Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11796 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#11796 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#99 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C22⋊C4 | C23×C7 |
| prime radical | C2×C7×#392 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C24×C7 ◁ C22×C7×C22⋊C4 ◁ G | |
1068 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 106 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)(10,18,20,13)(11,14)(12,17)(15,19,22,23)(16,21);(9,13)(10,19)(14,15)(16,22)(17,18)(20,23);(9,14)(13,15)(16,17)(18,22)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4784);