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) = C4×C13 · Schur multiplier (H2) = C1 |
| direct product | Dic3×C13 |
| unambiguous semidirect product | C3⋊(C4×C13) |
| derived, Sylow | C3 ⋊ C4×C13 |
| Hall | C3×C13 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C13 | D3 |
| commutator, hypocenter | C3 | C4×C13 |
| Frattini | C2 | D3×C13 |
| Fitting, socle, prime radical | C2×C3×C13 | C2 |
| cosocle | C2×C3 | C2×C13 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C13 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 156 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(3,4)(5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20);(1,3,2,4)(6,7)78 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×D13, #64601, #64603, … |
| maximal normal subgroup of | Dic3×D13, #64601, #64603, … |
| maximal quotient of | C13×C3⋊C8, C2×Dic3×C13, C13×Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(156, 3);