Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | D3×Q8×D7 |
| Fitting | C3×C7×Q8 ⋊ C22 |
| Hall | C3×C7 ⋊ C22×Q8 |
| Sylow | C7 ⋊ C2×D3×Q8 |
| Sylow | C3 ⋊ C2×Q8×D7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×D7 |
| commutator, socle, cosocle | C2×C3×C7 | C24 |
| Fitting | C3×C7×Q8 | C22 |
| prime radical | C2×D3×D7 | C22 |
| hypercenter | Q8 | D3×D7 |
| hypocenter | C3×C7 | C22×Q8 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
312 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
75 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D7×SD16, D7×#915, D3×#6478, … |
| maximal normal subgroup of | D3×D7×SD16, D7×#915, D3×#6478, … |
| maximal quotient of | D7×#887, D3×#6450, #13058553, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1160);