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×#19735 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
780 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)(9,13)(11,14)(15,19)(16,20)(18,21)(22,23)(26,32)(27,34)(29,35)(31,37)(33,38)(36,39);(8,9,14,20,12,16,11,13)(10,15,21,23,17,22,18,19)(24,25)(26,29)(27,31)(28,30)(32,35)(33,36)(34,37)(38,39);(8,10)(9,15)(11,18)(12,17)(13,19)(14,21)(16,22)(20,23)(24,26,27,32,28,33,34,38)(25,29,31,35,30,36,37,39);(9,16)(10,17)(13,20)(18,21)(25,30)(29,36)(31,37)(35,39)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12723);