Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#33701 |
| 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 | |
590 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)(18,31)(19,24)(20,26)(21,40)(22,42)(23,30)(25,34)(27,38)(28,41)(29,39)(32,44)(33,45)(35,43)(36,47)(37,46);(9,12)(13,15)(16,18,22,33,20,32,21,19,23,34,38,35,39,36,41,37)(17,24,28,44,29,45,27,31,30,46,40,47,26,43,42,25);(8,10,11,14)(9,13,12,15)(16,19,21,36,39,33,22,34,23,37,41,32,20,35,38,18)(17,25,27,45,26,47,28,24,30,31,42,43,29,44,40,46);(16,20,23,39)(17,26,30,29)(18,32,34,36)(19,35,37,33)(21,38,41,22)(24,43,46,45)(25,47,31,44)(27,28,42,40)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26689);