Finite Group DatabasebetaFinite Group Databasebeta| order | 234 |
| isoclinism | isoclinic class of C13⋊C3 |
| automorphism | Aut(G) = D3×AGL(1,𝔽13) (order 936) · Inn(G) = C13⋊C3 · Out(G) = C4×D3 |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C2×C3×C13⋊C3 |
| unambiguous semidirect product | (C2×C13)⋊C32 |
| unambiguous semidirect product | (C2×C3×C13)⋊C3 |
| Fitting, socle | C2×C3×C13 ⋊ C3 |
| derived, Sylow | C13 ⋊ C2×C32 |
| Hall | C2×C13 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C13⋊C3 |
| commutator, cosocle, hypocenter | C13 | C2×C32 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C13 | C3 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C2×C3 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 78 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2)(7,8,10)(9,12,15)(11,14,13)(16,18,17);(3,4,5)(6,7,9,8,11,15,12,16,18,10,13,17,14)42 conjugacy classes — view the full character table.
| maximal subgroup of | C3×Dic13⋊C3, Dic3×C13⋊C3, C3⋊(Dic13⋊C3), … |
| maximal normal subgroup of | C3×Dic13⋊C3, Dic3×C13⋊C3, C3⋊(Dic13⋊C3), … |
| maximal quotient of | C3×C4×C13⋊C3, C22×C3×C13⋊C3, C2×C9×C13⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(234, 10);