Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C7 · Schur multiplier (H2) = C213 |
| direct product | C22×C7×#529 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C7 | C23 |
| commutator, Frattini, cosocle | C22 | C26×C7 |
| socle | C24×C7 | C24 |
| prime radical | C22×C7×#121 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C7 ◁ G | |
8670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 6 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)(19,20);(6,9)(10,12)(13,15,16,19)(14,17,18,20);(14,18)(17,20);(5,7,8,11)(6,10,9,12)784 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 55651);