Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C22×C7×#484 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C7 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C25×C7 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C7 ◁ C24×C4×C7 ◁ G | |
2662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(21,22,23,24,25,26,27);(3,4);(5,6)(7,10)(8,9)(11,12)(13,14,15,17)(16,18,20,19);(6,9)(10,12)(14,17)(18,19);(5,7,8,11)(6,10,9,12)(14,18)(16,20)(17,19)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53493);