Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C29×C4 |
| direct product | C22×C7×#391 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25×C7 | C23 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C23 | C25×C7 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25×C7 ◁ G | |
6174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 1792 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(21,22,23,24,25,26,27);(3,4);(6,8)(9,10,12,11)(15,19)(17,20);(5,6,7,8)(9,11,12,10)(13,14,16,18)(15,17,19,20);(13,15,16,19)(14,17,18,20)784 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 27009);