Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9431 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#9432 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #386 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C7×#3180 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(9,13)(10,17)(11,14)(16,21)(18,19)(20,22)(24,25)(26,33,34,37)(27,30,35,36)(28,31)(29,32)(38,39);(8,9,14,21,12,16,11,13)(10,15,17,22,19,23,18,20)(24,26,28,36,29,34,38,30)(25,27,31,37,32,35,39,33);(8,10)(9,15)(11,18)(12,19)(13,20)(14,17)(16,23)(21,22)(24,27)(25,30)(26,31)(28,37)(29,35)(32,36)(33,38)(34,39)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2420);