Finite Group DatabasebetaFinite Group Databasebeta| order | 684 |
| isoclinism | isoclinic class of D19⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| direct product | C2×#4294⋊C2 |
| unambiguous semidirect product | #4294⋊C22 |
| unambiguous semidirect product | (C2×#4294)⋊C2 |
| derived, Sylow | C19 ⋊ C22×C9 |
| Hall | #4294 ⋊ C22 |
| Hall | C2×D19 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D19⋊C3 |
| commutator, hypocenter | C19 | C22×C9 |
| Frattini | C3 | C2×D19⋊C3 |
| Fitting, socle | C2×C3×C19 | C2×C3 |
| prime radical | C2×C3×D19 | C3 |
| cosocle | C3×C19 | C22×C3 |
| LCS | G ▷ C19 | |
| UCS | C1 ◁ C2×C3 | |
30 conjugacy classes of subgroups.
| transitive degree | μt(G) = 342 |
54 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#4294⋊C2, C4⋊(#4294⋊C2), #13062911, … |
| maximal normal subgroup of | C4×#4294⋊C2, C4⋊(#4294⋊C2), #13062911, … |
| maximal quotient of | #4294⋊Q8, C4×#4294⋊C2, C4⋊(#4294⋊C2), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(684, 7);