Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#34457 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C25 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C23 | C25×C7 |
| prime radical | C2×C7×#120 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ G | |
2500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3,4,5,6,7)(8,9,11,10)(12,13)(14,18)(15,17)(16,19)(21,24)(22,26);(12,13,15,17)(14,16,19,18)(20,21,23,24)(22,25,26,27);(20,21)(22,25)(23,24)(26,27);(8,10,11,9)(12,13)(14,16)(15,17)(18,19)(20,21)(22,25)(23,24)(26,27);(12,14)(13,16)(15,19)(17,18)(20,22)(21,25)(23,26)(24,27)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 27445);