Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13011 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×D8 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C22×C7 | C4×D8 |
| prime radical | C2×C7×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C7×C8 ◁ C2×C4×C7×C8 ◁ G | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(9,11)(12,13,18,25,19,26,16,23,20,27,36,42,28,37,17,24)(14,22,30,21,31,39,15,33,32,38,35,40,34,43,29,41);(8,9,10,11)(12,14,28,31,20,32,19,34)(13,21,37,33,27,40,26,41)(15,17,35,36,29,16,30,18)(22,23,39,25,38,24,43,42);(12,15)(13,22)(14,17)(16,32)(18,34)(19,30)(20,29)(21,23)(24,40)(25,33)(26,43)(27,38)(28,35)(31,36)(37,39)(41,42)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5999);