Finite Group DatabasebetaFinite Group Databasebeta| order | 234 |
| isoclinism | isoclinic class of C13⋊C3 |
| automorphism | Aut(G) = C3×AGL(1,𝔽13) (order 468) · Inn(G) = C13⋊C3 · Out(G) = C3×C4 |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| direct product | C2×C13⋊C9 |
| unambiguous semidirect product | (C2×C13)⋊C9 |
| derived, Sylow | C13 ⋊ C2×C9 |
| Hall | C2×C13 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C13⋊C3 |
| commutator, hypocenter | C13 | C2×C9 |
| Frattini | C3 | C2×C13⋊C3 |
| Fitting, socle, prime radical | C2×C3×C13 | C3 |
| cosocle | C3×C13 | C2×C3 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C2×C3 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 234 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,7,11,5,10,6,8,4,9)(13,14,16)(15,18,21)(17,20,19)(22,24,23);(12,13,15,14,17,21,18,22,24,16,19,23,20)42 conjugacy classes — view the full character table.
| maximal subgroup of | Dic13⋊C9, C4×C13⋊C9, C2×D13⋊C9, … |
| maximal normal subgroup of | Dic13⋊C9, C4×C13⋊C9, C2×D13⋊C9, … |
| maximal quotient of | C4×C13⋊C9, C22×C13⋊C9, C2×C13⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(234, 2);