Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C9 · Schur multiplier (H2) = C32 |
| direct product | Q8×#6903 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C27 | C22×C32 |
| commutator, socle | C2×C3 | C22×C32×C9 |
| Frattini, cosocle | C2×C9 | C22×C33 |
| prime radical | C2×H(3,𝔽3) | C22×C9 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C27 ◁ G | |
330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2,3,4)(5,8,7,6)(9,10,14,12,16,26,22,32,47,13,17,27,23,33,48,41,56,70,24,34,49,42,57,71,63,77,85)(11,15,25,20,30,45,39,54,68,21,31,46,40,55,69,62,76,84,19,29,44,38,53,67,61,75,83)(18,28,43,36,51,65,59,73,81,35,50,64,58,72,80,78,86,88,37,52,66,60,74,82,79,87,89);(1,5,3,7)(2,6,4,8)(9,11,18)(10,15,28)(12,20,36)(13,21,35)(14,25,43)(16,30,51)(17,31,50)(19,37,24)(22,39,59)(23,40,58)(26,45,65)(27,46,64)(29,52,34)(32,54,73)(33,55,72)(38,60,42)(41,62,78)(44,66,49)(47,68,81)(48,69,80)(53,74,57)(56,76,86)(61,79,63)(67,82,71)(70,84,88)(75,87,77)(83,89,85);(11,19,21)(15,29,31)(18,35,37)(20,38,40)(25,44,46)(28,50,52)(30,53,55)(36,58,60)(39,61,62)(43,64,66)(45,67,69)(51,72,74)(54,75,76)(59,78,79)(65,80,82)(68,83,84)(73,86,87)(81,88,89)495 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 2242);