Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C23 |
| direct product | C7×#12786 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×C22⋊C4 |
| commutator | C23 | C22×C7×C8 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C22×C7 | C2×C22⋊C8 |
| prime radical | C24×C7 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(9,11)(12,13,15,20,16,21,26,33,17,19,25,31,27,32,37,41)(14,18,23,29,24,30,36,40,22,28,34,38,35,39,42,43);(8,9,10,11)(12,14)(13,18)(15,23)(16,24)(17,22)(19,28)(20,29)(21,30)(25,34)(26,36)(27,35)(31,38)(32,39)(33,40)(37,42)(41,43);(13,19)(14,22)(15,25)(21,32)(24,35)(26,37)(29,38)(40,43)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5774);