Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19733 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#19741 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| prime radical | C7×#3207 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(10,16)(11,18)(13,19)(15,21)(17,22)(20,23)(24,25,30,33)(26,34,38,35)(27,28,32,29)(31,36,39,37);(8,9)(10,13)(11,15)(12,14)(16,19)(17,20)(18,21)(22,23)(24,26,28,36,30,38,29,37)(25,31,27,35,33,39,32,34);(8,10,11,16,12,17,18,22)(9,13,15,19,14,20,21,23)(24,27,29,25,30,32,28,33)(26,35,37,31,38,34,36,39);(9,14)(13,20)(15,21)(19,23)(24,28,30,29)(25,32,33,27)(26,36,38,37)(31,34,39,35)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12729);