Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#8039 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C22⋊C4 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C23×C7 | C2×C22⋊C4 |
| prime radical | C24×C7 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ C22×C42×C7 ◁ G | |
706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7)(8,9,10,11)(13,17)(14,15)(18,19)(20,21)(22,29)(23,27)(24,32)(25,28)(26,30)(31,34)(33,35);(12,13,15,18,16,19,14,17)(20,22)(21,26)(23,30)(24,33)(25,31)(27,34)(28,35)(29,32);(20,23,25,24)(21,27,28,32)(22,30,31,33)(26,34,35,29)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1027);