Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#8936 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #120 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | #385 |
| prime radical | C23×C7 | #123 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
356 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)(11,18)(14,22)(15,19)(16,20)(21,23)(24,25,28,32,29,27,36,31)(26,30,34,38,35,33,39,37);(8,9)(10,14)(11,19)(12,15)(13,18)(16,21)(17,23)(20,22)(24,26,28,34,29,35,36,39)(25,30,32,38,27,33,31,37);(8,10,12,17)(9,14,15,23)(11,20,18,16)(13,21,19,22)(24,27)(25,29)(26,33)(28,31)(30,35)(32,36)(34,37)(38,39)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1924);