Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C9 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C4×#6869 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | H(3,𝔽3) |
| commutator | C32 | C2×C3×C4×C9 |
| Frattini, cosocle | C2×C3×C9 | C22×C32 |
| socle | C22×C3 | C2×C32⋊C9 |
| prime radical | C22×C33 | C2×C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ C2×C3×C4×C9 ◁ G | |
248 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2)(7,8,11,9,14,19,15,23,27,10,12,18,16,20,26,24,28,32,17,13,21,25,22,29,31,30,33);(3,4,5,6)(8,12,13)(14,20,22)(23,28,30)408 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 600);