Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8×C27 · Schur multiplier (H2) = C32 |
| direct product | C8×C32⋊C27 |
| unambiguous semidirect product | (C32×C8)⋊C27 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8×C9 | C32 |
| commutator | C3 | C3×C8×C27 |
| Frattini, cosocle | C3×C4×C9 | C2×C32 |
| socle | C2×C32 | C3×C4×C9 |
| prime radical | C2×C33 | C4×C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C8×C9 ◁ G | |
140 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)(11,14,16)(13,17,15)(18,19,22,20,23,28,25,30,35,21,24,29,26,31,36,33,38,41,27,32,37,34,39,42,40,43,44);(10,13,14)(12,16,17)792 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 81);