Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C29 |
| direct product | C2×C7×#2957 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C4 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C2×C4×D4 |
| prime radical | C22×C7×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C7×M16 ◁ G | |
2642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 41 points:
(1,2)(35,36,37,38,39,40,41);(3,4,8,13,9,12,7,21)(5,14,16,10,17,25,22,24)(6,11,20,26,18,27,19,31)(15,28,30,23,29,33,32,34);(3,5)(4,10)(6,15)(7,22)(8,16)(9,17)(11,23)(12,24)(13,25)(14,21)(18,29)(19,32)(20,30)(26,33)(27,34)(28,31);(3,6,8,20,9,18,7,19)(4,11,13,26,12,27,21,31)(5,15,16,30,17,29,22,32)(10,23,25,33,24,34,14,28);(4,12)(6,18)(10,24)(13,21)(14,25)(15,29)(19,20)(30,32)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53169);