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) = C3×C9 |
| direct product | Q8×#6887 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | C22×C32 |
| commutator | C2×C3 | C22×C32×C9 |
| Frattini, socle, cosocle | C2×C32 | C22×C33 |
| prime radical | C2×C33 | C22×C32 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2,3,4)(5,8,7,6)(9,10,12,11,13,15,14,16,17)(18,19,22)(20,24,30)(21,25,29)(23,31,28)(26,33,38)(27,34,37)(32,39,36)(35,41,43)(40,44,42);(1,5,3,7)(2,6,4,8)(19,23,25)(22,29,31)(24,32,34)(30,37,39)(33,40,41)(38,43,44);(18,20,26,21,27,35,28,36,42)(19,24,33,25,34,41,23,32,40)(22,30,38,29,37,43,31,39,44)495 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 2226);