Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C2 |
| direct product | C3×D7×C16 |
| Fitting | C3×C7×C16 ⋊ C2 |
| derived, Sylow | C7 ⋊ C2×C3×C16 |
| Hall | C3×C7 ⋊ C2×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C16 | D7 |
| commutator, hypocenter | C7 | C2×C3×C16 |
| Frattini | C8 | C2×C3×D7 |
| Fitting | C3×C7×C16 | C2 |
| socle | C2×C3×C7 | C2×C8 |
| prime radical | C2×C3×D7 | C8 |
| cosocle | C7×C8 | C22×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C16 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 336 |
240 conjugacy classes — view the full character table.
| maximal subgroup of | D7×C3⋊C32, #13050843, C3×D7×C32, … |
| maximal normal subgroup of | D7×C3⋊C32, #13050843, C3×D7×C32, … |
| maximal quotient of | C3×D7×C32, C3×#89242, C3×C16×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 136);