Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19697 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23×C8 |
| direct product | C7×#19701 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(9,12)(10,14)(13,15)(16,17,22,26)(18,27,30,31)(19,24,23,29)(20,25,21,28);(9,13)(12,15)(16,18,21,24,22,30,20,29)(17,23,25,27,26,19,28,31);(8,9,10,12,11,13,14,15)(16,19)(17,24)(18,28)(20,27)(21,31)(22,23)(25,30)(26,29);(16,20,22,21)(17,25,26,28)(18,29,30,24)(19,31,23,27)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12689);