Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C27 · Schur multiplier (H2) = C2×C32 |
| direct product | C2×C4×C32⋊C27 |
| unambiguous semidirect product | (C2×C32×C4)⋊C27 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C9 | C32 |
| commutator | C3 | C2×C3×C4×C27 |
| Frattini, cosocle | C2×C3×C9 | C22×C32 |
| socle | C22×C32 | C2×C3×C9 |
| prime radical | C22×C33 | C2×C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C9 ◁ G | |
280 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 2 generators on 42 points:
(1,2)(7,8,10)(9,12,14)(11,15,13)(16,17,20,18,21,26,23,28,33,19,22,27,24,29,34,31,36,39,25,30,35,32,37,40,38,41,42);(3,4,5,6)(8,11,12)(10,14,15)792 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 596);