Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7457 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C1 |
| direct product | C7×#7457 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1368 |
| commutator | C16 | C2×C7×C8 |
| Frattini, cosocle | #346 | C22×C7 |
| prime radical | C22×C7 | #365 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C4×C7×C8 ◁ C7×#346 ◁ G | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,3,4,5,6,7)(9,14,15,24,17,26,29,27)(10,18,19,32)(11,21)(12,23)(16,30,28,25,31,38,37,39)(20,35,34,33)(22,36);(8,9,10,14,11,16,18,25,12,15,20,27,21,28,33,30,13,17,19,26,22,31,32,39,23,29,34,24,36,37,35,38)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 445);