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×C3×C9 |
| direct product | C2×C4×#6887 |
| 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×C33 | C2×C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C9 ◁ G | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 3 generators on 42 points:
(1,2)(7,8,10,9,11,13,12,14,15)(16,17,20)(18,22,28)(19,23,27)(21,29,26)(24,31,36)(25,32,35)(30,37,34)(33,39,41)(38,42,40);(3,4,5,6)(17,21,23)(20,27,29)(22,30,32)(28,35,37)(31,38,39)(36,41,42);(16,18,24,19,25,33,26,34,40)(17,22,31,23,32,39,21,30,38)(20,28,36,27,35,41,29,37,42)792 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 2166);