Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C16 · Schur multiplier (H2) = C22 |
| direct product | C7×SD16⋊C16 |
| unambiguous semidirect product | SD16⋊(C7×C16) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C2×D4 |
| commutator | C4 | C22×C7×C16 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C22×C7 | C8×D4 |
| prime radical | C2×C7×D4 | C2×C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ C4×C7×C16 ◁ G | |
232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,12)(11,13)(14,15)(16,17,19,23,20,24,27,30,21,18,22,25,28,26,29,31);(9,12)(10,15)(13,14);(9,13)(12,14)(16,18,19,25,20,26,27,31,21,17,22,23,28,24,29,30)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5043);