Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7535 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C4 |
| direct product | C7×#7535 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1409 |
| commutator | C4×C8 | C2×C4×C7 |
| Frattini, cosocle | #492 | C22×C7 |
| prime radical | C7×#127 | D4 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7 ◁ C2×C4×C7 ◁ C42×C7 ◁ C7×#492 ◁ G | |
150 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 150 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)(10,17,18,13)(11,22,19,16)(12,15)(14,20,23,21);(9,13,15,23)(10,18)(11,19)(14,16,17,22)(20,21)175 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 523);