Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#8784 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C2×D4 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle, prime radical | C23×C7 | C2×C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ C43×C7 ◁ G | |
422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 1792 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,13)(11,12)(14,15)(17,22)(18,23)(19,20)(24,25,26,28,27,29,30,31);(9,12)(13,15)(16,17,20,18,21,23,19,22);(8,10,11,14)(9,13,12,15)(16,18,19,17,21,22,20,23)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1772);