Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C16 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×D5⋊C16 |
| unambiguous semidirect product | D5⋊(C32×C16) |
| derived, Sylow | C5 ⋊ C2×C32×C16 |
| Hall | C32×C5 ⋊ C2×C16 |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 720 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3)(7,8,9,11,10)(12,14,16,23,17,15,22,24)(13,18,20,26,21,19,25,27);(4,5,6)(8,9,10,11)(12,13,15,19,16,20,24,27,17,21,14,18,22,25,23,26)360 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 2558);