Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8541 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C27 |
| direct product | C7×#8541 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #379 |
| commutator | C24 | C22×C4×C7 |
| Frattini, cosocle | C25 | C23×C7 |
| prime radical | C22×C7×#120 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C25×C7 ◁ G | |
2618 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 2618 conjugacy classes of subgroups — too many to draw. Only its 174 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)(8,9,10,15)(11,13,16,20)(12,14,17,21)(18,19,22,23);(9,13)(10,16)(14,19)(17,22);(9,14)(10,17)(13,19)(16,22)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1529);