Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C33 |
| direct product | C3×Q8×AΣL(1,𝔽27) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C22×H(3,𝔽3) |
| commutator, Frattini, cosocle | C2×C32 | C22×C33 |
| prime radical | C2×C3×AΣL(1,𝔽27) | C22 |
| LCS | G ▷ C2×C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C33×Q8 ◁ G | |
756 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3)(13,14,15,16)(17,20,19,18);(4,5,7)(6,8,11)(9,12,10)(13,17,15,19)(14,18,16,20);(5,8,9)255 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 2243);