Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8×C9 · Schur multiplier (H2) = C32 |
| direct product | C8×#6868 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C8 | H(3,𝔽3) |
| commutator | C32 | C3×C8×C9 |
| Frattini, cosocle | C33×C4 | C2×C32 |
| socle | C2×C32 | C4×H(3,𝔽3) |
| prime radical | C2×C33 | C32×C4 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C8 ◁ C33×C8 ◁ G | |
144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2,3,4,5,6,7,8)(9,10,12,11,13,15,14,16,17)(18,19,22)(20,26,30)(21,25,31)(23,32,28)(24,35,29)(27,37,40)(33,43,44)(34,42,39)(36,41,38);(18,20,27,29,39,44,21,28,38)(19,23,33,24,26,36,25,34,37)(22,30,40,35,42,43,31,32,41)408 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 84);