Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7453 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×#7453 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1365 |
| commutator | C2×C16 | C2×C4×C7 |
| Frattini, cosocle | C4×C16 | C22×C7 |
| prime radical | C22×C7 | #361 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C4×C7×C8 ◁ C4×C7×C16 ◁ G | |
102 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)(8,9,11,18,13,31,10,22,12,30,23,34,24,39,25,17,15,21,29,16,33,20,27,38,14,19,28,37,32,35,26,36);(9,16,21,18)(10,23)(11,25)(12,14)(13,32)(17,20,36,31)(19,37,30,34)(22,35,38,39)(24,33)(26,29)(27,28)259 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 441);