Finite Group DatabasebetaFinite Group Databasebeta| order | 272 |
| isoclinism | isoclinic class of AGL(1,𝔽17) |
| automorphism | Aut(G) = AGL(1,𝔽17) (order 272) · Inn(G) = G · Out(G) = C1 |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C1 |
| affine general linear | AGL(1,𝔽17) |
| Fitting, socle, derived, Sylow | C17 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C17 | C16 |
| prime radical | D17 | C8 |
| cosocle | #3633 | C2 |
| LCS | G ▷ C17 | |
| UCS | C1 | |
10 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 17 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,3,4,8,5,9,12,15,6,10,13,16,11,17,7,14);(1,2,7,3,11,9,14,16,13,4,8,15,17,5,10,12,6)Polycyclic presentation:
2^4,17|a^2=b,b^2=c,c^2=d,e^a=e^3,e^b=e^9,e^c=e^13,e^d=e^1617 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(272, 50);