Finite Group DatabasebetaFinite Group Databasebeta| order | 684 |
| isoclinism | isoclinic class of D19⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #4294⋊C4 |
| derived, Sylow | C19 ⋊ C4×C9 |
| Hall | #4294 ⋊ C4 |
| Hall | Dic19 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | D19⋊C3 |
| commutator, hypocenter | C19 | C4×C9 |
| Fitting, socle, prime radical, cosocle | C2×C3×C19 | C2×C3 |
| LCS | G ▷ C19 | |
| UCS | C1 ◁ C2×C3 | |
18 conjugacy classes of subgroups.
| transitive degree | μt(G) = 684 |
Polycyclic presentation:
2,3,2,3,19|a^2=c,b^3=d,e^a=e^18,e^b=e^754 conjugacy classes — view the full character table.
| maximal subgroup of | #4294⋊Q8, C4×#4294⋊C2, C2×#4294⋊C4, … |
| maximal normal subgroup of | #4294⋊Q8, C4×#4294⋊C2, C2×#4294⋊C4, … |
| maximal quotient of | #4294⋊C8, C2×#4294⋊C4, C5×#4294⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(684, 1);