Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C16 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#12012 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7×C8 | D4 |
| commutator | C4 | C22×C7×C16 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C22×C7 | C2×C4⋊C8 |
| prime radical | C23×C7 | C4⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7×C8 ◁ C2×C4×C7×C8 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,17)(11,15)(12,21)(13,16)(14,19)(18,22)(20,23)(24,25,26,29,27,30,32,35,28,31,33,36,34,37,38,39);(8,10,12,19,13,20,11,18)(9,14,15,17,16,22,21,23);(8,11,13,12)(9,15,16,21)(10,18,20,19)(14,17,22,23)784 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5000);