Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7426 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#7426 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1351 |
| commutator | C2×C22⋊C4 | C2×C4×C7 |
| Frattini, cosocle | #379 | C22×C7 |
| prime radical | C25×C7 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C23×C7 ◁ C24×C7 ◁ C7×#379 ◁ G | |
332 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 42 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)(10,16)(11,19)(12,15)(13,17,22,20)(14,21)(18,23);(8,10)(9,13)(11,20,21,17)(12,18)(14,16,19,23)(15,22)175 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 414);