Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C210 |
| direct product | C2×C7×#2896 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C2×D4 |
| commutator | C4 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C23×C7 | C22×D4 |
| prime radical | C23×C7×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ C24×C4×C7 ◁ G | |
5198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2)(23,24,25,26,27,28,29);(3,4,5,6)(7,8)(9,16)(10,15)(11,19)(12,14)(13,17)(18,22)(20,21);(7,9)(8,13)(10,17)(11,20)(12,18)(14,21)(15,22)(16,19);(8,14)(13,21)(15,19)(16,22);(7,10,12,11)(8,15,14,19)(9,17,18,20)(13,22,21,16)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53071);