Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#13197 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | #121 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1954 |
| prime radical | C2×C7×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C22×C7×C8 ◁ G | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,20,14,19,30,35,15,21,29,17,12,18,11,28)(10,22,26,38,24,34,23,37,27,36,31,39,25,16,32,33);(8,10,11,23,12,25,29,26,15,27,30,32,14,24,13,31)(9,16,28,38,18,36,17,33,21,34,35,39,19,22,20,37);(8,11,12,29,15,30,14,13)(9,17,19,28,21,20,18,35)(10,23,25,26,27,32,24,31)(16,33,22,38,34,37,36,39)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6185);