Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of D3×D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| direct product | D3×C8×D7 |
| Fitting | C3×C7×C8 ⋊ C22 |
| derived, Hall | C3×C7 ⋊ C22×C8 |
| Sylow | C7 ⋊ C2×D3×C8 |
| Sylow | C3 ⋊ C2×C8×D7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | D3×D7 |
| commutator, hypocenter | C3×C7 | C22×C8 |
| Frattini | C4 | C2×D3×D7 |
| Fitting | C3×C7×C8 | C22 |
| socle | C2×C3×C7 | C22×C4 |
| prime radical | C2×D3×D7 | C4 |
| cosocle | C3×C4×C7 | C23 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C8 | |
152 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 152 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.
| transitive degree | μt(G) = 168 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D7×C16, D7×#798, D3×#6372, … |
| maximal normal subgroup of | D3×D7×C16, D7×#798, D3×#6372, … |
| maximal quotient of | D3×D7×C16, D7×#798, D3×#6372, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 387);