Finite Group DatabasebetaFinite Group Databasebeta| order | 4617 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C19 · Schur multiplier (H2) = C32 |
| direct product | C19×#6866 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C19 | H(3,𝔽3) |
| commutator | C32 | C3×C9×C19 |
| Frattini, cosocle | C33 | C32×C19 |
| prime radical | C34×C19 | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C19 ◁ C33×C19 ◁ G | |
180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 513 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,22,24,26,25,27,28)(29,30,32)(31,33,36)(34,37,35);(30,33,34)969 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4617, 13);