Finite Group DatabasebetaFinite Group Databasebeta| order | 156 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×C3×C4×D3 (order 144) · Inn(G) = D3 · Out(G) = C2×C3×C4 |
| holomorph | G ⋊ Aut(G) = #3811×AGL(1,𝔽13) |
| homology | Abelianization (H1) = C22×C13 · Schur multiplier (H2) = C2 |
| direct product | C2×D3×C13 |
| unambiguous semidirect product | C3⋊(C22×C13) |
| unambiguous semidirect product | (C2×C3)⋊(C2×C13) |
| Fitting, socle | C2×C3×C13 ⋊ C2 |
| derived, Sylow | C3 ⋊ C22×C13 |
| Hall | C3×C13 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C13 | D3 |
| commutator, cosocle, hypocenter | C3 | C22×C13 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C13 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C13 | |
20 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)(4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18);(1,2)(3,4,5)78 conjugacy classes — view the full character table.
| maximal subgroup of | D3×Dic13, #64602, #64604, … |
| maximal normal subgroup of | D3×Dic13, #64602, #64604, … |
| maximal quotient of | C13×Dic6, C4×D3×C13, C13×D12, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(156, 16);