Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#10798 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C7 | C23 |
| commutator | C23 | C2×C42×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle, prime radical | C24×C7 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C7 ◁ G | |
870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 1792 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,14,15)(17,18)(20,21,24,27,23,26,25,22);(16,17,19,18)(20,22,25,26,23,27,24,21);(9,11)(16,18,19,17)(21,26)(22,27)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 3786);