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×#17834 |
| 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)(11,18)(14,19)(15,20)(17,22)(21,23)(24,25,28,31,29,32,27,37)(26,33,35,38,36,39,34,30);(8,9,11,13,12,15,18,20)(10,14,17,19,16,21,22,23)(24,26,27,34,29,36,28,35)(25,30,31,33,32,38,37,39);(8,10)(9,14)(11,17)(12,16)(13,19)(15,21)(18,22)(20,23);(10,16)(14,21)(17,22)(19,23)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 10822);