Finite Group DatabasebetaFinite Group Databasebeta| order | 234 |
| isoclinism | isoclinic class of D39 |
| automorphism | Aut(G) = C2×D3×AGL(1,𝔽13) (order 1872) · Inn(G) = D39 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | C3×D39 |
| unambiguous semidirect product | C3⋊(C3×D13) |
| Fitting, socle, Hall | C32×C13 ⋊ C2 |
| derived | C3×C13 ⋊ C2×C3 |
| Sylow | C13 ⋊ C3×D3 |
| Sylow | C32 ⋊ D13 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D39 |
| commutator, cosocle, hypocenter | C3×C13 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C32×C13 | C2 |
| LCS | G ▷ C3×C13 | |
| UCS | C1 ◁ C3 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 78 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(5,6)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19);(4,5,6)(7,8,10,12,14,16,18,19,17,15,13,11,9)63 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×D13, D3×D39, (C3⋊D3)⋊D13, … |
| maximal normal subgroup of | C3×D3×D13, D3×D39, (C3⋊D3)⋊D13, … |
| maximal quotient of | C3×Dic39, C2×C3×D39, C3×D117, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(234, 13);