Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7×C8 · Schur multiplier (H2) = C24 |
| direct product | C7×#7882 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C7 | C2×D4 |
| commutator | C4 | C2×C4×C7×C8 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | C2×#95 |
| prime radical | C24×C7 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C7 ◁ C22×C42×C7 ◁ G | |
666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,11,14,13,15,12,10)(16,17)(18,24)(19,27)(20,23)(21,26)(22,28)(25,30)(29,31);(8,10,12,15,13,14,11,9)(16,18)(17,21,23,29)(19,26)(20,25)(22,24,27,30)(28,31);(18,25)(21,29)(24,30)(26,31)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 870);