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×#34348 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C24 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C23 | C25×C7 |
| socle | C23×C7 | C25 |
| prime radical | C24×C7 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,13)(14,20)(15,19)(16,18)(17,21)(22,26)(23,25)(24,27)(28,29)(30,33)(31,38)(32,35)(34,40)(36,43)(37,42)(39,41);(12,14)(13,17)(15,22)(16,21)(18,20)(19,24)(23,27)(25,26)(28,30,32,37)(29,33,35,42)(31,39,40,36)(34,43,38,41);(14,21)(17,20)(22,27)(24,26)(28,31,32,40)(29,34,35,38)(30,36,37,39)(33,41,42,43);(13,18)(14,21)(19,25)(22,27)(29,35)(33,42)(34,38)(41,43);(12,15,16,23)(13,19,18,25)(14,22,21,27)(17,24,20,26)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 27336);