Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4×C9 · Schur multiplier (H2) = C2×C34 |
| direct product | C2×C4×C9×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C9 | C32 |
| commutator | C3 | C2×C32×C4×C9 |
| Frattini, cosocle | C2×C32 | C22×C33 |
| socle | C22×C32 | C2×C33 |
| prime radical | C22×C3×H(3,𝔽3) | C2×C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C9 ◁ G | |
808 conjugacy classes of subgroups; only the 456 normal subgroups are shown.
This group has 808 conjugacy classes of subgroups — too many to draw. Only its 456 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(7,8,9,10,11,12,13,14,15);(3,4,5,6)(16,17,19)(18,21,23)(20,24,22);(17,20,21)(19,23,24)792 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 2167);