Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13561 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13561 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C2×AΣL(1,𝔽16) |
| commutator | C2×D4 | C22×C4×C7 |
| Frattini, cosocle | #127 | C23×C7 |
| prime radical | C2×C7×#453 | C2 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C23×C7 ◁ C22×C4×C7 ◁ C2×C7×#127 ◁ G | |
550 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 550 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 23 points:
(1,2,3,4,5,6,7)(8,9)(10,17,19,13)(11,15,21,14)(12,16)(18,23,20,22);(8,10)(9,13)(11,20)(12,18)(14,17)(15,23)(16,22)(19,21);(10,18)(13,22)(17,23)(19,20)196 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6549);