Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C24×C4 |
| direct product | C7×#18333 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C7×#392 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(9,14)(10,18)(11,22)(15,28)(16,26)(17,23)(19,34)(20,21)(24,37)(27,32)(29,35)(30,39)(31,38)(33,36);(8,9,12,17)(10,15,20,30)(11,23,22,14)(13,16,25,31)(18,32,21,29)(19,35,36,27)(24,38,37,26)(28,33,39,34);(11,24)(14,26)(19,34)(22,37)(23,38)(27,39)(28,35)(33,36);(8,10,13,21)(9,15,16,29)(11,19,24,34)(12,20,25,18)(14,27,26,39)(17,30,31,32)(22,36,37,33)(23,35,38,28)259 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 11321);