Finite Group DatabasebetaFinite Group Databasebeta| order | 684 |
| isoclinism | isoclinic class of D19 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3 |
| direct product | C32×Dic19 |
| derived, Sylow | C19 ⋊ C32×C4 |
| Hall | C32×C19 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | D19 |
| commutator, hypocenter | C19 | C32×C4 |
| Frattini | C2 | C32×D19 |
| Fitting, socle, prime radical | C2×C32×C19 | C2 |
| cosocle | C2×C19 | C2×C32 |
| LCS | G ▷ C19 | |
| UCS | C1 ◁ C2×C32 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 684 |
198 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×Dic19, C3×D57⋊C4, C3×#90697, … |
| maximal normal subgroup of | C3×D3×Dic19, C3×D57⋊C4, C3×#90697, … |
| maximal quotient of | C32×C19⋊C8, C2×C32×Dic19, C32×C5×Dic19, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(684, 22);