Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C34 · Schur multiplier (H2) = C37 |
| direct product | C32×Q8×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C33 | C22×C32 |
| commutator, Frattini, cosocle | C2×C3 | C22×C34 |
| prime radical | C2×C32×H(3,𝔽3) | C22 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ G | |
2700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(16,17,18,19)(20,23,22,21);(4,5,6)(16,20,18,22)(17,21,19,23);(7,8,10)(9,12,14)(11,15,13);(8,11,12)(10,14,15)495 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 3429);