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) = C3 |
| direct product | C8×#6872 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C9 | H(3,𝔽3) |
| commutator | C32 | C3×C8×C9 |
| Frattini, cosocle | C3×C4×C9 | C2×C32 |
| socle | C2×C3 | C4×C32⋊C9 |
| prime radical | C2×H(3,𝔽3) | C4×C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C8×C9 ◁ C3×C8×C9 ◁ G | |
104 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)(9,10,14,12,17,29,24,35,51,13,18,28,25,36,50,46,60,73,26,16,32,47,34,56,70,59,77)(11,19,37,22,39,62,44,64,81,23,33,55,45,58,76,69,79,87,21,15,27,43,31,49,68,54,72)(20,38,48,41,63,71,66,82,85,40,57,61,65,78,80,83,88,89,42,30,52,67,53,74,84,75,86);(9,11,20)(10,15,30)(12,22,41)(13,23,40)(14,27,48)(16,33,57)(17,31,53)(18,19,38)(21,42,26)(24,44,66)(25,45,65)(28,37,61)(29,49,71)(32,55,52)(34,58,78)(35,54,75)(36,39,63)(43,67,47)(46,69,83)(50,62,80)(51,72,85)(56,76,74)(59,79,88)(60,64,82)(68,84,70)(73,81,89)(77,87,86)408 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 88);