Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7533 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C8 |
| direct product | C7×#7533 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1409 |
| commutator | C4×C8 | C2×C4×C7 |
| Frattini, cosocle | #493 | C22×C7 |
| prime radical | C7×#493 | C22 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7 ◁ C2×C4×C7 ◁ C42×C7 ◁ C7×#493 ◁ G | |
156 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3,4,5,6,7)(8,9)(10,15,19,14)(11,16,12,21)(13,17)(18,23,20,22);(9,14,17,23)(10,18)(11,12)(15,21,22,16)(19,20)133 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 521);