Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1402 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#7486 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C42×C7 |
| Frattini, cosocle | #532 | C22×C7 |
| socle | C2×C7 | #1390 |
| prime radical | C7×#532 | C22 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C22×C4×C7 ◁ C7×#532 ◁ G | |
286 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 66 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)(8,9,11,15)(10,19,20,17)(12,16,22,23)(13,18,21,14);(8,9,12,16,13,18,20,17)(10,14,11,19,22,15,21,23)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 474);