Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C7×#22552 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C24 |
| commutator | C22 | C23×C7×C8 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C23×C7 | C23×C4 |
| prime radical | C25×C7 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
1280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7)(9,11)(12,13)(14,16)(15,17)(18,19)(20,21,22,24,23,25,26,27);(8,9)(10,11)(12,14,15,18)(13,16,17,19);(13,17)(16,19);(9,11)700 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 15540);