Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#19777 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| prime radical | C7×#3294 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C42×C7 ◁ G | |
648 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)(11,19)(14,20)(15,16)(17,23)(21,22)(24,25)(26,29)(27,30)(28,32)(31,35)(33,36)(34,38)(37,39);(8,9,12,16)(10,14,18,22)(11,15,19,13)(17,21,23,20)(24,26,28,34)(25,29,32,38)(27,33,35,39)(30,36,31,37);(8,10,12,18)(9,14,16,22)(11,17,19,23)(13,20,15,21)(25,30)(27,35)(29,36)(31,32)(33,39)(37,38);(9,15)(11,19)(13,16)(14,21)(17,23)(20,22)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12765);