Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#17324 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C7 | C24 |
| commutator | C22 | C22×C42×C7 |
| Frattini, cosocle | C42 | C24×C7 |
| socle | C22×C7 | C22×C42 |
| prime radical | C24×C7 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42×C7 ◁ G | |
742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,11,15,12,14,18,21)(10,13,17,20,16,19,22,23)(24,25)(26,29)(27,31)(28,30)(32,36)(33,35)(34,37)(38,39);(8,10)(9,13)(11,17)(12,16)(14,19)(15,20)(18,22)(21,23)(24,26,27,32,28,33,34,38)(25,29,31,36,30,35,37,39);(10,16)(13,19)(17,22)(20,23)(25,30)(29,35)(31,37)(36,39);(9,14)(13,19)(15,21)(20,23)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 10312);