Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#60307 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C24 |
| 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×C7×C8 ◁ G | |
1820 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)(16,17,20,25,21,23,31,36)(18,22,28,34,26,32,39,43)(19,24,29,37,30,35,41,45)(27,33,40,44,38,42,46,47);(8,10)(9,12)(11,15)(13,14)(16,18)(17,22)(19,27)(20,28)(21,26)(23,32)(24,33)(25,34)(29,40)(30,38)(31,39)(35,42)(36,43)(37,44)(41,46)(45,47);(10,15)(12,14)(17,23)(18,26)(24,35)(25,36)(27,38)(28,39)(37,45)(40,46);(9,13)(12,14)(18,26)(22,32)(27,38)(28,39)(33,42)(34,43)(40,46)(44,47);(16,19,20,29,21,30,31,41)(17,24,25,37,23,35,36,45)(18,27,28,40,26,38,39,46)(22,33,34,44,32,42,43,47)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53295);