Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#8093 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | C2×#95 |
| prime radical | C24×C7 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C42×C7 ◁ G | |
608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(9,13)(10,11)(14,15)(16,17,20,23,21,24,19,28)(18,25,27,22,26,31,29,30);(8,9,11,15,12,14,10,13)(16,18)(17,22)(19,29)(20,27)(21,26)(23,31)(24,30)(25,28);(9,14)(13,15)(18,26)(22,30)(25,31)(27,29)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1081);