Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#61021 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G | |
1724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
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)(26,30)(27,32)(28,31)(29,33)(34,37)(35,38)(36,39);(8,10)(9,13)(11,18)(12,17)(14,16)(15,20)(19,23)(21,22)(24,26,29,28)(25,30,33,31)(27,35,36,34)(32,38,39,37);(10,17)(13,16)(18,23)(20,22)(25,31)(26,28)(30,33)(32,38)(34,35)(37,39);(9,14)(13,16)(15,21)(20,22)(24,27,29,36)(25,32,33,39)(26,34,28,35)(30,37,31,38);(8,11,12,19)(9,15,14,21)(10,18,17,23)(13,20,16,22)(27,36)(32,39)(34,35)(37,38)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54009);