Finite Group DatabasebetaFinite Group Databasebeta| order | 156 |
| isoclinism | isoclinic class of AGL(1,𝔽13) |
| automorphism | Aut(G) = AGL(1,𝔽13) (order 156) · Inn(G) = G · Out(G) = C1 |
| holomorph | G ⋊ Aut(G) = AGL(1,𝔽13)2 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| affine general linear | AGL(1,𝔽13) |
| unambiguous semidirect product | #261⋊C3 |
| Fitting, socle, derived, Sylow | C13 ⋊ C3×C4 |
| Hall | #261 ⋊ C3 |
| Hall | C13⋊C3 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C13 | C3×C4 |
| prime radical | D13⋊C3 | C2 |
| cosocle | D13 | C2×C3 |
| LCS | G ▷ C13 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 13 |
A minimal faithful permutation representation — 2 generators on 13 points:
(2,7,11,8,4,13,5,6,9,3,10,12);(1,2,6,4,11,8,12,13,3,9,10,7,5)Polycyclic presentation:
2,3,2,13|a^2=c,d^a=d^8,d^b=d^3,d^c=d^1213 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(156, 7);