Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19671 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#19683 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| prime radical | C7×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
724 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)(8,9)(10,17)(11,15)(12,21)(13,16)(14,18)(19,22)(20,23)(24,25,28,31)(26,32,33,39)(27,35,36,30)(29,37,38,34);(8,10,12,18,13,19,11,20)(9,14,15,17,16,23,21,22)(26,33)(29,38)(32,39)(34,37);(24,26,27,34,28,33,36,37)(25,29,30,32,31,38,35,39);(8,11,13,12)(9,15,16,21)(10,18,19,20)(14,22,23,17)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12671);