Finite Group DatabasebetaFinite Group Databasebeta| order | 156 |
| isoclinism | isoclinic class of #261 |
| automorphism | Aut(G) = C2×AGL(1,𝔽13) (order 312) · Inn(G) = #261 · Out(G) = C2×C3 |
| holomorph | G ⋊ Aut(G) = D3×#72214122 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C3×#261 |
| Fitting, socle, Hall | C3×C13 ⋊ C4 |
| derived, Sylow | C13 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #261 |
| commutator, hypocenter | C13 | C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C3×C13 | C4 |
| prime radical | C3×D13 | C2 |
| cosocle | D13 | C2×C3 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C3 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 39 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,6,4,3)(5,13,9,8)(7,10,12,11)(14,15,16);(1,2,5,8,12,6,10,11,3,7,13,9,4)21 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#261, C2×C3×#261, #261⋊C9, … |
| maximal normal subgroup of | D3×#261, C2×C3×#261, #261⋊C9, … |
| maximal quotient of | C3×#1057, C2×C3×#261, C9×#261, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(156, 9);