×#709 |
| Sylow | C5 ⋊ C3×#1157 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | D5×D7 |
| commutator, cosocle | C2×C5×C7 | C23×C3 |
| Frattini | C2 | C2×C3×D5×D7 |
| Fitting | C3×C4×C5×C7 | C22 |
| socle | C2×C3×C5×C7 | C23 |
| hypocenter | C5×C7 | C3×#41 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C3×C4 | |
160 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 840 |
222 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 658);