Finite Group DatabasebetaFinite Group Databasebeta| order | 234 |
| isoclinism | isoclinic class of C3⋊(D13⋊C3) |
| automorphism | Aut(G) = D3×AGL(1,𝔽13) (order 936) · Inn(G) = G · Out(G) = C4 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(D13⋊C3) |
| unambiguous semidirect product | (C13⋊C3)⋊D3 |
| unambiguous semidirect product | D39⋊C3 |
| Fitting, socle, derived | C3×C13 ⋊ C2×C3 |
| Hall | C3×C13⋊C3 ⋊ C2 |
| Sylow | C13 ⋊ C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C13 | C2×C3 |
| LCS | G ▷ C3×C13 | |
| UCS | C1 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 39 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,3)(5,9,10,6,7,11)(8,15,14,12,13,16);(1,2,3)(4,5,8,7,11,14,13,15,16,10,9,12,6)Polycyclic presentation:
2,3^2,13|c^a=c^2,d^a=d^12,d^b=d^315 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D13⋊C3, C2×C3⋊(D13⋊C3), #10622036, … |
| maximal normal subgroup of | D3×D13⋊C3, C2×C3⋊(D13⋊C3), C3×C3⋊(D13⋊C3), … |
| maximal quotient of | C3⋊(Dic13⋊C3), C2×C3⋊(D13⋊C3), #10622036, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(234, 9);