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) = C29 |
| direct product | C22×C7×#441 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C7 | C2×D4 |
| commutator | C4 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C7 ◁ C22×C42×C7 ◁ G | |
2790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 1792 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2)(25,26,27,28,29,30,31);(3,4);(5,6,8,7)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(5,7,8,6)(10,15)(14,23)(16,20)(17,24)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53141);