Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C8 · Schur multiplier (H2) = C3 |
| direct product | C8×#6899 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C8 | C33 |
| commutator | C32 | C33×C8 |
| Frattini, cosocle | C32×C4 | C2×C33 |
| socle | C2×C32 | C33×C4 |
| prime radical | C2×C33 | C32×C4 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C8 ◁ G | |
220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2,3,4,5,6,7,8)(10,13,11)(14,15,16)(18,19,23)(20,27,39)(21,25,33)(22,26,34)(24,35,30)(28,38,42)(29,40,32)(31,37,43)(36,44,41);(9,10,14,17,11,15,12,13,16)(18,20,28,22,29,41,31,30,21)(19,24,36,26,27,25,37,40,38)(23,32,33,34,35,42,43,39,44);(9,11,16,17,13,14,12,10,15)(18,21,30,31,41,29,22,28,20)(19,25,24,37,36,40,26,38,27)(23,33,35,43,44,32,34,42,39)408 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 787);