Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#17836 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C42 | C24×C7 |
| socle | C22×C7 | C22×C4⋊C4 |
| prime radical | C2×C7×#41 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C4×C7×#41 ◁ G | |
616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(9,13)(11,14)(15,19)(16,20)(18,21)(22,23)(24,25,28,31,29,32,27,37)(26,33,35,38,36,39,34,30);(8,9,14,20,12,16,11,13)(10,15,21,23,17,22,18,19)(24,26,28,35,29,36,27,34)(25,30,37,39,32,38,31,33);(8,10)(9,15)(11,18)(12,17)(13,19)(14,21)(16,22)(20,23);(10,17)(15,22)(18,21)(19,23)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 10824);