Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28×C4 |
| direct product | C7×#60647 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G | |
3386 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)(9,11)(13,17)(15,18)(19,23)(20,24)(22,25)(26,27);(12,13,18,24,16,20,15,17)(14,19,25,27,21,26,22,23);(8,9,10,11);(12,14)(13,19)(15,22)(16,21)(17,23)(18,25)(20,26)(24,27);(14,21)(19,26)(22,25)(23,27)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53635);