Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#61015 |
| 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 | |
1768 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)(24,25,28,32)(26,33,29,30)(27,35,37,31)(34,38,36,39);(8,10)(9,13)(11,18)(12,17)(14,16)(15,20)(19,23)(21,22)(24,26,28,29)(25,30,32,33)(27,36,37,34)(31,39,35,38);(10,17)(13,16)(18,23)(20,22)(25,30)(26,29)(31,38)(32,33)(34,36)(35,39);(9,14)(13,16)(15,21)(20,22)(24,27,28,37)(25,31,32,35)(26,34,29,36)(30,38,33,39);(8,11,12,19)(9,15,14,21)(10,18,17,23)(13,20,16,22)(27,37)(31,35)(34,36)(38,39)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54003);