Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#13568 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C4×C7 |
| Frattini, cosocle | #124 | C23×C7 |
| socle | C2×C7 | C2×AΣL(1,𝔽16) |
| prime radical | C7×#3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C22×C4×C7 ◁ C7×#532 ◁ G | |
498 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 498 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)(8,9,11,15)(10,13,18,20)(12,17,22,23)(14,19,16,21);(8,9)(10,13)(11,20)(12,17)(14,22)(15,21)(16,19)(18,23);(8,10,12,19)(9,13,17,16)(11,18,22,21)(14,15,20,23)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6556);