Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#33697 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C7 |
| socle | C22×C7 | C22×D8 |
| prime radical | C7×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 47 points:
(1,2,3,4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,17)(18,25)(19,24)(20,26)(21,28)(22,27)(23,30)(29,37)(31,43)(32,39)(33,44)(34,40)(35,42)(36,45)(38,46)(41,47);(9,12)(13,15)(16,18,22,32,20,31,36,35,23,33,38,47,37,34,21,19)(17,24,28,40,29,41,46,44,30,42,45,43,26,39,27,25);(8,10,11,14)(9,13,12,15)(16,19,21,34,37,47,38,33,23,35,36,31,20,32,22,18)(17,25,27,39,26,43,45,42,30,44,46,41,29,40,28,24);(16,20,23,37)(17,26,30,29)(18,31,33,34)(19,32,35,47)(21,22,36,38)(24,39,42,41)(25,43,44,40)(27,45,46,28)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26685);