Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#8461 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #379 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C7×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,17,13,16,11,15)(10,18,19,23,20,14,21,22)(25,29,31,34)(30,38,39,37);(8,10)(9,14)(11,19)(12,21)(13,20)(15,22)(16,18)(17,23)(24,25)(26,30)(27,34)(28,31)(29,36)(32,37)(33,39)(35,38);(8,11,13,12)(9,15,16,17)(10,19,20,21)(14,22,18,23)(24,26,28,33)(25,30,31,39)(27,35,36,32)(29,37,34,38)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1449);