Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6909 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C4 · Schur multiplier (H2) = C2×C33 |
| direct product | C2×C4×#6910 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C3×H(3,𝔽3) |
| commutator | C32 | C2×C33×C4 |
| Frattini, cosocle | C2×C32 | C22×C33 |
| socle | C22×C3 | C2×C3×H(3,𝔽3) |
| prime radical | C22×C3×H(3,𝔽3) | C2×C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C33×C4 ◁ G | |
544 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 264 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 — 3 generators on 33 points:
(1,2)(7,8,11)(9,15,20)(10,14,21)(12,22,18)(13,25,19)(16,27,30)(17,24,31)(23,33,29)(26,32,28);(3,4,5,6)(7,9,16,10,18,28,19,17,29)(8,12,23,14,24,27,13,15,26)(11,20,30,21,22,32,25,31,33);(9,17,18)(12,15,24)(16,28,29)(20,31,22)(23,27,26)(30,32,33)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 2189);