Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#60327 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C25 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C24×C4 |
| prime radical | C7×#3529 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ G | |
1812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 47 points:
(1,2,3,4,5,6,7)(8,9)(10,14)(11,13)(12,15)(16,17,20,24,21,25,32,39)(18,22,28,35,26,33,40,44)(19,23,31,38,30,37,43,47)(27,34,41,45,29,36,42,46);(8,10)(9,12)(11,15)(13,14)(16,18)(17,22)(19,29)(20,28)(21,26)(23,36)(24,35)(25,33)(27,30)(31,42)(32,40)(34,37)(38,46)(39,44)(41,43)(45,47);(10,15)(12,14)(18,26)(19,30)(22,33)(23,37)(28,40)(31,43)(35,44)(38,47);(9,13)(12,14)(18,26)(22,33)(27,29)(28,40)(34,36)(35,44)(41,42)(45,46);(16,19,20,31,21,30,32,43)(17,23,24,38,25,37,39,47)(18,27,28,41,26,29,40,42)(22,34,35,45,33,36,44,46)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53315);