Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C32 · Schur multiplier (H2) = C2 |
| direct product | C7×#7455 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C16 | D4 |
| commutator | C4 | C2×C7×C32 |
| Frattini, cosocle | C4×C16 | C22×C7 |
| socle, prime radical | C22×C7 | C4⋊C16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C16 ◁ C4×C7×C16 ◁ G | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 1792 |
A minimal faithful permutation representation — 2 generators on 47 points:
(1,2,3,4,5,6,7)(9,12)(10,14)(13,15)(16,17,18,22,19,23,26,32,20,24,27,33,29,35,38,42,21,25,28,34,30,36,39,43,31,37,40,44,41,45,46,47);(8,9,10,12,11,13,14,15)784 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 443);