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) = C1 |
| direct product | C8×C81⋊C3 |
| unambiguous semidirect product | (C8×C81)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C27 | C32 |
| commutator | C3 | C3×C8×C27 |
| Frattini, cosocle | C4×C27 | C2×C32 |
| socle | C2×C3 | C3×C4×C27 |
| prime radical | C2×C32 | C4×C27 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C8×C27 ◁ G | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 2 generators on 89 points:
(1,2,3,4,5,6,7,8)(9,10,14,11,16,26,19,31,45,12,17,27,20,32,46,36,53,67,22,34,47,38,55,68,58,74,82,13,18,25,21,33,43,37,54,64,23,35,44,39,56,65,59,75,80,41,57,66,61,76,81,77,86,88,24,15,28,40,29,48,60,50,69,42,30,49,62,51,70,78,72,83,63,52,71,79,73,84,87,85,89);(10,15,18)(14,25,28)(16,29,33)(17,30,35)(26,43,48)(27,44,49)(31,50,54)(32,51,56)(34,52,57)(45,64,69)(46,65,70)(47,66,71)(53,72,75)(55,73,76)(67,80,83)(68,81,84)(74,85,86)(82,88,89)792 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 93);