Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#60300 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C25 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C24×C4 |
| prime radical | C7×#3475 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ G | |
1724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,16)(11,15)(12,14)(13,17)(18,22)(19,21)(20,23)(24,25,29,33,28,30,27,32)(26,34,35,39,36,31,37,38);(8,10)(9,13)(11,18)(12,17)(14,16)(15,20)(19,23)(21,22)(25,30)(31,34)(32,33)(38,39);(10,17)(13,16)(18,23)(20,22)(25,30)(31,34)(32,33)(38,39);(9,14)(13,16)(15,21)(20,22)(24,26)(25,31)(27,37)(28,36)(29,35)(30,34)(32,39)(33,38);(8,11,12,19)(9,15,14,21)(10,18,17,23)(13,20,16,22)(24,27,28,29)(25,32,30,33)(26,35,36,37)(31,38,34,39)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53288);