Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C2 |
| direct product | C7×#12060 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C2×D4 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle, prime radical | C22×C7 | C8×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ C4×C7×C16 ◁ G | |
204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 1792 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,16)(11,19)(12,14)(13,17)(15,21)(18,23)(20,22)(24,25,27,31,28,26,33,34,29,32,36,39,37,35,30,38);(8,10,12,18)(9,13,14,22)(11,20,21,17)(15,23,19,16);(9,14)(13,22)(15,19)(16,23)(24,26,27,34,28,32,33,39,29,35,36,38,37,25,30,31)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5048);