Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#9297 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #386 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C23×C7 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
454 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)(8,9,12,16,13,15,11,21)(10,17,20,22,19,23,18,14)(24,25,28,30,29,33,27,32)(26,34,36,31,37,38,35,39);(8,10)(9,14,15,22)(11,18)(12,20)(13,19)(16,17,21,23)(25,30)(27,28)(31,34)(32,33)(35,36)(38,39);(9,15)(11,12)(14,22)(18,20)(24,26,29,37)(25,31,33,39)(27,36,28,35)(30,34,32,38)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2285);