Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#60331 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C24 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C24×C4 |
| prime radical | C7×#3482 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ G | |
1792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,16)(11,14)(12,15)(13,18)(17,22)(19,21)(20,23)(24,25,29,30,28,32,26,31)(27,33,36,37,35,39,34,38);(8,10,12,18)(9,13,15,16)(11,19,20,17)(14,22,23,21)(24,26,28,29)(25,30,32,31)(27,34,35,36)(33,37,39,38);(8,11,12,20)(9,14,15,23)(10,17,18,19)(13,21,16,22)(24,26,28,29)(25,31,32,30)(27,34,35,36)(33,38,39,37);(9,15)(13,16)(14,23)(21,22)(25,32)(30,31)(33,39)(37,38);(24,27,29,36,28,35,26,34)(25,33,30,37,32,39,31,38)700 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53319);